Introduction
The problem in plain words. To classify objects one declares some of them to be “the same”. Very often the rule is given by a transformation: two matrices are the same if one is obtained from the other by a change of basis; two elements of a monoid are the same if each divides the other (Green’s relations [3 , 4 ] ); two sequences of rationals are the same if they get arbitrarily close, which is how the real numbers are built; two modules are stably the same if they become isomorphic after adding a common summand. Each of these relations is usually defined by its own formula, and each forgets information: an orbit relation does not record the stabilizers, and the divisibility relation does not record in how many essentially different ways one element divides another.
The idea. Let m m m m ~ \widetilde{m} m C \mathcal{C} C m m m compared to m ~ \widetilde{m} m m m m matches m ~ \widetilde{m} m D \mathcal{D} D D = ( C , D , σ , τ 1 , τ 2 ; U 1 , U 2 , W ) \mathcal{D} = (\mathcal{C}, \mathcal{D}, \sigma, \tau_1, \tau_2; \mathcal{U}_1, \mathcal{U}_2, \mathcal{W}) D = ( C , D , σ , τ 1 , τ 2 ; U 1 , U 2 , W ) comparison datum (Definition 2.2): D \mathcal{D} D σ \sigma σ C \mathcal{C} C D \mathcal{D} D τ 1 , τ 2 \tau_1,\tau_2 τ 1 , τ 2 D \mathcal{D} D U 1 , U 2 \mathcal{U}_1,\mathcal{U}_2 U 1 , U 2 W \mathcal{W} W K ( D ) \mathsf{K}(\mathcal{D}) K ( D ) comparison category , whose objects are the morphisms of C \mathcal{C} C D = C \mathcal{D} = \mathcal{C} D = C σ = τ i = Id \sigma= \tau_i = \operatorname{Id} σ = τ i = Id m → m ~ m \to\widetilde{m} m → m m ~ = u 1 m u 2 \widetilde{m} = u_1 m u_2 m = u 1 m u 2 K ( D ) \mathsf{K}(\mathcal{D}) K ( D ) levels : level − 1 -1 − 1 k ≥ 1 k \ge1 k ≥ 1
The structural picture. This paper shows that the many relations above are one construction seen through a small number of operations.
Table 2.
In these terms: Green’s relations are the transport relations of the Hom bimodule; orbit relations are transport relations of an action; limit relations are transport relations enriched by a cost; stabilization relations are transport relations of a filtered colimit of data. The constraints on which relations occur are not on the shape of K, which is arbitrary, but on s s s σ , τ 1 , τ 2 \sigma,\tau_1,\tau_2 σ , τ 1 , τ 2 § 10 .
Main structural results: normal form and families. (A) Normal form: K ( D ) \mathrm{K}(\mathcal{D}) K ( D ) ∫ V H \int_V H ∫ V H im ( s ) \operatorname{im}(s) im ( s ) Theorem 4.2 ); level − 1 -1 − 1 Realization: every small category is, up to opposite, a union of components of some K ( D ) \mathrm{K}(\mathcal{D}) K ( D ) D \mathcal{D} D Theorems 4.5 , 5.5 ); so the shape of K is unconstrained, and the higher levels of transport data are homotopy colimits (Proposition 5.8 ). (C) Green’s relations are transport relations: K ( Gr ( C ) ) \mathrm{K}(\operatorname{Gr}(\mathcal{C})) K ( Gr ( C )) − 1 -1 − 1 Theorem 6.5 ). (D) Limit relations: for lengths zero every limit relation is the pullback of closure equality along an orbit map (Theorem 7.5 ), which covers Hausdorff closure, approximate unitary equivalence and completions. (E) Stabilization: a strict stabilizer produces a datum D ∞ \mathcal{D}^{\infty} D ∞ K ( D ∞ ) = K ∞ \mathrm{K}(\mathcal{D}^{\infty})=\mathrm{K}_{\infty} K ( D ∞ ) = K ∞ Proposition 8.8 ). Examples and counterexamples are collected in § 11 .
The regularity axis. The matching of a comparison goes through a map σ \sigma σ C \mathcal{C} C D \mathcal{D} D σ \sigma σ σ \sigma σ Corollary 10.13 ); this says nothing. Classification by a rule that respects composition is only meaningful if σ \sigma σ How regular must σ , τ 1 , τ 2 \sigma,\tau_1,\tau_2 σ , τ 1 , τ 2 We grade data by seven classes ρ 0 ⊂ ρ 1 ⊂ ⋯ ⊂ ρ 6 \rho_0 \subset\rho_1 \subset\cdots\subset\rho_6 ρ 0 ⊂ ρ 1 ⊂ ⋯ ⊂ ρ 6
caption:
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Each class is contained in the next, and the question becomes: which relations occur in each class, are the inclusions strict, and what is the least class in which a given relation occurs?
Main structural results: regularity.
(F) Rigid and diagonal data are classified. Rigid relations are m ~ ∈ U 1 m U 2 \tilde{m}\in U_1mU_2 m ~ ∈ U 1 m U 2 m ∈ U 1 m ~ U 2 m\in U_1\tilde{m}U_2 m ∈ U 1 m ~ U 2 C / ≈ C/\!\approx C / ≈
(G) The classical families sit at σ = Id \sigma=\operatorname{Id} σ = Id , once the relation is viewed on the right category (Proposition 10.8 ); regularity is relative to the category on which the relation lives.
(H) Completely functorial data contain relations that no rigid or diagonal datum gives: constrained Green data, hence conjugacy and twisted conjugacy of any group (Theorem 10.17 ).
(I) A dichotomy. On a nil monoid a completely functorial strict datum has either { 1 } \{1\} { 1 } − 1 -1 − 1 { 1 , a } \{1,a\} { 1 , a } { 0 } \{0\} { 0 } { 1 , a , 0 } \{1,a,0\} { 1 , a , 0 } a 2 = 0 a^2=0 a 2 = 0 − 1 -1 − 1
(J) Closure properties under products, full pullbacks, opposites and disjoint unions (Proposition 10.9 ); and the first strictness statements along the axis (Proposition 10.6).
Examples and counterexamples are collected in § 11 .
Open problems. (1) Is every equivalence relation on Mor ( B M ) \operatorname{Mor}(BM) Mor ( B M ) M M M E 0 E_0 E 0 Z / n \mathbb{Z}/n Z / n n ≤ 4 n\le4 n ≤ 4 − 1 -1 − 1 ρ 2 , ρ 3 , ρ 4 , ρ 5 \rho_2,\rho_3,\rho_4,\rho_5 ρ 2 , ρ 3 , ρ 4 , ρ 5 − 1 -1 − 1 ( ∞ , 2 ) (\infty,2) ( ∞ , 2 )
Reader’s guide. A reader interested in the idea only should read this introduction and § 4 and § 9 . § 2 and § 3 fix the definitions. § 5 –§ 8 treat the transport, Green, limit and stabilization relations and may be read independently. § 10 studies which relations occur for regular data; it uses § 2 –§ 6 , and the limit and stabilization data only in Proposition 10.8 .
Conventions. Composition is written from right to left, so u 1 m u 2 u_1mu_2 u 1 m u 2 u 2 u_2 u 2 m m m u 1 u_1 u 1 K = K ( D ) \mathrm{K}=\mathrm{K}(\mathcal{D}) K = K ( D ) ≃ k \simeq_k ≃ k k k k k = − 1 , 0 k=-1,0 k = − 1 , 0 k ≥ 1 k\ge1 k ≥ 1 π k \pi_k π k B \mathbf{B} B W \mathcal{W} W 1 \mathbf{1} 1 R , L , J , H , D \mathrm{R},\mathrm{L},\mathrm{J},\mathrm{H},\mathrm{D} R , L , J , H , D
Comparison data and the comparison category
Background on 2-categories
2.1. Background on 2-categories. We first recall the notions about 2-categories used throughout the paper; see [6 ] .
Definition 2.1. A 2-category D \mathcal{D} D p : X → Y p:X\to Y p : X → Y p , q : X → Y p,q:X\to Y p , q : X → Y α : p ⇒ q \alpha:p\Rightarrow q α : p ⇒ q vertically , β ∘ v α : p ⇒ r \beta\circ_v\alpha:p\Rightarrow r β ∘ v α : p ⇒ r α : p ⇒ q \alpha:p\Rightarrow q α : p ⇒ q β : q ⇒ r \beta:q\Rightarrow r β : q ⇒ r horizontally , β ∘ h α : p ′ p ⇒ q ′ q \beta\circ_h\alpha:p'p\Rightarrow q'q β ∘ h α : p ′ p ⇒ q ′ q α : p ⇒ q \alpha:p\Rightarrow q α : p ⇒ q p , q : X → Y p,q:X\to Y p , q : X → Y β : p ′ ⇒ q ′ \beta:p'\Rightarrow q' β : p ′ ⇒ q ′ p ′ , q ′ : Y → Z p',q':Y\to Z p ′ , q ′ : Y → Z 1 p : p ⇒ p 1_p:p\Rightarrow p 1 p : p ⇒ p 1 p ′ ∘ h 1 p = 1 p ′ p 1_{p'}\circ_h1_p=1_{p'p} 1 p ′ ∘ h 1 p = 1 p ′ p interchange law ( β ′ ∘ v β ) ∘ h ( α ′ ∘ v α ) = ( β ′ ∘ h α ′ ) ∘ v ( β ∘ h α ) (\beta'\circ_v\beta)\circ_h(\alpha'\circ_v\alpha)=(\beta'\circ_h\alpha')\circ_v(\beta\circ_h\alpha) ( β ′ ∘ v β ) ∘ h ( α ′ ∘ v α ) = ( β ′ ∘ h α ′ ) ∘ v ( β ∘ h α ) C a t \mathrm{Cat} Cat 2-functor sends objects, 1-morphisms and 2-morphisms to objects, 1-morphisms and 2-morphisms, respecting sources and targets, identities and both compositions. A 2-morphism is invertible if it has an inverse for ∘ v \circ_v ∘ v D \mathcal{D} D thin if between two parallel 1-morphisms there is at most one 2-morphism. A subcategory of a category is wide if it contains all objects (but possibly only some morphisms); it must contain identities and be closed under composition. Given α : p ⇒ q \alpha:p\Rightarrow q α : p ⇒ q p , q : X → Y p,q:X\to Y p , q : X → Y a : Y → Z a:Y\to Z a : Y → Z b : X ′ → X b:X'\to X b : X ′ → X whiskering of α \alpha α a a a b b b 1 a ∘ h α ∘ h 1 b : a p b ⇒ a q b 1_a\circ_h\alpha\circ_h1_b:apb\Rightarrow aqb 1 a ∘ h α ∘ h 1 b : a p b ⇒ a q b α \alpha α a a a b b b
2-categories and 2-functors are strict, and size issues are ignored. A category is a 2-category with identity 2-cells only. By the interchange law, a horizontal composite is a composite of whiskerings, β ∘ h α = ( 1 q ′ ∘ h α ) ∘ v ( β ∘ h 1 p ) \beta\circ_h\alpha=(1_{q'}\circ_h\alpha)\circ_v(\beta\circ_h1_p) β ∘ h α = ( 1 q ′ ∘ h α ) ∘ v ( β ∘ h 1 p )
Comparison data and the comparison category
2.2. Comparison data and the comparison category. We now give the basic definition of the paper.
Definition 2.2. A comparison datum D = ( C , D , σ , τ 1 , τ 2 ; U 1 , U 2 , W ) \mathcal{D}=(\mathcal{C},\mathcal{D},\sigma,\tau_1,\tau_2;\mathcal{U}_1,\mathcal{U}_2,\mathcal{W}) D = ( C , D , σ , τ 1 , τ 2 ; U 1 , U 2 , W ) C \mathcal{C} C D \mathcal{D} D τ 1 , τ 2 : C → D \tau_1,\tau_2:\mathcal{C}\to\mathcal{D} τ 1 , τ 2 : C → D σ \sigma σ σ ( A ) = τ 1 ( A ) = τ 2 ( A ) \sigma(A)=\tau_1(A)=\tau_2(A) σ ( A ) = τ 1 ( A ) = τ 2 ( A ) σ : Hom ( A , B ) → Hom ( σ A , σ B ) \sigma:\operatorname{Hom}(A,B)\to\operatorname{Hom}(\sigma A,\sigma B) σ : Hom ( A , B ) → Hom ( σ A , σ B ) U 1 , U 2 ⊆ C \mathcal{U}_1,\mathcal{U}_2\subseteq\mathcal{C} U 1 , U 2 ⊆ C W \mathcal{W} W D \mathcal{D} D
Four special types of comparison data will be used.
Definition 2.3. U 1 \mathcal{U}_1 U 1 U 2 \mathcal{U}_2 U 2 1 \mathbf{1} 1 basic datum is ( U 1 , U 2 , W ) = ( C , C , all 2-morphisms ) (\mathcal{U}_1,\mathcal{U}_2,\mathcal{W})=(\mathcal{C},\mathcal{C},\text{all 2-morphisms}) ( U 1 , U 2 , W ) = ( C , C , all 2-morphisms ) W = all \mathcal{W}=\text{all} W = all D \mathcal{D} D invertible version is ( Iso ( C ) , Iso ( C ) , invertible 2-morphisms ) (\operatorname{Iso}(\mathcal{C}),\operatorname{Iso}(\mathcal{C}),\text{invertible 2-morphisms}) ( Iso ( C ) , Iso ( C ) , invertible 2-morphisms ) strict version has W = 1 \mathcal{W}=\mathbf{1} W = 1 completely functorial if σ \sigma σ W \mathcal{W} W − 1 -1 − 1
A morphism of C \mathcal{C} C D \mathcal{D} D
Definition 2.4. For m : A → B m:A\to B m : A → B u 1 : B → B ~ u_1:B\to\widetilde{B} u 1 : B → B u 2 : A ~ → A u_2:\widetilde{A}\to A u 2 : A → A
Θ u 1 , u 2 ( m ) : = τ 1 ( u 1 ) ∘ σ ( m ) ∘ τ 2 ( u 2 ) : σ A ~ → σ B ~ . \Theta_{u_1,u_2}(m):=\tau_1(u_1)\circ\sigma(m)\circ\tau_2(u_2):\sigma\widetilde{A}\to\sigma\widetilde{B}. Θ u 1 , u 2 ( m ) := τ 1 ( u 1 ) ∘ σ ( m ) ∘ τ 2 ( u 2 ) : σ A → σ B . Comparisons between morphisms are the arrows of the following category.
Definition 2.6. K ( D ) \mathcal{K}(\mathcal{D}) K ( D ) C \mathcal{C} C m → m ~ m\to\widetilde{m} m → m m : A → B m:A\to B m : A → B m ~ : A ~ → B ~ \widetilde{m}:\widetilde{A}\to\widetilde{B} m : A → B ( u 1 , u 2 , Φ ) (u_1,u_2,\Phi) ( u 1 , u 2 , Φ ) u 1 ∈ U 1 u_1\in\mathcal{U}_1 u 1 ∈ U 1 u 2 ∈ U 2 u_2\in\mathcal{U}_2 u 2 ∈ U 2 u 1 : B → B ~ u_1:B\to\widetilde{B} u 1 : B → B u 2 : A ~ → A u_2:\widetilde{A}\to A u 2 : A → A Φ : Θ u 1 , u 2 ( m ) ⇒ σ ( m ~ ) \Phi:\Theta_{u_1,u_2}(m)\Rightarrow\sigma(\widetilde{m}) Φ : Θ u 1 , u 2 ( m ) ⇒ σ ( m ) W \mathcal{W} W
( u 1 ′ , u 2 ′ , Φ ′ ) ∘ ( u 1 , u 2 , Φ ) = ( u 1 ′ u 1 , u 2 u 2 ′ , Φ ′ ∘ v ( 1 τ 1 u 1 ′ ∘ h Φ ∘ h 1 τ 2 u 2 ′ ) ) , ( i d , i d , 1 σ m ) . (u'_1,u'_2,\Phi')\circ(u_1,u_2,\Phi)=\left(u'_1u_1,u_2u'_2,\Phi'\circ_{\mathrm v}(1_{\tau_1u'_1}\circ_{\mathrm h}\Phi\circ_{\mathrm h}1_{\tau_2u'_2})\right),\qquad(\mathrm{id},\mathrm{id},1_{\sigma m}). ( u 1 ′ , u 2 ′ , Φ ′ ) ∘ ( u 1 , u 2 , Φ ) = ( u 1 ′ u 1 , u 2 u 2 ′ , Φ ′ ∘ v ( 1 τ 1 u 1 ′ ∘ h Φ ∘ h 1 τ 2 u 2 ′ ) ) , ( id , id , 1 σ m ) . Proposition 2.7. K ( D ) \mathcal{K}(\mathcal{D}) K ( D )
Proof. Typing of the composition. Let m : A → B m:A\to B m : A → B m ~ : A ~ → B ~ \widetilde{m}:\widetilde{A}\to\widetilde{B} m : A → B m ^ : A ^ → B ^ \widehat{m}:\widehat{A}\to\widehat{B} m : A → B ( u 1 , u 2 , Φ ) : m → m ~ (u_1,u_2,\Phi):m\to\widetilde{m} ( u 1 , u 2 , Φ ) : m → m ( u 1 ′ , u 2 ′ , Φ ′ ) : m ~ → m ^ (u'_1,u'_2,\Phi'):\widetilde{m}\to\widehat{m} ( u 1 ′ , u 2 ′ , Φ ′ ) : m → m u 1 : B → B ~ u_1:B\to\widetilde{B} u 1 : B → B u 1 ′ : B ~ → B ^ u'_1:\widetilde{B}\to\widehat{B} u 1 ′ : B → B u 1 ′ u 1 : B → B ^ u'_1u_1:B\to\widehat{B} u 1 ′ u 1 : B → B u 2 : A ~ → A u_2:\widetilde{A}\to A u 2 : A → A u 2 ′ : A ^ → A ~ u'_2:\widehat{A}\to\widetilde{A} u 2 ′ : A → A u 2 u 2 ′ : A ^ → A u_2u'_2:\widehat{A}\to A u 2 u 2 ′ : A → A Φ : Θ u 1 , u 2 ( m ) ⇒ σ ( m ~ ) \Phi:\Theta_{u_1,u_2}(m)\Rightarrow\sigma(\widetilde{m}) Φ : Θ u 1 , u 2 ( m ) ⇒ σ ( m ) τ 1 ( u 1 ′ ) \tau_1(u'_1) τ 1 ( u 1 ′ ) τ 2 ( u 2 ′ ) \tau_2(u'_2) τ 2 ( u 2 ′ ) 2 2 2
τ 1 ( u 1 ′ ) Θ u 1 , u 2 ( m ) τ 2 ( u 2 ′ ) ⟹ τ 1 ( u 1 ′ ) σ ( m ~ ) τ 2 ( u 2 ′ ) = Θ u 1 ′ , u 2 ′ ( m ~ ) , \tau_1(u'_1)\Theta_{u_1,u_2}(m)\tau_2(u'_2)\Longrightarrow\tau_1(u'_1)\sigma(\widetilde{m})\tau_2(u'_2)=\Theta_{u'_1,u'_2}(\widetilde{m}), τ 1 ( u 1 ′ ) Θ u 1 , u 2 ( m ) τ 2 ( u 2 ′ ) ⟹ τ 1 ( u 1 ′ ) σ ( m ) τ 2 ( u 2 ′ ) = Θ u 1 ′ , u 2 ′ ( m ) , whose source is Θ u 1 ′ u 1 , u 2 u 2 ′ ( m ) \Theta_{u'_1u_1,u_2u'_2}(m) Θ u 1 ′ u 1 , u 2 u 2 ′ ( m ) τ 1 , τ 2 \tau_1,\tau_2 τ 1 , τ 2 Φ ′ \Phi' Φ ′ Φ ′ ∘ v ( 1 τ 1 u 1 ′ ∘ h Φ ∘ h 1 τ 2 u 2 ′ ) \Phi'\circ_{\mathrm v}(1_{\tau_1u'_1}\circ_{\mathrm h}\Phi\circ_{\mathrm h}1_{\tau_2u'_2}) Φ ′ ∘ v ( 1 τ 1 u 1 ′ ∘ h Φ ∘ h 1 τ 2 u 2 ′ ) 2 2 2 Θ u 1 ′ u 1 , u 2 u 2 ′ ( m ) ⇒ σ ( m ^ ) \Theta_{u'_1u_1,u_2u'_2}(m)\Rightarrow\sigma(\widehat{m}) Θ u 1 ′ u 1 , u 2 u 2 ′ ( m ) ⇒ σ ( m ) W \mathcal{W} W W \mathcal{W} W u 1 ′ u 1 ∈ U 1 u'_1u_1\in\mathcal{U}_1 u 1 ′ u 1 ∈ U 1 u 2 u 2 ′ ∈ U 2 u_2u'_2\in\mathcal{U}_2 u 2 u 2 ′ ∈ U 2 Θ i d , i d ( m ) = σ ( m ) \Theta_{\mathrm{id},\mathrm{id}}(m)=\sigma(m) Θ id , id ( m ) = σ ( m ) τ i \tau_i τ i 1 σ m ∈ W 1_{\sigma m}\in\mathcal{W} 1 σ m ∈ W
Associativity and units. Let ( u 1 ′ ′ , u 2 ′ ′ , Φ ′ ′ ) (u''_1,u''_2,\Phi'') ( u 1 ′′ , u 2 ′′ , Φ ′′ ) ( u 1 ′ ′ u 1 ′ u 1 , u 2 u 2 ′ u 2 ′ ′ , Φ ′ ′ ∘ v ( 1 ∘ h Φ ′ ∘ h 1 ) ∘ v ( 1 ∘ h Φ ∘ h 1 ) ) (u''_1u'_1u_1,u_2u'_2u''_2,\Phi''\circ_{\mathrm v}(1\circ_{\mathrm h}\Phi'\circ_{\mathrm h}1)\circ_{\mathrm v}(1\circ_{\mathrm h}\Phi\circ_{\mathrm h}1)) ( u 1 ′′ u 1 ′ u 1 , u 2 u 2 ′ u 2 ′′ , Φ ′′ ∘ v ( 1 ∘ h Φ ′ ∘ h 1 ) ∘ v ( 1 ∘ h Φ ∘ h 1 )) 1 1 1 ∘ v \circ_{\mathrm v} ∘ v a ′ a a'a a ′ a a a a a ′ a' a ′ 1 a ∘ h 1 a = 1 a ′ a 1_a\circ_{\mathrm h}1_a=1_{a'a} 1 a ∘ h 1 a = 1 a ′ a ∘ v \circ_{\mathrm v} ∘ v ( i d , i d , 1 σ m ) (\mathrm{id},\mathrm{id},1_{\sigma m}) ( id , id , 1 σ m ) K ( D ) \mathcal{K}(\mathcal{D}) K ( D )
The ladder of levels
Levels -1 and 0
We recall the notions of connected component and of classifying space of a category.
Definition 3.1. Let K be a category. Two objects x x x y y y connected component if there is a finite zigzag x = x 0 , x 1 , … , x n = y x=x_0,x_1,\ldots,x_n=y x = x 0 , x 1 , … , x n = y i i i x i − 1 → x i x_{i-1}\to x_i x i − 1 → x i x i → x i − 1 x_i\to x_{i-1} x i → x i − 1 π 0 K \pi_0K π 0 K
Definition 3.2. Let K be a small category. Its nerve N K NK N K n n n x 0 → x 1 → ⋯ → x n x_0\to x_1\to\cdots\to x_n x 0 → x 1 → ⋯ → x n classifying space ∣ N K ∣ |NK| ∣ N K ∣ n n n n n n N K NK N K Definition 3.1 ), its fundamental group π 1 ( ∣ N K ∣ , m ) \pi_1(|NK|,m) π 1 ( ∣ N K ∣ , m ) m m m m m m π n \pi_n π n n n n groupoid completion K [ K − 1 ] K[K^{-1}] K [ K − 1 ] ∣ N K ∣ |NK| ∣ N K ∣ Aut K ( m ) \operatorname{Aut}_K(m) Aut K ( m ) m → m m\to m m → m
The levels of a comparison datum are defined as follows.
Definition 3.3. Let K = K ( D ) K=K(\mathcal{D}) K = K ( D ) m ⊴ m ~ m\trianglelefteq\widetilde{m} m ⊴ m Hom K ( m , m ~ ) ≠ ∅ \operatorname{Hom}_K(m,\widetilde{m})\ne\varnothing Hom K ( m , m ) = ∅
Level − 1 -1 − 1 is the relation m ≃ − 1 m ~ : ⟺ m ⊴ m ~ ⊴ m m\simeq_{-1}\widetilde{m}:\Longleftrightarrow m\trianglelefteq\widetilde{m}\trianglelefteq m m ≃ − 1 m :⟺ m ⊴ m ⊴ m u 1 , u 2 , v 1 , v 2 u_1,u_2,v_1,v_2 u 1 , u 2 , v 1 , v 2 u i , v i ∈ U i u_i,v_i\in U_i u i , v i ∈ U i Θ u 1 , u 2 ( m ) ⇒ σ ( m ~ ) \Theta_{u_1,u_2}(m)\Rightarrow\sigma(\widetilde{m}) Θ u 1 , u 2 ( m ) ⇒ σ ( m ) Θ v 1 , v 2 ( m ~ ) ⇒ σ ( m ) \Theta_{v_1,v_2}(\widetilde{m})\Rightarrow\sigma(m) Θ v 1 , v 2 ( m ) ⇒ σ ( m ) W \mathcal{W} W
Level 0 is the relation m ≃ 0 m ~ : ⟺ m , m ~ m\simeq_0\widetilde{m}:\Longleftrightarrow m,\widetilde{m} m ≃ 0 m :⟺ m , m ≃ 0 \simeq_0 ≃ 0 π 0 K \pi_0K π 0 K
Level k ≥ 1 k\geq1 k ≥ 1 at m m m π k ( D , m ) : = π k ( ∣ N K ∣ , m ) \pi_k(\mathcal{D},m):=\pi_k(|NK|,m) π k ( D , m ) := π k ( ∣ N K ∣ , m ) m m m k ≥ 2 k\geq2 k ≥ 2 m ≃ 0 m ~ m\simeq_0\widetilde{m} m ≃ 0 m m m m m ~ \widetilde{m} m k = 1 k=1 k = 1
Level − 1 -1 − 1
Definition 3.4. For a category K let P K PK P K Hom P K ( x , y ) \operatorname{Hom}_{PK}(x,y) Hom P K ( x , y ) Hom K ( x , y ) ≠ ∅ \operatorname{Hom}_K(x,y)\ne\varnothing Hom K ( x , y ) = ∅ ( − 1 ) (-1) ( − 1 ) K → P K K\to PK K → P K
The next proposition identifies levels − 1 -1 − 1 K ( D ) K(\mathcal{D}) K ( D )
Proposition 3.5. Let K = K ( D ) K=K(\mathcal{D}) K = K ( D )
(a) m ≃ − 1 m ~ m\simeq_{-1}\widetilde{m} m ≃ − 1 m m ≅ m ~ m\cong\widetilde{m} m ≅ m P K PK P K Q ( m ) ≅ Q ( m ~ ) Q(m)\cong Q(\widetilde{m}) Q ( m ) ≅ Q ( m ) Q Q Q
(b) m ≃ 0 m ~ m\simeq_0\widetilde{m} m ≃ 0 m Q ( m ) = Q ( m ~ ) Q(m)=Q(\widetilde{m}) Q ( m ) = Q ( m ) Q Q Q
(c) The set Mor ( K ) / ≃ − 1 \operatorname{Mor}(K)/\simeq_{-1} Mor ( K ) / ≃ − 1 [ m ] ≤ [ m ~ ] [m]\leq[\widetilde{m}] [ m ] ≤ [ m ] m ⊴ m ~ m\trianglelefteq\widetilde{m} m ⊴ m ≃ 0 \simeq_0 ≃ 0
(d) ≃ − 1 = ≃ 0 \simeq_{-1}=\simeq_0 ≃ − 1 = ≃ 0 ⊴ \trianglelefteq ⊴
Proof. (a) Two objects of a preorder are isomorphic iff each is below the other. If m ≃ − 1 m ~ m\simeq_{-1}\widetilde{m} m ≃ − 1 m Q ( m ) ≤ Q ( m ~ ) ≤ Q ( m ) Q(m)\leq Q(\widetilde{m})\leq Q(m) Q ( m ) ≤ Q ( m ) ≤ Q ( m ) Q Q Q Q = ( K → P K ) Q=(K\to PK) Q = ( K → P K )
(b) A functor to a discrete category is constant on components, and K → π 0 K K\to\pi_0K K → π 0 K [ m ] ≤ [ m ~ ] [m] \le[\widetilde{m}] [ m ] ≤ [ m ] m → m ~ m \to\widetilde{m} m → m ≃ − 1 ⊆ ⊴ ∩ ⊴ o p ⊆ ≃ 0 \simeq_{-1} \subseteq\trianglelefteq\cap\trianglelefteq^{\mathrm{op}} \subseteq\simeq_0 ≃ − 1 ⊆ ⊴ ∩ ⊴ op ⊆ ≃ 0 ⊴ \trianglelefteq ⊴ ≃ − 1 \simeq_{-1} ≃ − 1 ≃ 0 \simeq_0 ≃ 0 ≃ − 1 = ≃ 0 \simeq_{-1} = \simeq_0 ≃ − 1 = ≃ 0 ⊴ ⊆ ≃ 0 = ≃ − 1 ⊆ ⊴ o p \trianglelefteq\subseteq\simeq_0 = \simeq_{-1} \subseteq\trianglelefteq^{\mathrm{op}} ⊴ ⊆ ≃ 0 = ≃ − 1 ⊆ ⊴ op □ \square □
Level − 1 -1 − 1
Proposition 3.6. Let T : K → K ′ T : \mathrm{K} \to\mathrm{K}' T : K → K ′ x ⊴ y x \trianglelefteq y x ⊴ y T x ⊴ T y Tx \trianglelefteq Ty T x ⊴ T y T T T ≃ − 1 \simeq_{-1} ≃ − 1 ≃ 0 \simeq_0 ≃ 0 π k ( ∣ N K ∣ , x ) → π k ( ∣ N K ′ ∣ , T x ) \pi_k(|NK|,x) \to\pi_k(|NK'|,Tx) π k ( ∣ N K ∣ , x ) → π k ( ∣ N K ′ ∣ , T x ) k ≥ 1 k \ge1 k ≥ 1
Proof. T T T Hom K ( x , y ) ≠ ∅ \operatorname{Hom}_{\mathrm{K}}(x,y) \ne\varnothing Hom K ( x , y ) = ∅ Hom K ′ ( T x , T y ) ≠ ∅ \operatorname{Hom}_{\mathrm{K}'}(Tx,Ty) \ne\varnothing Hom K ′ ( T x , T y ) = ∅ T x Tx T x ≃ − 1 \simeq_{-1} ≃ − 1 T T T ≃ 0 \simeq_0 ≃ 0 Proposition 3.5 (c). Finally, a natural transformation between two functors induces a homotopy between the induced maps of classifying spaces [9 ] , so an equivalence of categories induces a homotopy equivalence ∣ N K ∣ ≃ ∣ N K ′ ∣ |NK| \simeq|NK'| ∣ N K ∣ ≃ ∣ N K ′ ∣ □ \square □
Remark 3.7. Levels − 1 , 0 , 1 , 2 , … -1, 0, 1, 2, \ldots − 1 , 0 , 1 , 2 , … − 1 -1 − 1 ∣ N K ∣ |NK| ∣ N K ∣ − 1 -1 − 1 direction of arrows, so it is invariant under equivalence of categories K (Proposition 3.6) but not under weak homotopy equivalence (Example 11.1(a)); this is why it sits below the homotopy levels, and why it coincides with level 0 for reversible data. The poset of Proposition 3.5 (c) is the extra structure carried by level − 1 -1 − 1
Level 1 and the higher levels
We now turn to level 1.
Proposition 3.8. Let K = K ( D ) K = K(\mathcal{D}) K = K ( D ) m m m
(a) π 1 ( ∣ N K ∣ , m ) ≅ Aut K [ K − 1 ] ( m ) \pi_1(|NK|,m) \cong\operatorname{Aut}_{K[K^{-1}]}(m) π 1 ( ∣ N K ∣ , m ) ≅ Aut K [ K − 1 ] ( m ) m m m m m m
(b) There is a canonical group homomorphism Aut K ( m ) → π 1 ( ∣ N K ∣ , m ) \operatorname{Aut}_K(m) \to\pi_1(|NK|,m) Aut K ( m ) → π 1 ( ∣ N K ∣ , m )
(c) If K is connected, functors K → B G K \to BG K → B G π 1 ( ∣ N K ∣ , m ) → G \pi_1(|NK|,m) \to G π 1 ( ∣ N K ∣ , m ) → G
Proof. (a) is the identification of K [ K − 1 ] K[K^{-1}] K [ K − 1 ] Definition 3.2 ). (b) An arrow a : m → m a : m \to m a : m → m m m m ℓ ( a ) \ell(a) ℓ ( a ) m m m ∣ N K ∣ |NK| ∣ N K ∣ id m \operatorname{id}_m id m a , b : m → m a,b : m \to m a , b : m → m m → a m → b m m \xrightarrow{a} m \xrightarrow{b} m m a m b m a a a b b b b a ba ba ℓ ( b a ) \ell(ba) ℓ ( ba ) ℓ ( a ) ⋅ ℓ ( b ) \ell(a) \cdot\ell(b) ℓ ( a ) ⋅ ℓ ( b ) ℓ ( a ) \ell(a) ℓ ( a ) ℓ ( b ) \ell(b) ℓ ( b ) [ ℓ ( b a ) ] = [ ℓ ( a ) ] [ ℓ ( b ) ] [\ell(ba)] = [\ell(a)][\ell(b)] [ ℓ ( ba )] = [ ℓ ( a )] [ ℓ ( b )] π 1 \pi_1 π 1 a a a [ ℓ ( a ) ] [ ℓ ( a − 1 ) ] = [ ℓ ( a − 1 a ) ] = 1 [\ell(a)][\ell(a^{-1})] = [\ell(a^{-1}a)] = 1 [ ℓ ( a )] [ ℓ ( a − 1 )] = [ ℓ ( a − 1 a )] = 1 [ ℓ ( a − 1 ) ] = [ ℓ ( a ) ] − 1 [\ell(a^{-1})] = [\ell(a)]^{-1} [ ℓ ( a − 1 )] = [ ℓ ( a ) ] − 1 φ ( a ) : = [ ℓ ( a ) ] \varphi(a) := [\ell(a)] φ ( a ) := [ ℓ ( a )] φ ( b a ) = [ ℓ ( a − 1 b − 1 ) ] = [ ℓ ( b − 1 ) ] [ ℓ ( a − 1 ) ] = φ ( b ) φ ( a ) \varphi(ba) = [\ell(a^{-1}b^{-1})] = [\ell(b^{-1})][\ell(a^{-1})] = \varphi(b)\varphi(a) φ ( ba ) = [ ℓ ( a − 1 b − 1 )] = [ ℓ ( b − 1 )] [ ℓ ( a − 1 )] = φ ( b ) φ ( a ) φ : Aut K ( m ) → π 1 ( ∣ N K ∣ , m ) \varphi: \operatorname{Aut}_K(m) \to\pi_1(|NK|,m) φ : Aut K ( m ) → π 1 ( ∣ N K ∣ , m ) ∣ N K ∣ |NK| ∣ N K ∣ B Aut ( x ) B\operatorname{Aut}(x) B Aut ( x ) K → B G K \to BG K → B G N K → N B G NK \to NBG N K → N B G ∣ N B G ∣ = K ( G , 1 ) |NBG| = K(G,1) ∣ N B G ∣ = K ( G , 1 ) π 1 → G \pi_1 \to G π 1 → G [2 ] . □ \square □
By (c), level 1 is the home of the invariants that are multiplicative under composition of comparisons : a character χ : π 1 → G \chi: \pi_1 \to G χ : π 1 → G
The higher levels vanish on contractible components.
Proposition 3.9. If a connected component of K K K ∣ N K ∣ |NK| ∣ N K ∣ k ≥ 1 k \ge1 k ≥ 1
Proof. A category with an initial or terminal object has a contractible classifying space [9 ] . □
The ladder
The results of this section are summarized in the following theorem.
Theorem 3.10. Let D \mathcal{D} D K = K ( D ) K = K(\mathcal{D}) K = K ( D )
(a) The relation ⊴ \trianglelefteq ⊴ ≃ − 1 \simeq_{-1} ≃ − 1 ≃ 0 \simeq_{0} ≃ 0 ≃ − 1 ⊆ ≃ 0 \simeq_{-1} \subseteq\simeq_{0} ≃ − 1 ⊆ ≃ 0 ≃ − 1 = ≃ 0 \simeq_{-1} = \simeq_{0} ≃ − 1 = ≃ 0 ⊴ \trianglelefteq ⊴ ≃ − 1 \simeq_{-1} ≃ − 1 ≃ 0 \simeq_{0} ≃ 0
(b) m ≃ − 1 m ~ m \simeq_{-1} \widetilde{m} m ≃ − 1 m m , m ~ m,\widetilde{m} m , m K K K m ≃ 0 m ~ m \simeq_{0} \widetilde{m} m ≃ 0 m K K K
(c) For k ≥ 1 k \ge1 k ≥ 1 k k k π k ( ∣ N K ∣ , m ) \pi_k(|NK|,m) π k ( ∣ N K ∣ , m ) Aut K ( m ) → π 1 ( ∣ N K ∣ , m ) \operatorname{Aut}_K(m) \to\pi_1(|NK|,m) Aut K ( m ) → π 1 ( ∣ N K ∣ , m ) K K K K → B G K \to BG K → B G π 1 → G \pi_1 \to G π 1 → G
(d) If K K K k ≥ 1 k \ge1 k ≥ 1
Proof. ⊴ \trianglelefteq ⊴ ≃ − 1 \simeq_{-1} ≃ − 1 ≃ 0 \simeq_{0} ≃ 0 ≃ − 1 ⊆ ≃ 0 \simeq_{-1} \subseteq\simeq_{0} ≃ − 1 ⊆ ≃ 0 ∣ N K ∣ |NK| ∣ N K ∣ π 0 K \pi_0K π 0 K N K NK N K Proposition 3.5 , part (c) is Proposition 3.8, and part (d) is Proposition 3.9 . □
Thin reduction
Levels − 1 -1 − 1 0 0 0
Proposition 3.11. Let D W \mathcal{D}_{\mathcal{W}} D W D \mathcal{D} D p ⇒ q p \Rightarrow q p ⇒ q D \mathcal{D} D p ⇒ q p \Rightarrow q p ⇒ q W \mathcal{W} W ⊴ \trianglelefteq ⊴ ≃ − 1 \simeq_{-1} ≃ − 1 ≃ 0 \simeq_{0} ≃ 0 D W \mathcal{D}_{\mathcal{W}} D W k ≥ 1 k \ge1 k ≥ 1
Proof. D W \mathcal{D}_{\mathcal{W}} D W W \mathcal{W} W m → m ~ m \to\widetilde{m} m → m K ( D ) K(\mathcal{D}) K ( D ) u 1 , u 2 u_1,u_2 u 1 , u 2 Θ u 1 , u 2 ( m ) ⇒ σ ( m ~ ) \Theta_{u_1,u_2}(m) \Rightarrow\sigma(\widetilde{m}) Θ u 1 , u 2 ( m ) ⇒ σ ( m ) W \mathcal{W} W D W \mathcal{D}_{\mathcal{W}} D W − 1 -1 − 1 0 0 0 Φ \Phi Φ
Normal form: every comparison category is a Grothendieck construction
The comparison category looks complicated, because a comparison combines two modifications and a 2-morphism. It is in fact a very classical object, and this explains the shape of the four families treated later. We first record the action behind a datum.
Definition 4.1. Let D \mathcal{D} D V = V ( D ) V = V(\mathcal{D}) V = V ( D ) ( A , B ) (A,B) ( A , B ) C \mathcal{C} C ( A , B ) → ( A ~ , B ~ ) (A,B) \to(\widetilde{A},\widetilde{B}) ( A , B ) → ( A , B ) ( u 2 , u 1 ) (u_2,u_1) ( u 2 , u 1 ) u 2 : A ~ → A u_2 : \widetilde{A} \to A u 2 : A → A U 2 \mathcal{U}_2 U 2 u 1 : B → B ~ u_1 : B \to\widetilde{B} u 1 : B → B U 1 \mathcal{U}_1 U 1 ( u 2 ′ , u 1 ′ ) ( u 2 , u 1 ) = ( u 2 u 2 ′ , u 1 ′ u 1 ) (u'_2,u'_1)(u_2,u_1) = (u_2u'_2,u'_1u_1) ( u 2 ′ , u 1 ′ ) ( u 2 , u 1 ) = ( u 2 u 2 ′ , u 1 ′ u 1 ) H = H D : V → Cat H = H_{\mathcal{D}} : V \to\operatorname{Cat} H = H D : V → Cat ( A , B ) (A,B) ( A , B ) H ( A , B ) H(A,B) H ( A , B ) σ A → σ B \sigma A \to\sigma B σ A → σ B D \mathcal{D} D
are the 2-morphisms of W \mathcal{W} W ( u 2 , u 1 ) (u_2,u_1) ( u 2 , u 1 ) p ↦ τ 1 ( u 1 ) p τ 2 ( u 2 ) p \mapsto\tau_1(u_1)\,p\,\tau_2(u_2) p ↦ τ 1 ( u 1 ) p τ 2 ( u 2 ) s : Mor ( C ) → Ob ∫ V H s:\operatorname{Mor}(\mathcal{C}) \to\operatorname{Ob}\int_V H s : Mor ( C ) → Ob ∫ V H s ( m ) = ( ( A , B ) , σ m ) s(m)=((A,B),\sigma_m) s ( m ) = (( A , B ) , σ m ) m : A → B m:A\to B m : A → B ∫ V H \int_V H ∫ V H ( v , x ) (v,x) ( v , x ) x ∈ H ( v ) x\in H(v) x ∈ H ( v ) ( v , x ) → ( v ′ , x ′ ) (v,x)\to(v',x') ( v , x ) → ( v ′ , x ′ ) ( a , φ ) (a,\varphi) ( a , φ ) a : v → v ′ a:v\to v' a : v → v ′ V V V φ : H ( a ) x → x ′ \varphi:H(a)x\to x' φ : H ( a ) x → x ′ H ( v ′ ) H(v') H ( v ′ )
H H H τ 1 , τ 2 \tau_1,\tau_2 τ 1 , τ 2 H ( a ) H(a) H ( a ) W \mathcal{W} W V V V H H H σ \sigma σ σ A \sigma A σ A s s s
Theorem 4.2. K ( D ) K(\mathcal{D}) K ( D ) Mor ( C ) \operatorname{Mor}(\mathcal{C}) Mor ( C ) Hom ( m , m ~ ) = Hom ∫ V H ( s ( m ) , s ( m ~ ) ) \operatorname{Hom}(m,\widetilde{m})=\operatorname{Hom}_{\int_V H}(s(m),s(\widetilde{m})) Hom ( m , m ) = Hom ∫ V H ( s ( m ) , s ( m )) ∫ V H \int_V H ∫ V H K ( D ) → ∫ V H K(\mathcal{D})\to\int_V H K ( D ) → ∫ V H m ↦ s ( m ) m\mapsto s(m) m ↦ s ( m ) im ( s ) \operatorname{im}(s) im ( s )
Proof. An arrow s ( m ) → s ( m ~ ) s(m)\to s(\widetilde{m}) s ( m ) → s ( m ) ∫ V H \int_V H ∫ V H ( ( u 2 , u 1 ) , Φ ) ((u_2,u_1),\Phi) (( u 2 , u 1 ) , Φ ) u 2 : A ~ → A u_2:\widetilde{A}\to A u 2 : A → A U 2 \mathcal{U}_2 U 2 u 1 : B → B ~ u_1:B\to\widetilde{B} u 1 : B → B U 1 \mathcal{U}_1 U 1 Φ : τ 1 ( u 1 ) σ ( m ) τ 2 ( u 2 ) ⇒ σ ( m ~ ) \Phi:\tau_1(u_1)\sigma(m)\tau_2(u_2)\Rightarrow\sigma(\widetilde{m}) Φ : τ 1 ( u 1 ) σ ( m ) τ 2 ( u 2 ) ⇒ σ ( m ) W \mathcal{W} W Θ u 1 , u 2 ( m ) ⇒ σ ( m ~ ) \Theta_{u_1,u_2}(m)\Rightarrow\sigma(\widetilde{m}) Θ u 1 , u 2 ( m ) ⇒ σ ( m ) ( u 1 , u 2 , Φ ) (u_1,u_2,\Phi) ( u 1 , u 2 , Φ ) ∫ V H \int_V H ∫ V H ( ( u 2 , u 1 ) , Φ ) ((u_2,u_1),\Phi) (( u 2 , u 1 ) , Φ ) ( ( u 2 ′ , u 1 ′ ) , Φ ′ ) ((u'_2,u'_1),\Phi') (( u 2 ′ , u 1 ′ ) , Φ ′ ) ( ( u 2 u 2 ′ , u 1 ′ u 1 ) , Φ ′ ∘ V H ( u 2 ′ , u 1 ′ ) Φ ) ((u_2u'_2,u'_1u_1),\Phi'\circ V_H(u'_2,u'_1)\Phi) (( u 2 u 2 ′ , u 1 ′ u 1 ) , Φ ′ ∘ V H ( u 2 ′ , u 1 ′ ) Φ ) H ( u 2 ′ , u 1 ′ ) Φ = 1 τ 1 u 1 ′ ∘ h Φ ∘ h 1 τ 2 u 2 ′ H(u'_2,u'_1)\Phi=1_{\tau_1u'_1}\circ h\Phi\circ h1_{\tau_2u'_2} H ( u 2 ′ , u 1 ′ ) Φ = 1 τ 1 u 1 ′ ∘ h Φ ∘ h 1 τ 2 u 2 ′ m , m ′ m,m' m , m ′ s ( m ) = s ( m ′ ) s(m)=s(m') s ( m ) = s ( m ′ ) σ m = σ m ′ \sigma_m=\sigma_{m'} σ m = σ m ′ ( id , id , 1 ) (\operatorname{id},\operatorname{id},1) ( id , id , 1 ) m → m ′ m\to m' m → m ′ K ( D ) K(\mathcal{D}) K ( D )
Call D \mathcal{D} D saturated if im ( s ) \operatorname{im}(s) im ( s ) ∫ V H \int_V H ∫ V H ∫ V H \int_V H ∫ V H im ( s ) \operatorname{im}(s) im ( s )
Corollary 4.3. (a) m ≃ − 1 m ~ m\simeq_{-1}\widetilde{m} m ≃ − 1 m s ( m ) s(m) s ( m ) s ( m ~ ) s(\widetilde{m}) s ( m ) ∫ V H \int_V H ∫ V H − 1 -1 − 1 s s s
(b) If D \mathcal{D} D ≃ 0 \simeq_0 ≃ 0 s s s π 0 ∫ V H \pi_0\int_V H π 0 ∫ V H π k ( D , m ) = π k ( hocolim V ∣ N H ∣ , s ( m ) ) \pi_k(\mathcal{D},m)=\pi_k(\operatorname{hocolim}_V |NH|,s(m)) π k ( D , m ) = π k ( hocolim V ∣ N H ∣ , s ( m )) k ≥ 1 k\ge1 k ≥ 1 V V V
(c) If D \mathcal{D} D H H H H ( A , B ) H(A,B) H ( A , B ) V V V ∐ A , B Hom D ( σ A , σ B ) \coprod_{A,B}\operatorname{Hom}_{\mathcal{D}}(\sigma A,\sigma B) ∐ A , B Hom D ( σ A , σ B )
Proof. (a) Arrows of K ( D ) K(\mathcal{D}) K ( D ) m , m ~ m,\widetilde{m} m , m ∫ V H \int_V H ∫ V H s ( m ) , s ( m ~ ) s(m),s(\widetilde{m}) s ( m ) , s ( m ) N ∫ V H N\int_V H N ∫ V H
Corollary 4.4. Let D = Gr ( C ) \mathcal{D}=\operatorname{Gr}(\mathcal{C}) D = Gr ( C ) σ = τ i = Id \sigma=\tau_i=\operatorname{Id} σ = τ i = Id U i = C \mathcal{U}_i=\mathcal{C} U i = C W = 1 \mathcal{W}=1 W = 1 V = C o p × C V=\mathcal{C}^{\mathrm{op}}\times\mathcal{C} V = C op × C H ( A , B ) = Hom C ( A , B ) H(A,B)=\operatorname{Hom}_{\mathcal{C}}(A,B) H ( A , B ) = Hom C ( A , B ) ( u 2 , u 1 ) ⋅ p = u 1 p u 2 (u_2,u_1)\cdot p=u_1pu_2 ( u 2 , u 1 ) ⋅ p = u 1 p u 2 s s s Ob ∫ V H \operatorname{Ob}\int_V H Ob ∫ V H D \mathcal{D} D K ( Gr ( C ) ) = ∫ V Hom C K(\operatorname{Gr}(\mathcal{C}))=\int_V\operatorname{Hom}_{\mathcal{C}} K ( Gr ( C )) = ∫ V Hom C J \mathcal{J} J C \mathcal{C} C
Conversely, a transport datum realizes any such ∫ V H \int_V H ∫ V H C = V o p \mathcal{C}=V^{\mathrm{op}} C = V op F = H F=H F = H
Theorem 4.5. Every comparison datum is a restriction of a transport datum: K ( D ) K(\mathcal{D}) K ( D ) K ( T ) K(\mathcal{T}) K ( T ) T \mathcal{T} T H : C o p = V → Cat H:\mathcal{C}^{\mathrm{op}}=V\to\operatorname{Cat} H : C op = V → Cat C ′ = V o p \mathcal{C}'=V^{\mathrm{op}} C ′ = V op
s ( m ) s(m) s ( m ) m ∈ Mor ( C ) m \in\operatorname{Mor}(\mathcal{C}) m ∈ Mor ( C ) K ( D ) \mathsf{K}(\mathcal{D}) K ( D ) D \mathcal{D} D s s s σ \sigma σ
Proof. Take C ′ = V o p \mathcal{C}' = \mathcal{V}^{\mathrm{op}} C ′ = V op U = C ′ \mathcal{U} = \mathcal{C}' U = C ′ C ′ o p = V \mathcal{C}'^{\mathrm{op}} = \mathcal{V} C ′ op = V F = H F = H F = H W ( A , B ) \mathcal{W}_{(A,B)} W ( A , B ) R F C ′ ≃ ∫ V H \mathcal{R}_{F}^{\mathcal{C}'} \simeq\int_{\mathcal{V}} H R F C ′ ≃ ∫ V H K ( T ) \mathsf{K}(\mathcal{T}) K ( T )
Thus the four families reduce to the normal form as follows. Green data are transport data (Corollary 4.4); limit data are transport data enriched by a cost, and stabilized data are filtered colimits of data (Sections 7 and 8). No comparison category is excluded (Theorem 4.5(a)); what the four families have in common, and what distinguishes the relations they produce from arbitrary ones, is the shape of s s s σ \sigma σ σ \sigma σ s s s C \mathcal{C} C σ \sigma σ
Transport relations
Transport data compare observations : an observation on an object is pulled back along admissible morphisms, and two observations are related when each is controlled by a pullback of the other. They cover orbit relations of actions and pullback relations such as mutual reachability in a category, and for them the levels k ≥ 1 k \ge1 k ≥ 1
Systems of observations
Definition 5.1. A preorder ( P , ⊑ ) (P,\sqsubseteq) ( P , ⊑ ) thin category : a category with at most one morphism between two objects (an arrow x → y x \to y x → y x ⊑ y x \sqsubseteq y x ⊑ y x ≃ y x \simeq y x ≃ y x ⊑ y ⊑ x x \sqsubseteq y \sqsubseteq x x ⊑ y ⊑ x reachability preorder of a category K \mathcal{K} K a ≲ b a \lesssim b a ≲ b a → b a \to b a → b K \mathcal{K} K K \mathcal{K} K C a t \mathbf{Cat} Cat contravariant functor F : C o p → C a t F : \mathcal{C}^{\mathrm{op}} \to\mathbf{Cat} F : C op → Cat A A A F ( A ) F(A) F ( A ) w : A ~ → A w : \widetilde{A} \to A w : A → A C \mathcal{C} C F ( w ) : F ( A ) → F ( A ~ ) F(w) : F(A) \to F(\widetilde{A}) F ( w ) : F ( A ) → F ( A ) F ( id ) = id F(\operatorname{id}) = \operatorname{id} F ( id ) = id F ( w w ′ ) = F ( w ′ ) ∘ F ( w ) F(ww') = F(w') \circ F(w) F ( w w ′ ) = F ( w ′ ) ∘ F ( w ) w w ′ ww' w w ′
Transport relations are defined from the following structure.
Definition 5.2. A system of observations is a functor F : C o p → C a t F : \mathcal{C}^{\mathrm{op}} \to\mathbf{Cat} F : C op → Cat W A ⊆ Mor F ( A ) \mathcal{W}_A \subseteq\operatorname{Mor} F(A) W A ⊆ Mor F ( A ) F ( w ) W A ⊆ W A ~ F(w)\mathcal{W}_A \subseteq\mathcal{W}_{\widetilde{A}} F ( w ) W A ⊆ W A w : A ~ → A w : \widetilde{A} \to A w : A → A w ∗ : = F ( w ) w^* := F(w) w ∗ := F ( w ) U ⊆ C \mathcal{U} \subseteq\mathcal{C} U ⊆ C ∫ U F \int_{\mathcal{U}} F ∫ U F ( A , x ) (A,x) ( A , x ) x ∈ F ( A ) x \in F(A) x ∈ F ( A ) ( A , x ) → ( A ~ , x ~ ) (A,x) \to(\widetilde{A},\widetilde{x}) ( A , x ) → ( A , x ) ( w , φ ) (w,\varphi) ( w , φ ) w : A ~ → A w : \widetilde{A} \to A w : A → A U \mathcal{U} U φ : w ∗ x → x ~ \varphi: w^*x \to\widetilde{x} φ : w ∗ x → x W A ~ \mathcal{W}_{\widetilde{A}} W A ( w ′ , φ ′ ) ( w , φ ) = ( w w ′ , φ ′ ∘ w ′ ∗ φ ) (w',\varphi')(w,\varphi) = (ww',\varphi' \circ w'^*\varphi) ( w ′ , φ ′ ) ( w , φ ) = ( w w ′ , φ ′ ∘ w ′ ∗ φ ) transport levels are the levels of ∫ U F \int_{\mathcal{U}} F ∫ U F − 1 -1 − 1 x ≃ F − 1 x ′ x \simeq_{F^{-1}} x' x ≃ F − 1 x ′ 0 0 0 π 0 ∫ U F \pi_0\int_{\mathcal{U}} F π 0 ∫ U F k ≥ 1 k \ge1 k ≥ 1 π k ∣ N ∫ U F ∣ \pi_k\lvert N\int_{\mathcal{U}} F\rvert π k ∣ N ∫ U F ∣ W A = { x ⊑ y ⊑ x } \mathcal{W}_A = \{x \sqsubseteq y \sqsubseteq x\} W A = { x ⊑ y ⊑ x } ≃ \simeq ≃ − 1 -1 − 1 x ≃ x ′ ⟺ ∃ w , w ~ ∈ U : w ∗ x ≃ x ~ , w ~ ∗ x ′ ≃ x x \simeq x' \Longleftrightarrow\exists w,\widetilde{w} \in\mathcal{U}: w^*x \simeq\widetilde{x},\ \widetilde{w}^*x' \simeq x x ≃ x ′ ⟺ ∃ w , w ∈ U : w ∗ x ≃ x , w ∗ x ′ ≃ x ∫ U F \int_{\mathcal{U}} F ∫ U F
Remark 5.3. Think of A A A x ∈ F ( A ) x \in F(A) x ∈ F ( A ) observation on A A A φ : x → y \varphi: x \to y φ : x → y F ( A ) F(A) F ( A ) w ∗ x w^*x w ∗ x A ~ \widetilde{A} A A ~ \widetilde{A} A w : A ~ → A w : \widetilde{A} \to A w : A → A
An arrow ( A , x ) → ( A ~ , x ~ ) (A,x)\to(\widetilde{A},\widetilde{x}) ( A , x ) → ( A , x ) w w w x x x x ~ \widetilde{x} x
To realize transport relations as levels of a comparison datum, observations are turned into morphisms.
Definition 5.4. Let F F F C C C ⋆ \star ⋆ x ∈ F ( A ) x\in F(A) x ∈ F ( A ) x : A → ⋆ x:A\to\star x : A → ⋆ C F C_F C F Ob C ⊔ { ⋆ } \operatorname{Ob} C\sqcup\{\star\} Ob C ⊔ { ⋆ } Hom ( A , B ) = Hom C ( A , B ) \operatorname{Hom}(A,B)=\operatorname{Hom}_C(A,B) Hom ( A , B ) = Hom C ( A , B ) A , B ∈ Ob C A,B\in\operatorname{Ob} C A , B ∈ Ob C Hom ( A , ⋆ ) = Ob F ( A ) \operatorname{Hom}(A,\star)=\operatorname{Ob} F(A) Hom ( A , ⋆ ) = Ob F ( A ) Hom ( ⋆ , ⋆ ) = { id } \operatorname{Hom}(\star,\star)=\{\operatorname{id}\} Hom ( ⋆ , ⋆ ) = { id } Hom ( ⋆ , A ) = ∅ \operatorname{Hom}(\star,A)=\varnothing Hom ( ⋆ , A ) = ∅ x ∘ w : = w ∗ x x\circ w:=w^*x x ∘ w := w ∗ x x ∈ Hom ( A , ⋆ ) x\in\operatorname{Hom}(A,\star) x ∈ Hom ( A , ⋆ ) w ∈ Hom C ( A ~ , A ) w\in\operatorname{Hom}_C(\widetilde{A},A) w ∈ Hom C ( A , A ) ( x ∘ w ) ∘ w ′ = x ∘ ( w w ′ ) (x\circ w)\circ w'=x\circ(ww') ( x ∘ w ) ∘ w ′ = x ∘ ( w w ′ ) w ′ ∗ w ∗ = ( w w ′ ) ∗ w'^*w^*=(ww')^* w ′ ∗ w ∗ = ( w w ′ ) ∗
The 2-category D F D_F D F x ⇒ y x\Rightarrow y x ⇒ y F ( A ) F(A) F ( A ) x , y : A → ⋆ x,y:A\to\star x , y : A → ⋆ F ( A ) F(A) F ( A ) φ : x ⇒ y \varphi:x\Rightarrow y φ : x ⇒ y w w w F ( w ) φ : w ∗ x ⇒ w ∗ y F(w)\varphi:w^*x\Rightarrow w^*y F ( w ) φ : w ∗ x ⇒ w ∗ y id ⋆ \operatorname{id}_\star id ⋆ F ( w ) F(w) F ( w ) F ( w w ′ ) = F ( w ′ ) F ( w ) F(ww')=F(w')F(w) F ( w w ′ ) = F ( w ′ ) F ( w ) D F D_F D F not thin when F ( A ) F(A) F ( A )
The transport levels are indeed levels of a comparison datum.
Theorem 5.5. For the datum D F U = ( C F , D F , Id , Id , Id ; U 1 = 1 , U 2 = U ∪ { id ⋆ } , W = { identity 2-cells } ∪ ⨆ A W A ) \mathcal{D}^{\mathcal{U}}_F=(C_F,D_F,\operatorname{Id},\operatorname{Id},\operatorname{Id};\mathcal{U}_1=1,\mathcal{U}_2=\mathcal{U}\cup\{\operatorname{id}_\star\},\mathcal{W}=\{\text{identity 2-cells}\}\cup\bigsqcup_A\mathcal{W}_A) D F U = ( C F , D F , Id , Id , Id ; U 1 = 1 , U 2 = U ∪ { id ⋆ } , W = { identity 2-cells } ∪ ⨆ A W A ) K ( D F U ) K(\mathcal{D}^{\mathcal{U}}_F) K ( D F U ) x : A → ⋆ x:A\to\star x : A → ⋆ ∫ U F \int_{\mathcal{U}}F ∫ U F K ( D F U ) K(\mathcal{D}^{\mathcal{U}}_F) K ( D F U ) D F U \mathcal{D}^{\mathcal{U}}_F D F U x x x F = ∗ F=\ast F = ∗ U \mathcal{U} U U op \mathcal{U}^{\operatorname{op}} U op K ( D ) K(\mathcal{D}) K ( D )
Proof. The class W \mathcal{W} W F ( w ) W A ⊆ W A ~ F(w)\mathcal{W}_A\subseteq\mathcal{W}_{\widetilde{A}} F ( w ) W A ⊆ W A Definition 5.2 , so D F U \mathcal{D}^{\mathcal{U}}_F D F U σ = Id \sigma=\operatorname{Id} σ = Id
For objects x : A → ⋆ x:A\to\star x : A → ⋆ x ~ : A ~ → ⋆ \widetilde{x}:\widetilde{A}\to\star x : A → ⋆ B = B ~ = ⋆ B=\widetilde{B}=\star B = B = ⋆ u 1 = id ⋆ u_1=\operatorname{id}_\star u 1 = id ⋆ u 2 = w ∈ U u_2=w\in\mathcal{U} u 2 = w ∈ U w : A ~ → A w:\widetilde{A}\to A w : A → A Θ u 1 , u 2 ( x ) = id ⋆ ∘ x ∘ w = w ∗ x \Theta_{u_1,u_2}(x)=\operatorname{id}_\star\circ x\circ w=w^*x Θ u 1 , u 2 ( x ) = id ⋆ ∘ x ∘ w = w ∗ x Φ : w ∗ x ⇒ x ~ \Phi:w^*x\Rightarrow\widetilde{x} Φ : w ∗ x ⇒ x W A ~ \mathcal{W}_{\widetilde{A}} W A x → x ~ x\to\widetilde{x} x → x K K K ( w , φ ) (w,\varphi) ( w , φ ) Definition 5.2 .
The composition of two arrows ( id ⋆ , w , φ ) : x → x ~ (\operatorname{id}_\star,w,\varphi):x\to\widetilde{x} ( id ⋆ , w , φ ) : x → x ( id ⋆ , w ′ , φ ′ ) : x ~ → x ^ (\operatorname{id}_\star,w',\varphi'):\widetilde{x}\to\widehat{x} ( id ⋆ , w ′ , φ ′ ) : x → x Definition 2.6 is ( id ⋆ , w w ′ , φ ′ ∘ v ( 1 ∘ h φ ∘ h 1 w ′ ) ) (\operatorname{id}_\star,ww',\varphi'\circ_{\mathrm v}(1\circ_{\mathrm h}\varphi\circ_{\mathrm h}1_{w'})) ( id ⋆ , w w ′ , φ ′ ∘ v ( 1 ∘ h φ ∘ h 1 w ′ )) w ′ w' w ′ F ( w ′ ) = w ′ ∗ F(w')=w'^* F ( w ′ ) = w ′ ∗ ( w w ′ , φ ′ ∘ w ′ ∗ φ ) (ww',\varphi'\circ w'^*\varphi) ( w w ′ , φ ′ ∘ w ′ ∗ φ ) ∫ U F \int_{\mathcal{U}}F ∫ U F w : A ~ → A w:\widetilde{A}\to A w : A → A w ′ : A ^ → A ~ w':\widehat{A}\to\widetilde{A} w ′ : A → A w w ′ : A ^ → A ww':\widehat{A}\to A w w ′ : A → A φ : w ∗ x → x ~ \varphi:w^*x\to\widetilde{x} φ : w ∗ x → x F ( A ~ ) F(\widetilde{A}) F ( A ) w ′ ∗ φ : w ′ ∗ w ∗ x = ( w w ′ ) ∗ x → w ′ ∗ x ~ w'^*\varphi:w'^*w^*x=(ww')^*x\to w'^*\widetilde{x} w ′ ∗ φ : w ′ ∗ w ∗ x = ( w w ′ ) ∗ x → w ′ ∗ x F ( A ^ ) F(\widehat{A}) F ( A ) φ ′ : w ′ ∗ x ~ → x ^ \varphi':w'^*\widetilde{x}\to\widehat{x} φ ′ : w ′ ∗ x → x W A ^ \mathcal{W}_{\widehat{A}} W A K K K Proposition 2.7 , so is ∫ U F \int_{\mathcal{U}}F ∫ U F F = ∗ F=\ast F = ∗ ∫ U F = U op \int_{\mathcal{U}}F=\mathcal{U}^{\operatorname{op}} ∫ U F = U op
Finally we show that the full subcategory is a union of components. Since U 1 = 1 \mathcal{U}_1=1 U 1 = 1 ( u 1 , u 2 , Φ ) : m → m ~ (u_1,u_2,\Phi):m\to\widetilde{m} ( u 1 , u 2 , Φ ) : m → m K K K u 1 = id u_1=\operatorname{id} u 1 = id B = B ~ B=\widetilde{B} B = B ⋆ \star ⋆ id ⋆ \operatorname{id}_\star id ⋆ Ob C \operatorname{Ob} C Ob C id ⋆ \operatorname{id}_\star id ⋆ x : A → ⋆ x:A\to\star x : A → ⋆ U 2 = U ∪ { id ⋆ } \mathcal{U}_2=\mathcal{U}\cup\{\operatorname{id}_\star\} U 2 = U ∪ { id ⋆ } ⋆ \star ⋆ A ≠ ⋆ A\ne\star A = ⋆ □ \square □
The higher levels of transport data are unconstrained, already for completely functorial strict data.
Corollary 5.6. Let U \mathcal{U} U D \mathcal{D} D Theorem 5.5 for the constant functor F = ∗ F=\ast F = ∗ W A \mathcal{W}_A W A A A A U \mathcal{U} U k ≥ 1 k\ge1 k ≥ 1 k k k D \mathcal{D} D x A : A → ⋆ x_A:A\to\star x A : A → ⋆ π k ( ∣ N U ∣ , A ) \pi_k(|N\mathcal{U}|,A) π k ( ∣ N U ∣ , A ) T T T k ≥ 1 k\ge1 k ≥ 1 ∣ T ∣ |T| ∣ T ∣
Proof. By Theorem 5.5 the levels at x A x_A x A ∫ U F = U o p \int_{\mathcal{U}} F = \mathcal{U}^{\mathrm{op}} ∫ U F = U op ∣ N U o p ∣ |N\mathcal{U}^{\mathrm{op}}| ∣ N U op ∣ ∣ N U ∣ |N\mathcal{U}| ∣ N U ∣ T T T U \mathcal{U} U T T T ∣ T ∣ |T| ∣ T ∣ [2 ] . □
Higher levels as homotopy colimits
The higher levels are computed with homotopy colimits.
Definition 5.7. For a small category I \mathcal{I} I Ψ : I → T o p \Psi: \mathcal{I} \to\mathrm{Top} Ψ : I → Top homotopy colimit hocolim I Ψ \operatorname{hocolim}_{\mathcal{I}} \Psi hocolim I Ψ Ψ ( i ) \Psi(i) Ψ ( i ) Ψ ( i → j ) \Psi(i \to j) Ψ ( i → j ) n n n ∐ i 0 → ⋯ → i n Ψ ( i 0 ) \coprod_{i_0 \to\cdots\to i_n} \Psi(i_0) ∐ i 0 → ⋯ → i n Ψ ( i 0 ) Ψ : I → C a t \Psi: \mathcal{I} \to\mathrm{Cat} Ψ : I → Cat Grothendieck construction ∫ I Ψ \int_{\mathcal{I}} \Psi ∫ I Ψ ( i , x ) (i,x) ( i , x ) x ∈ Ψ ( i ) x \in\Psi(i) x ∈ Ψ ( i ) ( i , x ) → ( j , y ) (i,x) \to(j,y) ( i , x ) → ( j , y ) ( a , φ ) (a,\varphi) ( a , φ ) a : i → j a : i \to j a : i → j I \mathcal{I} I φ : Ψ ( a ) x → y \varphi: \Psi(a)x \to y φ : Ψ ( a ) x → y Ψ ( j ) \Psi(j) Ψ ( j ) [10 ] states that hocolim I ∣ N Ψ ∣ ≃ ∣ N ∫ I Ψ ∣ \operatorname{hocolim}_{\mathcal{I}} |N\Psi| \simeq|N\int_{\mathcal{I}} \Psi| hocolim I ∣ N Ψ∣ ≃ ∣ N ∫ I Ψ∣ I = U o p \mathcal{I} = \mathcal{U}^{\mathrm{op}} I = U op Ψ = F \Psi= F Ψ = F ( A , x ) → ( A ~ , x ~ ) (A,x) \to(\widetilde{A},\widetilde{x}) ( A , x ) → ( A , x ) ( w , φ ) (w,\varphi) ( w , φ ) w : A ~ → A w : \widetilde{A} \to A w : A → A U \mathcal{U} U A → A ~ A \to\widetilde{A} A → A U o p \mathcal{U}^{\mathrm{op}} U op φ : F ( w ) x → x ~ \varphi: F(w)x \to\widetilde{x} φ : F ( w ) x → x ∫ U F \int_{\mathcal{U}} F ∫ U F
The higher transport levels are given by Thomason’s theorem.
Proposition 5.8. ∣ N ∫ U F ∣ ≃ hocolim U o p ∣ N F ∣ |N\int_{\mathcal{U}} F| \simeq\operatorname{hocolim}_{\mathcal{U}^{\mathrm{op}}} |NF| ∣ N ∫ U F ∣ ≃ hocolim U op ∣ N F ∣ [10 ] ). In particular level 0 0 0 π 0 \pi_0 π 0 k ≥ 1 k \geq1 k ≥ 1 k k k
Proof. Apply Thomason’s theorem to the covariant functor F : U o p → C a t F : \mathcal{U}^{\mathrm{op}} \to\mathrm{Cat} F : U op → Cat
Thus the higher levels are realized, for strict data, by the homotopy colimit of the observation system; what remains open is the ( ∞ , 2 ) (\infty,2) ( ∞ , 2 )
Compatibility with composition
Proposition 5.9. Let F F F U ⊆ C \mathcal{U} \subseteq\mathcal{C} U ⊆ C a : A ′ → A a : A' \to A a : A ′ → A w ∈ U ( A ) w \in\mathcal{U}(A) w ∈ U ( A ) w ′ ∈ U ( A ′ ) w' \in\mathcal{U}(A') w ′ ∈ U ( A ′ ) w a = a w ′ wa = aw' w a = a w ′ x ≃ − 1 x ~ x \simeq_{-1} \widetilde{x} x ≃ − 1 x F ( A ) F(A) F ( A ) a ∗ x ≃ − 1 a ∗ x ~ a^*x \simeq_{-1} a^*\widetilde{x} a ∗ x ≃ − 1 a ∗ x F ( A ′ ) F(A') F ( A ′ )
Proof. F ( w a ) = F ( a ) F ( w ) = a ∗ w ∗ F(wa) = F(a)F(w) = a^*w^* F ( w a ) = F ( a ) F ( w ) = a ∗ w ∗ F ( a w ′ ) = w ′ ∗ a ∗ F(aw') = w'^*a^* F ( a w ′ ) = w ′ ∗ a ∗ w a = a w ′ wa = aw' w a = a w ′ a ∗ w ∗ = w ′ ∗ a ∗ a^*w^* = w'^*a^* a ∗ w ∗ = w ′ ∗ a ∗ w ∗ x ≅ x ~ w^*x \cong\widetilde{x} w ∗ x ≅ x a ∗ a^* a ∗ ≅ \cong ≅ w ′ ∗ ( a ∗ x ) = a ∗ ( w ∗ x ) ≅ a ∗ x ~ w'^*(a^*x) = a^*(w^*x) \cong a^*\widetilde{x} w ′ ∗ ( a ∗ x ) = a ∗ ( w ∗ x ) ≅ a ∗ x w ′ w' w ′ a ∗ x → a ∗ x ~ a^*x \to a^*\widetilde{x} a ∗ x → a ∗ x
Actions
The basic examples of transport relations are orbit relations.
Definition 5.10. A (right) action of a monoid M M M Z Z Z ( x , m ) ↦ x m (x,m) \mapsto xm ( x , m ) ↦ x m x 1 = x x1 = x x 1 = x ( x m ) m ′ = x ( m m ′ ) (xm)m' = x(mm') ( x m ) m ′ = x ( m m ′ ) orbit of x x x x M = { x m : m ∈ M } xM = \{xm : m \in M\} x M = { x m : m ∈ M } G G G g ⋅ x g \cdot x g ⋅ x action groupoid Z / / G Z//G Z // G x ∈ Z x \in Z x ∈ Z g ∈ G g \in G g ∈ G g : x → g ⋅ x g : x \to g \cdot x g : x → g ⋅ x x x x Stab G ( x ) = { g ∈ G : g ⋅ x = x } \operatorname{Stab}_G(x) = \{g \in G : g \cdot x = x\} Stab G ( x ) = { g ∈ G : g ⋅ x = x } homotopy quotient E G × G Z EG \times_G Z E G × G Z E G × Z EG \times Z E G × Z G G G E G EG E G G G G Z Z Z ∣ N ( Z / / G ) ∣ |N(Z//G)| ∣ N ( Z // G ) ∣ B Stab G ( x ) B\operatorname{Stab}_G(x) B Stab G ( x ) π 0 \pi_0 π 0 π 1 \pi_1 π 1 x x x Stab G ( x ) \operatorname{Stab}_G(x) Stab G ( x )
Orbit relations of actions are level − 1 -1 − 1
Corollary 5.11. (a) Let a monoid M M M Z Z Z F F F U = B M \mathcal{U} = BM U = B M W = 1 \mathcal{W} = 1 W = 1 x ≃ − 1 x ′ ⟺ x M = x ′ M x \simeq_{-1} x' \Longleftrightarrow xM = x'M x ≃ − 1 x ′ ⟺ x M = x ′ M ≃ 0 \simeq_0 ≃ 0 x ∼ x m x \sim xm x ∼ x m
(b) Let a group G G G Z Z Z F ( g ) x = g − 1 x F(g)x=g^{-1}x F ( g ) x = g − 1 x ∫ F \int F ∫ F ≃ − 1 = ≃ 0 \simeq_{-1}=\simeq_0 ≃ − 1 = ≃ 0 ≃ G \simeq_G ≃ G x x x Stab G ( x ) \operatorname{Stab}_G(x) Stab G ( x ) k ≥ 2 k\geq2 k ≥ 2 E G × G Z EG\times_G Z E G × G Z
Proof. (a) For a discrete F F F F ( A ) F(A) F ( A ) W = 1 \mathcal{W}=1 W = 1 x → x ′ x\to x' x → x ′ ∫ U F \int_{\mathcal{U}}F ∫ U F m ∈ M m\in M m ∈ M F ′ ( m ) x = x m = x ′ F'(m)x=xm=x' F ′ ( m ) x = x m = x ′ F ( m m ′ ) x = ( x m ) m ′ = F ( m ′ ) F ( m ) x F(mm')x=(xm)m'=F(m')F(m)x F ( m m ′ ) x = ( x m ) m ′ = F ( m ′ ) F ( m ) x ≃ \simeq ≃ x ≃ − 1 x ′ x\simeq_{-1}x' x ≃ − 1 x ′ x ′ ∈ x M x'\in xM x ′ ∈ x M x ∈ x ′ M x\in x'M x ∈ x ′ M x ′ = x m x'=xm x ′ = x m x ′ M = x m M ⊆ x M x'M=xmM\subseteq xM x ′ M = x m M ⊆ x M x M = x ′ M xM=x'M x M = x ′ M x ∼ x m x\sim xm x ∼ x m G G G g : x → x ′ g:x\to x' g : x → x ′ g − 1 : x ′ → x g^{-1}:x'\to x g − 1 : x ′ → x ∫ F \int F ∫ F ≃ − 1 = ≃ 0 \simeq_{-1}=\simeq_0 ≃ − 1 = ≃ 0 ∫ F \int F ∫ F F ( g ) x = g − 1 x F(g)x=g^{-1}x F ( g ) x = g − 1 x ( g h ) − 1 x = h − 1 ( g − 1 x ) (gh)^{-1}x=h^{-1}(g^{-1}x) ( g h ) − 1 x = h − 1 ( g − 1 x )
Pullback relations
Another basic family of transport relations consists of pullback relations: two points are related when their images under a map into a preorder are equivalent (Example 11.3).
Green's relations
Green’s relations are the oldest example of equivalence relations on morphisms defined by mutual divisibility. They are level − 1 -1 − 1
Operations on relations
We use the following operations on equivalence relations.
Definition 6.1. For two equivalence relations E , E ′ E,E' E , E ′ X X X E ∩ E ′ E\cap E' E ∩ E ′ x , y x,y x , y x E y xEy x E y x E ′ y xE'y x E ′ y E ∨ E ′ E\vee E' E ∨ E ′ x ( E ∨ E ′ ) y x(E\vee E')y x ( E ∨ E ′ ) y x = x 0 , x 1 , … , x n = y x=x_0,x_1,\ldots,x_n=y x = x 0 , x 1 , … , x n = y E E E E ′ E' E ′ E ∘ E ′ E\circ E' E ∘ E ′ { ( x , z ) : ∃ y , x E ′ y and y E z } \{(x,z):\exists y,\ xE'y\text{ and }yEz\} {( x , z ) : ∃ y , x E ′ y and y E z } E ∨ E ′ E\vee E' E ∨ E ′
The Green data
We recall Green’s relations.
Definition 6.2. A monoid is a set M M M B M \mathbf{B}M B M M M M m ∈ M m\in M m ∈ M m M = { m x : x ∈ M } mM=\{mx:x\in M\} m M = { m x : x ∈ M } M m = { x m : x ∈ M } Mm=\{xm:x\in M\} M m = { x m : x ∈ M } M m M = { x m y : x , y ∈ M } MmM=\{xmy:x,y\in M\} M m M = { x m y : x , y ∈ M } m m m [3 , 4 ] are
m R m ~ ⟺ m M = m ~ M , m L m ~ ⟺ M m = M m ~ , m\mathrel{R}\widetilde{m}\Longleftrightarrow mM=\widetilde{m}M,\qquad m\mathrel{L}\widetilde{m}\Longleftrightarrow Mm=M\widetilde{m}, m R m ⟺ m M = m M , m L m ⟺ M m = M m , m J m ~ ⟺ M m M = M m ~ M , m\mathrel{J}\widetilde{m}\Longleftrightarrow MmM=M\widetilde{m}M, m J m ⟺ M m M = M m M , together with H = R ∩ L H=R\cap L H = R ∩ L D = R ∨ L D=R\vee L D = R ∨ L m R m mRm m R m m ~ = m x \widetilde{m}=mx m = m x m = m ~ y m=\widetilde{m}y m = m y x , y x,y x , y L L L J J J D D D H H H J J J
Green’s relations are level − 1 -1 − 1
Definition 6.3. For a category C C C
Gr ( C ) = ( C , C , Id , Id , Id ; U 1 = U 2 = C , W = 1 ) , \operatorname{Gr}(C) = (C,C,\operatorname{Id},\operatorname{Id},\operatorname{Id};\ U_1=U_2=C,\ W=1), Gr ( C ) = ( C , C , Id , Id , Id ; U 1 = U 2 = C , W = 1 ) , and let Gr R ( C ) \operatorname{Gr}_{R}(C) Gr R ( C ) Gr L ( C ) \operatorname{Gr}_{L}(C) Gr L ( C ) U 1 U_1 U 1 U 2 U_2 U 2 1 \mathbf{1} 1
Proposition 6.4. (a) Hom K ( m , m ~ ) = { ( u 1 , u 2 ) : m ~ = u 1 m u 2 } \operatorname{Hom}_{K}(m,\widetilde{m})=\{(u_1,u_2):\widetilde{m}=u_1mu_2\} Hom K ( m , m ) = {( u 1 , u 2 ) : m = u 1 m u 2 } Gr ( C ) \operatorname{Gr}(C) Gr ( C ) m ≃ − 1 m ~ m\simeq_{-1}\widetilde{m} m ≃ − 1 m m ~ = u 1 m u 2 \widetilde{m}=u_1mu_2 m = u 1 m u 2 m = v 1 m ~ v 2 m=v_1\widetilde{m}v_2 m = v 1 m v 2 u i , v i u_i,v_i u i , v i
(b) For C = B M C=\mathbf{BM} C = BM − 1 -1 − 1 Gr ( C ) \operatorname{Gr}(C) Gr ( C ) J J J [3 , 4 ] .
(c) For C = B M C=\mathbf{BM} C = BM − 1 -1 − 1 Gr R \operatorname{Gr}_{R} Gr R R R R − 1 -1 − 1 Gr L \operatorname{Gr}_{L} Gr L L L L
Proof. Here σ = τ i = Id \sigma=\tau_i=\operatorname{Id} σ = τ i = Id Θ u 1 , u 2 ( m ) = u 1 m u 2 \Theta_{u_1,u_2}(m)=u_1mu_2 Θ u 1 , u 2 ( m ) = u 1 m u 2 u 1 m u 2 ⇒ m ~ u_1mu_2\Rightarrow\widetilde{m} u 1 m u 2 ⇒ m W = 1 W=1 W = 1 u 1 m u 2 = m ~ u_1mu_2=\widetilde{m} u 1 m u 2 = m C = B M C=\mathbf{BM} C = BM m ~ ∈ M m M \widetilde{m}\in MmM m ∈ M m M m ≃ − 1 m ~ m\simeq_{-1}\widetilde{m} m ≃ − 1 m m ~ ∈ M m M \widetilde{m}\in MmM m ∈ M m M m ∈ M m ~ M m\in M\widetilde{m}M m ∈ M m M M m M = M m ~ M MmM=M\widetilde{m}M M m M = M m M U 1 = 1 U_1=\mathbf{1} U 1 = 1 u 1 = 1 u_1=1 u 1 = 1 m ~ ∈ m M \widetilde{m}\in mM m ∈ m M m ∈ m ~ M m\in\widetilde{m}M m ∈ m M m M = m ~ M mM=\widetilde{m}M m M = m M U 2 = 1 U_2=\mathbf{1} U 2 = 1
The whole ladder of levels of Green data can be computed.
Theorem 6.5. Let C C C Tw ( C ) \operatorname{Tw}(C) Tw ( C ) C C C m → m ~ m\to\widetilde{m} m → m ( f , g ) (f,g) ( f , g ) m = g m ~ f m=g\widetilde{m}f m = g m f
(a) K ( Gr ( C ) ) ≅ Tw ( C ) o p K(\operatorname{Gr}(C))\cong\operatorname{Tw}(C)^{\mathrm{op}} K ( Gr ( C )) ≅ Tw ( C ) op
(b) Level 0 of Gr ( C ) \operatorname{Gr}(C) Gr ( C ) m m m m ~ \widetilde{m} m C C C k ≥ 1 k\geq1 k ≥ 1 k k k m : A → B m:A\to B m : A → B π k ( ∣ N C ∣ , A ) \pi_k(|NC|,A) π k ( ∣ N C ∣ , A ) M M M M g p = π 1 ( B M ) M^{\mathrm{gp}}=\pi_1(BM) M gp = π 1 ( B M ) m m m
(c) K ( Gr R ( C ) ) ≅ ∐ B ( C / B ) o p K(\operatorname{Gr}_{R}(C))\cong\coprod_B(C/B)^{\mathrm{op}} K ( Gr R ( C )) ≅ ∐ B ( C / B ) op K ( Gr L ( C ) ) ≅ ∐ A ( A / C ) K(\operatorname{Gr}_{L}(C))\cong\coprod_A(A/C) K ( Gr L ( C )) ≅ ∐ A ( A / C ) k ≥ 1 k\geq1 k ≥ 1 R R R L L L
Proof. (a) By Proposition 6.4(a) an arrow m → m ~ m\to\widetilde{m} m → m K ( Gr ( C ) ) K(\operatorname{Gr}(C)) K ( Gr ( C )) ( u 1 , u 2 ) (u_1,u_2) ( u 1 , u 2 ) m ~ = u 1 m u 2 \widetilde{m}=u_1mu_2 m = u 1 m u 2 m ~ → m \widetilde{m}\to m m → m Tw ( C ) \operatorname{Tw}(C) Tw ( C ) ( f , g ) = ( u 2 , u 1 ) (f,g)=(u_2,u_1) ( f , g ) = ( u 2 , u 1 ) ( u 1 ′ , u 2 ′ ) ( u 1 , u 2 ) = ( u 1 ′ u 1 , u 2 u 2 ′ ) (u'_1,u'_2)(u_1,u_2)=(u'_1u_1,u_2u'_2) ( u 1 ′ , u 2 ′ ) ( u 1 , u 2 ) = ( u 1 ′ u 1 , u 2 u 2 ′ ) Tw ( C ) o p \operatorname{Tw}(C)^{\mathrm{op}} Tw ( C ) op Tw ( C ) \operatorname{Tw}(C) Tw ( C ) N C NC N C n n n ( 2 n + 1 ) (2n+1) ( 2 n + 1 ) C C C ∣ N C ∣ |NC| ∣ N C ∣ [1 ] ; the nerve of the opposite category has the homeomorphic realization. The vertex m m m m m m A A A π 0 \pi_0 π 0 π 1 ( B M ) \pi_1(BM) π 1 ( B M ) M M M Gr R \operatorname{Gr}_{R} Gr R u 1 = id B u_1=\operatorname{id}_B u 1 = id B B = B ~ B=\widetilde{B} B = B m → m ~ m\to\widetilde{m} m → m u 2 u_2 u 2 m ~ = m u 2 \widetilde{m}=mu_2 m = m u 2 m ~ → m \widetilde{m}\to m m → m C / B C/B C / B C / B C/B C / B id B \operatorname{id}_B id B ( C / B ) o p (C/B)^{\mathrm{op}} ( C / B ) op Gr L \operatorname{Gr}_{L} Gr L A / C A/C A / C id A \operatorname{id}_A id A
Thus R R R L L L J J J C C C M M M − 1 -1 − 1 0 0 0 1 1 1 M M M M M M B M BM B M t = 0 t=0 t = 0 k ≥ 1 k\geq1 k ≥ 1
The relation H = R ∩ L H=R\cap L H = R ∩ L − 1 -1 − 1
Definition 6.7. For a monoid M M M Gr H ( M ) = ( B M , B ( M × M ) , σ , τ 1 , τ 2 ; B M , B M , 1 ) \operatorname{Gr}_{\mathrm{H}}(M) = (\mathbf{B}M,\mathbf{B}(M \times M),\sigma,\tau_1,\tau_2;\mathbf{B}M,\mathbf{B}M,1) Gr H ( M ) = ( B M , B ( M × M ) , σ , τ 1 , τ 2 ; B M , B M , 1 ) B ( M × M ) \mathbf{B}(M \times M) B ( M × M ) σ , τ 1 , τ 2 : M → M × M \sigma,\tau_1,\tau_2 : M \to M \times M σ , τ 1 , τ 2 : M → M × M σ ( m ) = ( m , m ) \sigma(m) = (m,m) σ ( m ) = ( m , m ) τ 1 ( u ) = ( u , 1 ) \tau_1(u) = (u,1) τ 1 ( u ) = ( u , 1 ) τ 2 ( u ) = ( 1 , u ) \tau_2(u) = (1,u) τ 2 ( u ) = ( 1 , u )
Proposition 6.8. Level − 1 -1 − 1 Gr H ( M ) \operatorname{Gr}_{\mathrm{H}}(M) Gr H ( M )
Proof. Since W = 1 W = 1 W = 1 m → m ~ m \to\widetilde{m} m → m ( u 1 , u 2 ) (u_1,u_2) ( u 1 , u 2 ) ( u 1 , 1 ) ( m , m ) ( 1 , u 2 ) = ( m ~ , m ~ ) (u_1,1)(m,m)(1,u_2) = (\widetilde{m},\widetilde{m}) ( u 1 , 1 ) ( m , m ) ( 1 , u 2 ) = ( m , m ) m ~ = u 1 m = m u 2 \widetilde{m} = u_1m = mu_2 m = u 1 m = m u 2 m ⪯ − 1 m ~ m \preceq_{-1} \widetilde{m} m ⪯ − 1 m m ~ ∈ M m ∩ m M \widetilde{m} \in Mm \cap mM m ∈ M m ∩ m M m ≃ − 1 m ~ m \simeq_{-1} \widetilde{m} m ≃ − 1 m m ~ ∈ M m ∩ m M \widetilde{m} \in Mm \cap mM m ∈ M m ∩ m M m ∈ M m ~ ∩ m ~ M m \in M\widetilde{m} \cap\widetilde{m}M m ∈ M m ∩ m M m ~ ∈ M m \widetilde{m} \in Mm m ∈ M m m ∈ M m ~ m \in M\widetilde{m} m ∈ M m M m = M m ~ Mm = M\widetilde{m} M m = M m m M = m ~ M mM = \widetilde{m}M m M = m M m H m ~ m \mathrel{\mathrm{H}} \widetilde{m} m H m
Invertible comparisons
The levels k ≥ 1 k \ge1 k ≥ 1 C \mathcal{C} C m m m m m m
Definition 6.10. For a category C \mathcal{C} C Gr × ( C ) = ( C , C , Id , Id , Id ; Iso ( C ) , Iso ( C ) , 1 ) \operatorname{Gr}^{\times}(\mathcal{C}) = (\mathcal{C},\mathcal{C},\operatorname{Id},\operatorname{Id},\operatorname{Id};\operatorname{Iso}(\mathcal{C}),\operatorname{Iso}(\mathcal{C}),1) Gr × ( C ) = ( C , C , Id , Id , Id ; Iso ( C ) , Iso ( C ) , 1 )
Proposition 6.11. K ( Gr × ( C ) ) K(\operatorname{Gr}^{\times}(\mathcal{C})) K ( Gr × ( C )) ≃ − 1 = ≃ 0 \simeq_{-1} = \simeq_0 ≃ − 1 = ≃ 0 m ~ = u 1 m u 2 \widetilde{m} = u_1 m u_2 m = u 1 m u 2 u 1 , u 2 u_1,u_2 u 1 , u 2 k ≥ 2 k \ge2 k ≥ 2 m : A → B m : A \to B m : A → B { ( u 1 , u 2 ) ∈ Aut ( B ) × Aut ( A ) : u 1 m u 2 = m } \{(u_1,u_2) \in\operatorname{Aut}(B) \times\operatorname{Aut}(A) : u_1mu_2 = m\} {( u 1 , u 2 ) ∈ Aut ( B ) × Aut ( A ) : u 1 m u 2 = m } ( u 1 ′ , u 2 ′ ) ( u 1 , u 2 ) = ( u 1 ′ u 1 , u 2 u 2 ′ ) (u'_1,u'_2)(u_1,u_2) = (u'_1u_1,u_2u'_2) ( u 1 ′ , u 2 ′ ) ( u 1 , u 2 ) = ( u 1 ′ u 1 , u 2 u 2 ′ )
Proof. If ( u 1 , u 2 ) : m → m ~ (u_1,u_2) : m \to\widetilde{m} ( u 1 , u 2 ) : m → m m ~ = u 1 m u 2 \widetilde{m} = u_1mu_2 m = u 1 m u 2 u 1 , u 2 u_1,u_2 u 1 , u 2 m = u 1 − 1 m ~ u 2 − 1 m = u_1^{-1}\widetilde{m}u_2^{-1} m = u 1 − 1 m u 2 − 1 ( u 1 − 1 , u 2 − 1 ) : m ~ → m (u_1^{-1},u_2^{-1}) : \widetilde{m} \to m ( u 1 − 1 , u 2 − 1 ) : m → m ∣ N K ∣ |\mathcal{N}K| ∣ N K ∣ B Aut K ( x ) B\operatorname{Aut}_K(x) B Aut K ( x ) k ≥ 2 k \ge2 k ≥ 2 Aut K ( m ) \operatorname{Aut}_K(m) Aut K ( m ) m m m ( u 1 , u 2 ) (u_1,u_2) ( u 1 , u 2 ) m ~ = m \widetilde{m} = m m = m u 1 ∈ Aut ( B ) u_1 \in\operatorname{Aut}(B) u 1 ∈ Aut ( B ) u 2 ∈ Aut ( A ) u_2 \in\operatorname{Aut}(A) u 2 ∈ Aut ( A ) u 1 m u 2 = m u_1mu_2 = m u 1 m u 2 = m
Congruences
We now study when level − 1 -1 − 1 C \mathcal{C} C
Definition 6.12. Two morphisms are parallel if they have the same domain and the same codomain (in a monoid all elements are parallel). Let D \mathcal{D} D ( C , C , Id , Id , Id ; U 1 , U 2 , 1 ) (\mathcal{C},\mathcal{C},\operatorname{Id},\operatorname{Id},\operatorname{Id};\mathcal{U}_1,\mathcal{U}_2,1) ( C , C , Id , Id , Id ; U 1 , U 2 , 1 ) ≃ − 1 \simeq_{-1} ≃ − 1 left congruence if m ≃ − 1 m ~ m \simeq_{-1} \widetilde{m} m ≃ − 1 m a m ≃ − 1 a m ~ am \simeq_{-1} a\widetilde{m} am ≃ − 1 a m a a a right congruence analogously, and a congruence if it is both; a congruence is exactly what is needed to form a quotient category. For an object B B B U i ( B ) \mathcal{U}_i(B) U i ( B ) B B B U i \mathcal{U}_i U i m , m ~ : A → B m,\widetilde{m} : A \to B m , m : A → B m ≃ − 1 m ~ m \simeq_{-1} \widetilde{m} m ≃ − 1 m B B B A A A U ( A ) \mathcal{U}(A) U ( A )
Proposition 6.13. Let D \mathcal{D} D
(a) If for every a : B → B ′ a : B \to B' a : B → B ′ u ∈ U 1 ( B ) u \in\mathcal{U}_1(B) u ∈ U 1 ( B ) u ′ ∈ U 1 ( B ′ ) u' \in\mathcal{U}_1(B') u ′ ∈ U 1 ( B ′ ) a u = u ′ a au = u'a a u = u ′ a ≃ − 1 \simeq_{-1} ≃ − 1 a : A ′ → A a : A' \to A a : A ′ → A u ∈ U 2 ( A ) u \in\mathcal{U}_2(A) u ∈ U 2 ( A ) u ′ ∈ U 2 ( A ′ ) u' \in\mathcal{U}_2(A') u ′ ∈ U 2 ( A ′ ) u a = a u ′ ua = au' u a = a u ′ ≃ − 1 \simeq_{-1} ≃ − 1
(b) In particular R \mathrm{R} R U 1 = 1 \mathcal{U}_1 = 1 U 1 = 1 L \mathrm{L} L C = B G \mathcal{C} = \mathbf{B}G C = B G u ′ = a u a − 1 u' = aua^{-1} u ′ = a u a − 1
Proof. (a) For parallel m , m ~ : A → B m,\widetilde{m} : A \to B m , m : A → B m ~ = u 1 m u 2 \widetilde{m} = u_1mu_2 m = u 1 m u 2 u 1 ∈ U 1 ( B ) u_1 \in\mathcal{U}_1(B) u 1 ∈ U 1 ( B ) u 2 ∈ U 2 ( A ) u_2 \in\mathcal{U}_2(A) u 2 ∈ U 2 ( A ) a : B → B ′ a : B \to B' a : B → B ′ u 1 ′ ∈ U 1 ( B ′ ) u'_1 \in\mathcal{U}_1(B') u 1 ′ ∈ U 1 ( B ′ ) a u 1 = u 1 ′ a au_1 = u'_1a a u 1 = u 1 ′ a a m ~ = a u 1 m u 2 = u 1 ′ a m u 2 = a\widetilde{m} = au_1mu_2 = u'_1amu_2 = a m = a u 1 m u 2 = u 1 ′ am u 2 =
u 1 ′ ( a m ) u 2 u'_1(am)u_2 u 1 ′ ( am ) u 2 a m ⊴ a m ~ am \unlhd a\tilde m am ⊴ a m ~ m ~ a = u 1 m u 2 a = u 1 m a u 2 ′ \tilde m a = u_1m u_2a = u_1ma u'_2 m ~ a = u 1 m u 2 a = u 1 ma u 2 ′ u 1 = 1 u_1 = 1 u 1 = 1 u 1 ′ = i d u'_1 = \mathrm{id} u 1 ′ = id u ′ = a u a − 1 u' = aua^{-1} u ′ = a u a − 1 a u = u ′ a au = u'a a u = u ′ a
Green’s relation J is not compatible with composition in general (Example 11.5).
Limit relations
Many relations are not exact but hold “in the limit”: operators that are approximately unitarily equivalent, sets with the same closure, sequences that become arbitrarily close. They are obtained by replacing the existence of a comparison by the vanishing of the infimum of its cost. The main result of this section is that all such relations of zero length are pullbacks of closure equality along orbit maps (Theorem 7.5 ).
Gauge data
Definition 7.1. A comparison datum with lengths γ i : U i → [ 0 , ∞ ] \gamma_i : U_i \to[0,\infty] γ i : U i → [ 0 , ∞ ] γ i ( i d ) = 0 \gamma_i(\mathrm{id}) = 0 γ i ( id ) = 0 γ i ( u u ′ ) ≤ γ i ( u ) + γ i ( u ′ ) \gamma_i(uu') \leq\gamma_i(u) + \gamma_i(u') γ i ( u u ′ ) ≤ γ i ( u ) + γ i ( u ′ ) c : W → [ 0 , ∞ ] c : W \to[0,\infty] c : W → [ 0 , ∞ ] c ( 1 ) = 0 c(1) = 0 c ( 1 ) = 0 c ( β ∘ v α ) ≤ c ( α ) + c ( β ) c(\beta\circ_v \alpha) \leq c(\alpha) + c(\beta) c ( β ∘ v α ) ≤ c ( α ) + c ( β ) u ∈ U 1 u \in U_1 u ∈ U 1 u ′ ∈ U 2 u' \in U_2 u ′ ∈ U 2 ε > 0 \varepsilon> 0 ε > 0 δ > 0 \delta> 0 δ > 0 c ( α ) < δ ⇒ c ( 1 τ 1 u ∘ h α ∘ h 1 τ 2 u ′ ) < ε c(\alpha) < \delta\Rightarrow c(1_{\tau_1u} \circ_h \alpha\circ_h 1_{\tau_2u'}) < \varepsilon c ( α ) < δ ⇒ c ( 1 τ 1 u ∘ h α ∘ h 1 τ 2 u ′ ) < ε c ( 1 τ 1 u ∘ h α ∘ h 1 τ 2 u ′ ) ≤ c ( α ) c(1_{\tau_1u} \circ_h \alpha\circ_h 1_{\tau_2u'}) \leq c(\alpha) c ( 1 τ 1 u ∘ h α ∘ h 1 τ 2 u ′ ) ≤ c ( α ) d ( m , m ~ ) : = inf { γ 1 ( u 1 ) + γ 2 ( u 2 ) + c ( Φ ) } d(m,\tilde m) := \inf\{\gamma_1(u_1) + \gamma_2(u_2) + c(\Phi)\} d ( m , m ~ ) := inf { γ 1 ( u 1 ) + γ 2 ( u 2 ) + c ( Φ )} Hom K ( m , m ~ ) \operatorname{Hom}_{K}(m,\tilde m) Hom K ( m , m ~ ) d + ( m , m ~ ) = d ( m , m ~ ) + d ( m ~ , m ) d^+(m,\tilde m) = d(m,\tilde m) + d(\tilde m,m) d + ( m , m ~ ) = d ( m , m ~ ) + d ( m ~ , m ) γ i ( u ) \gamma_i(u) γ i ( u ) c ( Φ ) c(\Phi) c ( Φ ) Φ \Phi Φ ∥ x − y ∥ \lVert x-y\rVert ∥ x − y ∥
Proposition 7.2. For a gauge datum , m ⊴ lim m ~ : ⟺ d ( m , m ~ ) = 0 m \unlhd^{\lim} \tilde m :\Longleftrightarrow d(m,\tilde m) = 0 m ⊴ l i m m ~ :⟺ d ( m , m ~ ) = 0 ≃ − 1 lim : = { d + = 0 } \simeq^{\lim}_{-1} := \{d^+ = 0\} ≃ − 1 l i m := { d + = 0 } d + d^+ d + Mor ( C ) \operatorname{Mor}(C) Mor ( C ) ≃ − 1 f i n : = { d + < ∞ } \simeq^{\mathrm{fin}}_{-1} := \{d^+ < \infty\} ≃ − 1 fin := { d + < ∞ } ≃ − 1 lim ⊆ ≃ − 1 f i n \simeq^{\lim}_{-1} \subseteq\simeq^{\mathrm{fin}}_{-1} ≃ − 1 l i m ⊆ ≃ − 1 fin ≃ − 1 f i n \simeq^{\mathrm{fin}}_{-1} ≃ − 1 fin − 1 -1 − 1
Proof. Reflexivity: the identity arrow has cost 0. Transitivity of ⊴ lim \unlhd^{\lim} ⊴ l i m d ( m , m ~ ) = d ( m ~ , m ^ ) = 0 d(m,\tilde m) = d(\tilde m,\hat m) = 0 d ( m , m ~ ) = d ( m ~ , m ^ ) = 0 ε > 0 \varepsilon> 0 ε > 0 ( u 1 ′ , u 2 ′ , Φ ′ ) : m ~ → m ^ (u'_1,u'_2,\Phi') : \tilde m \to\hat m ( u 1 ′ , u 2 ′ , Φ ′ ) : m ~ → m ^ < ε / 3 < \varepsilon/3 < ε /3 δ > 0 \delta> 0 δ > 0 ε / 3 \varepsilon/3 ε /3 u 1 ′ , u 2 ′ u'_1,u'_2 u 1 ′ , u 2 ′ ε / 3 \varepsilon/3 ε /3 ( u 1 , u 2 , Φ ) : m → m ~ (u_1,u_2,\Phi) : m \to\tilde m ( u 1 , u 2 , Φ ) : m → m ~ < δ < \delta < δ γ 1 ( u 1 ) + γ 1 ( u 1 ′ ) + γ 2 ( u 2 ) + γ 2 ( u 2 ′ ) + c ( Φ ′ ) + c ( 1 ∘ h Φ ∘ h 1 ) < δ + ε / 3 + ε / 3 ≤ ε \gamma_1(u_1) + \gamma_1(u'_1) + \gamma_2(u_2) + \gamma_2(u'_2) + c(\Phi') + c(1 \circ_h \Phi\circ_h 1) < \delta+ \varepsilon/3 + \varepsilon/3 \leq\varepsilon γ 1 ( u 1 ) + γ 1 ( u 1 ′ ) + γ 2 ( u 2 ) + γ 2 ( u 2 ′ ) + c ( Φ ′ ) + c ( 1 ∘ h Φ ∘ h 1 ) < δ + ε /3 + ε /3 ≤ ε c ( β ∘ v α ) ≤ c ( α ) + c ( β ) c(\beta\circ_v \alpha) \leq c(\alpha) + c(\beta) c ( β ∘ v α ) ≤ c ( α ) + c ( β ) d ( m , m ^ ) = 0 d(m,\hat m) = 0 d ( m , m ^ ) = 0 ≃ − 1 lim \simeq^{\lim}_{-1} ≃ − 1 l i m d ( m , m ^ ) ≤ d ( m , m ~ ) + d ( m ~ , m ^ ) d(m,\hat m) \leq d(m,\tilde m) + d(\tilde m,\hat m) d ( m , m ^ ) ≤ d ( m , m ~ ) + d ( m ~ , m ^ ) d + d^+ d + ≃ − 1 f i n \simeq^{\mathrm{fin}}_{-1} ≃ − 1 fin { γ i < ∞ } \{\gamma_i < \infty\} { γ i < ∞ } { c < ∞ } \{c < \infty\} { c < ∞ } d ( m , m ~ ) < ∞ d(m,\tilde m) < \infty d ( m , m ~ ) < ∞ m → m ~ m \to\tilde m m → m ~
In Lawvere’s language [5 ] , a (non-symmetric) metric space is a category enriched over ( [ 0 , ∞ ] , ≥ , + ) ([0,\infty],\geq,+) ([ 0 , ∞ ] , ≥ , + ) d d d K ( Q ) K(\mathcal{Q}) K ( Q ) d d d − 1 -1 − 1 − 1 -1 − 1
Closure relations
Let a monoid M M M X X X ρ \rho ρ x ↦ x m x \mapsto xm x ↦ x m C = U = B M C = U = BM C = U = B M F ( ∗ ) = X F(*) = X F ( ∗ ) = X X X X ρ \rho ρ F ( m ) x = x m F(m)x = xm F ( m ) x = x m W = all W = \text{all} W = all γ = 0 \gamma= 0 γ = 0 F ( g ) x = g − 1 x F(g)x = g^{-1}x F ( g ) x = g − 1 x
Proposition 7.3. For the closure datum, d ( x , y ) = ρ ( y , x M ) = inf m ρ ( x m , y ) d(x,y)=\rho(y,xM)=\inf_m \rho(xm,y) d ( x , y ) = ρ ( y , x M ) = inf m ρ ( x m , y ) y ∈ x M y\in xM y ∈ x M ≃ − 1 l i m \simeq^{\mathrm{lim}}_{-1} ≃ − 1 lim x M ‾ = y M ‾ \overline{xM}=\overline{yM} x M = y M ( W = 1 ) (W=1) ( W = 1 ) M = G M=G M = G d d d ≃ − 1 f i n \simeq^{\mathrm{fin}}_{-1} ≃ − 1 fin ρ ( y , G x ) < ∞ \rho(y,Gx)<\infty ρ ( y , G x ) < ∞
Proof. An arrow x → y x\to y x → y ( m , φ ) (m,\varphi) ( m , φ ) φ : x m ⇒ y \varphi:xm\Rightarrow y φ : x m ⇒ y ρ ( x m , y ) \rho(xm,y) ρ ( x m , y ) d ( x , y ) = inf m ρ ( x m , y ) d(x,y)=\inf_m\rho(xm,y) d ( x , y ) = inf m ρ ( x m , y ) y ∈ x M y\in xM y ∈ x M y m ′ ∈ x M ym'\in xM y m ′ ∈ x M m ′ m' m ′ m ′ m' m ′ y M ⊆ x M yM\subseteq xM y M ⊆ x M y M ‾ ⊆ x M ‾ \overline{yM}\subseteq\overline{xM} y M ⊆ x M y ∈ y M ‾ ⊆ y M ‾ = x M ‾ y\in\overline{yM}\subseteq\overline{yM}=\overline{xM} y ∈ y M ⊆ y M = x M ρ ( g x , y ) = ρ ( x , g − 1 y ) \rho(gx,y)=\rho(x,g^{-1}y) ρ ( g x , y ) = ρ ( x , g − 1 y ) d ( y , x ) = d ( x , y ) d(y,x)=d(x,y) d ( y , x ) = d ( x , y )
The comparison category of the closure datum records the acting group, and contains the exact relation as its part of cost zero.
Proposition 7.4. In the setting of Proposition 7.3 let X ≠ ∅ X\ne\varnothing X = ∅ ρ \rho ρ M = G M=G M = G
(a) K ≅ Ind ( X ) × B G K\cong\operatorname{Ind}(X)\times BG K ≅ Ind ( X ) × B G Ind ( X ) \operatorname{Ind}(X) Ind ( X ) G G G k ≥ 2 k\ge2 k ≥ 2
(b) The arrows of cost 0 form the comparison category of the strict datum, the action groupoid; the inclusion induces Stab G ( x ) → G \operatorname{Stab}_G(x)\to G Stab G ( x ) → G
Proof. (a) An arrow x → y x\to y x → y ( g , φ ) (g,\varphi) ( g , φ ) g ∈ G g\in G g ∈ G φ \varphi φ Hom ( x , y ) ≅ G \operatorname{Hom}(x,y)\cong G Hom ( x , y ) ≅ G ∣ NInd ( X ) ∣ |\operatorname{NInd}(X)| ∣ NInd ( X ) ∣ { c = 0 } \{c=0\} { c = 0 } ∘ v \circ_v ∘ v ρ \rho ρ c ( φ ) = ρ ( g − 1 x , y ) = 0 c(\varphi)=\rho(g^{-1}x,y)=0 c ( φ ) = ρ ( g − 1 x , y ) = 0 y = g − 1 x y=g^{-1}x y = g − 1 x
Closure relations are not special: they are all the limit relations of zero length. To say so, let Y = ∐ A , B Ob H ( A , B ) \mathcal{Y}=\coprod_{A,B}\operatorname{Ob}H(A,B) Y = ∐ A , B Ob H ( A , B ) σ A → σ B \sigma_A\to\sigma_B σ A → σ B D \mathcal{D} D V V V p , q ∈ Y p,q\in\mathcal{Y} p , q ∈ Y ρ ( p , q ) = inf { c ( Φ ) : Φ : p ⇒ q in W } \rho(p,q)=\inf\{c(\Phi):\Phi:p\Rightarrow q\text{ in }\mathcal{W}\} ρ ( p , q ) = inf { c ( Φ ) : Φ : p ⇒ q in W } ρ = ∞ \rho=\infty ρ = ∞ S ⊆ Y S\subseteq\mathcal{Y} S ⊆ Y cl ( S ) = { q : inf p ∈ S ρ ( p , q ) = 0 } \operatorname{cl}(S)=\{q:\inf_{p\in S}\rho(p,q)=0\} cl ( S ) = { q : inf p ∈ S ρ ( p , q ) = 0 } m m m C C C O ( m ) = { Θ u 1 , u 2 ( m ) } = V ⋅ s ( m ) ⊆ Y O(m)=\{\Theta_{u_1,u_2}(m)\}=V\cdot s(m)\subseteq\mathcal{Y} O ( m ) = { Θ u 1 , u 2 ( m )} = V ⋅ s ( m ) ⊆ Y
Theorem 7.5. Let D \mathcal{D} D γ 1 = γ 2 = 0 \gamma_1=\gamma_2=0 γ 1 = γ 2 = 0
(a) ρ ( p , p ) = 0 \rho(p,p)=0 ρ ( p , p ) = 0 ρ ( p , r ) ≤ ρ ( p , q ) + ρ ( q , r ) \rho(p,r)\le\rho(p,q)+\rho(q,r) ρ ( p , r ) ≤ ρ ( p , q ) + ρ ( q , r ) cl \operatorname{cl} cl cl ( S ) \operatorname{cl}(S) cl ( S ) V V V S S S
(b) d ( m , m ~ ) = 0 d(m,\widetilde m)=0 d ( m , m ) = 0 σ ( m ~ ) ∈ cl O ( m ) \sigma(\widetilde m)\in\operatorname{cl}O(m) σ ( m ) ∈ cl O ( m ) cl O ( m ~ ) ⊆ cl O ( m ) \operatorname{cl}O(\widetilde m)\subseteq\operatorname{cl}O(m) cl O ( m ) ⊆ cl O ( m )
(c) m ≃ l i m − 1 m ~ m\simeq^{-1}_{\mathrm{lim}}\widetilde m m ≃ lim − 1 m cl O ( m ) = cl O ( m ~ ) \operatorname{cl}O(m)=\operatorname{cl}O(\widetilde m) cl O ( m ) = cl O ( m ) m ↦ O ( m ) m\mapsto O(m) m ↦ O ( m )
Proof. (a) The first two properties are c ( 1 ) = 0 c(1)=0 c ( 1 ) = 0 c c c ∘ v \circ_v ∘ v q ∈ cl ( cl S ) q\in\operatorname{cl}(\operatorname{cl}S) q ∈ cl ( cl S ) r r r ρ ( r , q ) < ϵ \rho(r,q)<\epsilon ρ ( r , q ) < ϵ r ′ ∈ S r'\in S r ′ ∈ S ρ ( r ′ , r ) < ϵ \rho(r',r)<\epsilon ρ ( r ′ , r ) < ϵ ρ ( r ′ , q ) < 2 ϵ \rho(r',q)<2\epsilon ρ ( r ′ , q ) < 2 ϵ S S S V V V q ∈ cl ( S ) q\in\operatorname{cl}(S) q ∈ cl ( S ) a ∈ V a\in V a ∈ V p n ∈ S p_n\in S p n ∈ S ρ ( p n , q ) → 0 \rho(p_n,q)\to0 ρ ( p n , q ) → 0 H ( a ) p n ∈ S H(a)p_n\in S H ( a ) p n ∈ S ρ ( H ( a ) p n , H ( a ) q ) → 0 \rho(H(a)p_n,H(a)q)\to0 ρ ( H ( a ) p n , H ( a ) q ) → 0 H ( a ) q ∈ cl ( S ) H(a)q\in\operatorname{cl}(S) H ( a ) q ∈ cl ( S ) γ = 0 \gamma=0 γ = 0 d ( m , m ~ ) = inf u ρ ( Θ u ( m ) , σ m ~ ) d(m,\widetilde m)=\inf_u\rho(\Theta_u(m),\sigma\widetilde m) d ( m , m ) = inf u ρ ( Θ u ( m ) , σ m ) u u u O ( m ) O(m) O ( m ) σ ( m ~ ) \sigma(\widetilde m) σ ( m ) ∞ \infty ∞ σ ( m ~ ) ∈ cl O ( m ) \sigma(\widetilde m)\in\operatorname{cl}O(m) σ ( m ) ∈ cl O ( m ) O ( m ~ ) = V σ ( m ~ ) ⊆ cl O ( m ) O(\widetilde m)=V\sigma(\widetilde m)\subseteq\operatorname{cl}O(m) O ( m ) = V σ ( m ) ⊆ cl O ( m ) cl O ( m ~ ) ⊆ cl O ( m ) \operatorname{cl}O(\widetilde m)\subseteq\operatorname{cl}O(m) cl O ( m ) ⊆ cl O ( m ) σ ( m ~ ) ∈ O ( m ~ ) \sigma(\widetilde m)\in O(\widetilde m) σ ( m ) ∈ O ( m )
The theorem contains, with one proof, every limit relation of zero length that appears in practice.
Corollary 7.6. The following are limit relations of zero length, hence pullbacks of closure equality: (a) equality of orbit closures for a continuous action of a monoid by uniformly continuous maps (take V = B M V = BM V = B M { T n x } \{T^n x\} { T n x } lim sup n ρ ( x n , y n ) \limsup_n \rho(x_n,y_n) lim sup n ρ ( x n , y n ) V V V ρ ( x , y ) = 0 = ρ ( y , x ) \rho(x,y) = 0 = \rho(y,x) ρ ( x , y ) = 0 = ρ ( y , x ) V V V
Lengths γ > 0 \gamma> 0 γ > 0 u u u d ( m , m ~ ) = 0 d(m,\widetilde m) = 0 d ( m , m ) = 0 ε \varepsilon ε u u u γ ( u ) < ε \gamma(u) < \varepsilon γ ( u ) < ε m m m ε \varepsilon ε m ~ \widetilde m m
Two further limit relations, equality of closures of subsets for the Hausdorff distance and approximate unitary equivalence of operators, are treated in Examples 11.6 and 11.7.
Cauchy sequences
Let ( X , ρ ) (X,\rho) ( X , ρ ) C = p t \mathcal{C} = \mathrm{pt} C = pt F ( ∗ ) = X N F(*) = X^{\mathbb{N}} F ( ∗ ) = X N c ( x , y ) = lim sup n ρ ( x n , y n ) c(x,y) = \limsup_n \rho(x_n,y_n) c ( x , y ) = lim sup n ρ ( x n , y n ) W = a l l \mathcal{W} = \mathrm{all} W = all lim sup ( a n + b n ) ≤ lim sup a n + lim sup b n \limsup(a_n+b_n) \le\limsup a_n + \limsup b_n lim sup ( a n + b n ) ≤ lim sup a n + lim sup b n d = c d = c d = c
Proposition 7.7. (a) x ≃ − 1 lim y x \simeq^{\lim}_{-1} y x ≃ − 1 l i m y ρ ( x n , y n ) → 0 \rho(x_n,y_n) \to0 ρ ( x n , y n ) → 0 x ≃ − 1 f i n y x \simeq^{\mathrm{fin}}_{-1} y x ≃ − 1 fin y ρ ( x n , y n ) \rho(x_n,y_n) ρ ( x n , y n ) n n n Cau ( X ) \operatorname{Cau}(X) Cau ( X ) ≃ − 1 f i n \simeq^{\mathrm{fin}}_{-1} ≃ − 1 fin ≃ − 1 lim \simeq^{\lim}_{-1} ≃ − 1 l i m
(b) Cau ( X ) / ≃ − 1 lim \operatorname{Cau}(X)/\simeq^{\lim}_{-1} Cau ( X ) / ≃ − 1 l i m ρ ˉ ( [ x ] , [ y ] ) = lim n ρ ( x n , y n ) \bar{\rho}([x],[y]) = \lim_n \rho(x_n,y_n) ρ ˉ ([ x ] , [ y ]) = lim n ρ ( x n , y n ) X ^ \widehat{X} X X X X X X X
(c) With the cost c 0 = 0 c_0 = 0 c 0 = 0 ∞ \infty ∞ − 1 -1 − 1 ⊆ ≃ − 1 lim ⊆ ≃ − 1 f i n \subseteq\simeq^{\lim}_{-1} \subseteq\simeq^{\mathrm{fin}}_{-1} ⊆ ≃ − 1 l i m ⊆ ≃ − 1 fin
Proof. (a) is the definition, Cauchy sequences being bounded. (b) For Cauchy x , y x,y x , y ∣ ρ ( x n , y n ) − ρ ( x m , y m ) ∣ ≤ ρ ( x n , x m ) + ρ ( y n , y m ) |\rho(x_n,y_n)-\rho(x_m,y_m)| \le\rho(x_n,x_m)+\rho(y_n,y_m) ∣ ρ ( x n , y n ) − ρ ( x m , y m ) ∣ ≤ ρ ( x n , x m ) + ρ ( y n , y m ) ρ ( x n , y n ) \rho(x_n,y_n) ρ ( x n , y n ) ρ ˉ \bar{\rho} ρ ˉ ρ ˉ ( [ x ] , [ x k ] ) = lim n ρ ( x n , x k ) → 0 \bar{\rho}([x],[x_k]) = \lim_n \rho(x_n,x_k) \to0 ρ ˉ ([ x ] , [ x k ]) = lim n ρ ( x n , x k ) → 0 k → ∞ k \to\infty k → ∞ X k = [ x ( k ) ] X_k = [x^{(k)}] X k = [ x ( k ) ] X ^ \widehat{X} X N j N_j N j ρ ( x n ( j ) , x m ( j ) ) < 1 / j \rho(x_n^{(j)},x_m^{(j)}) < 1/j ρ ( x n ( j ) , x m ( j ) ) < 1/ j n , m ≥ N j n,m \ge N_j n , m ≥ N j n k n_k n k n k ≥ N j n_k \ge N_j n k ≥ N j j ≤ k j \le k j ≤ k ρ ( x n k ( k ) , x n k ( j ) ) < ρ ˉ ( X k , X j ) + 1 / k \rho(x_{n_k}^{(k)},x_{n_k}^{(j)}) < \bar{\rho}(X_k,X_j) + 1/k ρ ( x n k ( k ) , x n k ( j ) ) < ρ ˉ ( X k , X j ) + 1/ k ρ ( x n ( k ) , x n ( j ) ) → ρ ˉ ( X k , X j ) \rho(x_n^{(k)},x_n^{(j)}) \to\bar{\rho}(X_k,X_j) ρ ( x n ( k ) , x n ( j ) ) → ρ ˉ ( X k , X j ) z k = x n k ( k ) z_k = x_{n_k}^{(k)} z k = x n k ( k ) k ≥ ℓ k \ge\ell k ≥ ℓ ρ ( z k , z ℓ ) ≤ ρ ˉ ( X k , X ℓ ) + 1 / k + 1 / ℓ \rho(z_k,z_\ell) \le\bar{\rho}(X_k,X_\ell) + 1/k + 1/\ell ρ ( z k , z ℓ ) ≤ ρ ˉ ( X k , X ℓ ) + 1/ k + 1/ ℓ z z z j j j ρ ˉ ( [ z ] , [ X j ] ) = lim k → ∞ ρ ( z k , x n k ( j ) ) ≤ lim sup k → ∞ ρ ˉ ( X k , X j ) \bar{\rho}([z],[X_j]) = \lim_{k\to\infty}\rho(z_k,x_{n_k}^{(j)}) \le\limsup_{k\to\infty}\bar{\rho}(X_k,X_j) ρ ˉ ([ z ] , [ X j ]) = lim k → ∞ ρ ( z k , x n k ( j ) ) ≤ lim sup k → ∞ ρ ˉ ( X k , X j ) n k , k → ∞ n_k,k \to\infty n k , k → ∞ x ( j ) x^{(j)} x ( j ) j → ∞ j \to\infty j → ∞ [ z ] [z] [ z ] c 0 c_0 c 0 d = c 0 ∈ { 0 , ∞ } d = c_0 \in\{0,\infty\} d = c 0 ∈ { 0 , ∞ }
Equivalence of Cauchy sequences is the relation by which the real numbers are constructed from the rationals (and the p p p p p p { 0 , ∞ } \{0,\infty\} { 0 , ∞ }
Limit levels and S S S
The limit preorder ≤ lim \leq^{\lim} ≤ l i m K lim \mathcal{K}^{\lim} K l i m Mor ( C ) \operatorname{Mor}(\mathcal{C}) Mor ( C ) m → m ~ m \to\widetilde m m → m d ( m , m ~ ) = 0 d(m,\widetilde m) = 0 d ( m , m ) = 0 K 0 \mathcal{K}_0 K 0 K \mathcal{K} K K lim \mathcal{K}^{\lim} K l i m − 1 -1 − 1 ≃ − 1 lim \simeq^{\lim}_{-1} ≃ − 1 l i m ≤ lim \leq^{\lim} ≤ l i m k ≥ 1 k \geq1 k ≥ 1 − 1 -1 − 1
Stabilization relations
Stabilization relations identify two objects when they become equivalent after adding the same (or arbitrary) auxiliary summands. They are level − 1 -1 − 1
Stabilization systems
Stabilization requires the following auxiliary notions.
Definition 8.1. A commutative monoid ( S , + , 0 ) (S,+,0) ( S , + , 0 ) + + + 0 0 0 symmetric submonoid T ⊆ S × S T\subseteq S\times S T ⊆ S × S ( 0 , 0 ) (0,0) ( 0 , 0 ) ( z , z ′ ) ↦ ( z ′ , z ) (z,z')\mapsto(z',z) ( z , z ′ ) ↦ ( z ′ , z ) ( z , z ′ ) (z,z') ( z , z ′ ) N × N \mathbb{N}\times\mathbb{N} N × N
Stabilization is axiomatized as follows.
Definition 8.2. Fix a datum with level − 1 -1 − 1 ≃ = ≃ − 1 \simeq=\simeq_{-1} ≃= ≃ − 1 stabilization system is a commutative monoid ( S , + , 0 ) (S,+,0) ( S , + , 0 ) Σ z : Mor C → Mor C \Sigma_z:\operatorname{Mor} C\to\operatorname{Mor} C Σ z : Mor C → Mor C ( z ∈ S ) (z\in S) ( z ∈ S ) m ≃ m ~ ⇒ Σ z m ≃ Σ z m ~ m\simeq\widetilde m\Rightarrow\Sigma_zm\simeq\Sigma_z\widetilde m m ≃ m ⇒ Σ z m ≃ Σ z m Σ 0 m ≃ m \Sigma_0m\simeq m Σ 0 m ≃ m Σ w Σ z m ≃ Σ w + z m \Sigma_w\Sigma_zm\simeq\Sigma_{w+z}m Σ w Σ z m ≃ Σ w + z m Σ z \Sigma_z Σ z z z z 0 0 0 z z z w w w w + z w+z w + z T ⊆ S × S T\subseteq S\times S T ⊆ S × S m ≃ T m ~ : ⟺ ∃ ( z , z ′ ) ∈ T : Σ z m ≃ Σ z ′ m ~ m\simeq_T\widetilde m:\Longleftrightarrow\exists(z,z')\in T:\Sigma_zm\simeq\Sigma_{z'}\widetilde m m ≃ T m :⟺ ∃ ( z , z ′ ) ∈ T : Σ z m ≃ Σ z ′ m
Proposition 8.3. ≃ T \simeq_T ≃ T ≃ \simeq ≃
Proof. Reflexive, contains ≃ \simeq ≃ ( 0 , 0 ) ∈ T (0,0)\in T ( 0 , 0 ) ∈ T T T T Σ z m ≃ Σ z ′ m ~ \Sigma_zm\simeq\Sigma_{z'}\widetilde m Σ z m ≃ Σ z ′ m Σ w m ~ ≃ Σ w ′ m ^ \Sigma_w\widetilde m\simeq\Sigma_{w'}\widehat m Σ w m ≃ Σ w ′ m ( z , z ′ ) , ( w , w ′ ) ∈ T (z,z'),(w,w')\in T ( z , z ′ ) , ( w , w ′ ) ∈ T Σ w \Sigma_w Σ w Σ z ′ \Sigma_{z'} Σ z ′ Σ w Σ z m ≃ Σ w Σ z ′ m ~ \Sigma_w\Sigma_zm\simeq\Sigma_w\Sigma_{z'}\widetilde m Σ w Σ z m ≃ Σ w Σ z ′ m Σ z ′ Σ w m ~ ≃ Σ z ′ Σ w ′ m ^ \Sigma_{z'}\Sigma_w\widetilde m\simeq\Sigma_{z'}\Sigma_{w'}\widehat m Σ z ′ Σ w m ≃ Σ z ′ Σ w ′ m Σ w Σ z ′ m ~ ≃ Σ w + z ′ m ~ = Σ z ′ + w m ~ ≃ Σ z ′ Σ w m ~ \Sigma_w\Sigma_{z'}\widetilde m\simeq\Sigma_{w+z'}\widetilde m=\Sigma_{z'+w}\widetilde m\simeq\Sigma_{z'}\Sigma_w\widetilde m Σ w Σ z ′ m ≃ Σ w + z ′ m = Σ z ′ + w m ≃ Σ z ′ Σ w m + + + Σ w Σ z m ≃ Σ w + z m \Sigma_w\Sigma_zm\simeq\Sigma_{w+z}m Σ w Σ z m ≃ Σ w + z m Σ z ′ Σ w ′ m ^ ≃ Σ z ′ + w ′ m ^ \Sigma_{z'}\Sigma_{w'}\widehat m\simeq\Sigma_{z'+w'}\widehat m Σ z ′ Σ w ′ m ≃ Σ z ′ + w ′ m ≃ \simeq ≃ Σ w + z m ≃ Σ z ′ + w ′ m ^ \Sigma_{w+z}m\simeq\Sigma_{z'+w'}\widehat m Σ w + z m ≃ Σ z ′ + w ′ m ( w + z , z ′ + w ′ ) = ( w , w ′ ) + ( z , z ′ ) ∈ T (w+z,z'+w')=(w,w')+(z,z')\in T ( w + z , z ′ + w ′ ) = ( w , w ′ ) + ( z , z ′ ) ∈ T
We recall the terminology used in Example 11.9.
Definition 8.4. (a) A module P P P R R R finitely generated projective if it is a direct summand of R n R^n R n n n n R R R K 0 ( R ) K_0(R) K 0 ( R ) x , y x,y x , y [ x ] = [ y ] [x]=[y] [ x ] = [ y ] K 0 ( R ) K_0(R) K 0 ( R ) x ⊕ P ≅ y ⊕ P x\oplus P\cong y\oplus P x ⊕ P ≅ y ⊕ P P P P quadratic space ( V , q ) (V,q) ( V , q ) q q q nondegenerate if its associated bilinear form has trivial radical; an isometry is a linear bijection preserving q q q ⊥ \perp ⊥ hyperbolic plane Hyp \operatorname{Hyp} Hyp ( F 2 , ( x , y ) ↦ x y ) (\mathbb{F}^2,(x,y)\mapsto xy) ( F 2 , ( x , y ) ↦ x y ) Hyp n = Hyp ⊥ ⋯ ⊥ Hyp \operatorname{Hyp}^n=\operatorname{Hyp}\perp\cdots\perp\operatorname{Hyp} Hyp n = Hyp ⊥ ⋯ ⊥ Hyp n n n finite pointed CW complex is a space built from finitely many cells and equipped with a base point; a pointed homotopy equivalence is a homotopy equivalence preserving base points up to homotopy; the suspension S X SX S X X × [ 0 , 1 ] X\times[0,1] X × [ 0 , 1 ] X × { 0 , 1 } ∪ { ∗ } × [ 0 , 1 ] X\times\{0,1\}\cup\{*\}\times[0,1] X × { 0 , 1 } ∪ { ∗ } × [ 0 , 1 ] S n S^n S n n n n
The stabilizations compatible with a comparison datum are the following.
Definition 8.5. A strict stabilizer of a comparison datum D \mathcal{D} D Σ = ( Σ C , Σ D ) \Sigma=(\Sigma_{\mathcal{C}},\Sigma_{\mathcal{D}}) Σ = ( Σ C , Σ D ) Σ C : C → C \Sigma_{\mathcal{C}}:\mathcal{C}\to\mathcal{C} Σ C : C → C Σ D : D → D \Sigma_{\mathcal{D}}:\mathcal{D}\to\mathcal{D} Σ D : D → D Σ D σ ( m ) = σ ( Σ C m ) \Sigma_{\mathcal{D}}\sigma(m)=\sigma(\Sigma_{\mathcal{C}}m) Σ D σ ( m ) = σ ( Σ C m ) m m m Σ D τ i = τ i Σ C \Sigma_{\mathcal{D}}\tau_i=\tau_i\Sigma_{\mathcal{C}} Σ D τ i = τ i Σ C i = 1 , 2 i=1,2 i = 1 , 2 Σ C ( U i ) ⊆ U i \Sigma_{\mathcal{C}}(\mathcal{U}_i)\subseteq\mathcal{U}_i Σ C ( U i ) ⊆ U i Σ D ( W ) ⊆ W \Sigma_{\mathcal{D}}(\mathcal{W})\subseteq\mathcal{W} Σ D ( W ) ⊆ W Σ \Sigma Σ K K K Σ n = Σ n \Sigma_n=\Sigma^n Σ n = Σ n S = N S=\mathbb{N} S = N
stable levels are m ≃ − 1 s t m ~ ⟺ Σ n m ≃ − 1 Σ n m ~ m \simeq^{\mathrm{st}}_{-1} \widetilde{m} \Longleftrightarrow\Sigma^n m \simeq_{-1} \Sigma^n\widetilde{m} m ≃ − 1 st m ⟺ Σ n m ≃ − 1 Σ n m n n n m ≃ 0 s t m ~ ⟺ Σ n m ≃ 0 Σ n m ~ m \simeq^{\mathrm{st}}_{0} \widetilde{m} \Longleftrightarrow\Sigma^n m \simeq_{0} \Sigma^n\widetilde{m} m ≃ 0 st m ⟺ Σ n m ≃ 0 Σ n m n n n ≃ T \simeq^{\mathcal{T}} ≃ T T \mathcal{T} T
A strict stabilizer acts on the comparison category.
Proposition 8.6. Let Σ \Sigma Σ D \mathcal{D} D Σ \Sigma Σ K ( D ) \mathsf{K}(\mathcal{D}) K ( D ) ( u 1 , u 2 , Φ ) ↦ ( Σ u 1 , Σ u 2 , Σ Φ ) (u_1,u_2,\Phi) \mapsto(\Sigma u_1,\Sigma u_2,\Sigma\Phi) ( u 1 , u 2 , Φ ) ↦ ( Σ u 1 , Σ u 2 , ΣΦ ) ( S 1 ) (S1) ( S 1 ) ( S 2 ) (S2) ( S 2 ) Σ 0 m = m \Sigma_0m=m Σ 0 m = m Σ w Σ z = Σ w + z \Sigma_w\Sigma_z=\Sigma_{w+z} Σ w Σ z = Σ w + z
Proof. We have Θ Σ u 1 , Σ u 2 ( Σ m ) = τ 1 ( Σ u 1 ) σ ( Σ m ) τ 2 ( Σ u 2 ) = Σ D ( τ 1 ( u 1 ) σ ( m ) τ 2 ( u 2 ) ) = Σ D Θ u 1 , u 2 ( m ) \Theta_{\Sigma u_1,\Sigma u_2}(\Sigma m)=\tau_1(\Sigma u_1)\sigma(\Sigma m)\tau_2(\Sigma u_2)=\Sigma_{\mathcal{D}}\bigl(\tau_1(u_1)\sigma(m)\tau_2(u_2)\bigr)=\Sigma_{\mathcal{D}}\Theta_{u_1,u_2}(m) Θ Σ u 1 , Σ u 2 ( Σ m ) = τ 1 ( Σ u 1 ) σ ( Σ m ) τ 2 ( Σ u 2 ) = Σ D ( τ 1 ( u 1 ) σ ( m ) τ 2 ( u 2 ) ) = Σ D Θ u 1 , u 2 ( m ) Σ D Φ : Σ D Θ u 1 , u 2 ( m ) ⇒ Σ D σ ( m ~ ) = σ ( Σ m ~ ) \Sigma_{\mathcal{D}}\Phi:\Sigma_{\mathcal{D}}\Theta_{u_1,u_2}(m)\Rightarrow\Sigma_{\mathcal{D}}\sigma(\widetilde{m})=\sigma(\Sigma\widetilde{m}) Σ D Φ : Σ D Θ u 1 , u 2 ( m ) ⇒ Σ D σ ( m ) = σ ( Σ m ) W W W Σ D \Sigma_{\mathcal{D}} Σ D Σ \Sigma Σ K → K \mathsf{K}\to\mathsf{K} K → K ≃ − 1 \simeq_{-1} ≃ − 1 Σ 0 = id \Sigma^0=\operatorname{id} Σ 0 = id Σ w Σ z = Σ w + z \Sigma^w\Sigma^z=\Sigma^{w+z} Σ w Σ z = Σ w + z
Stabilized level − 1 -1 − 1
Proposition 8.7. Let Σ \Sigma Σ D \mathcal{D} D K ∞ = colim ( K → Σ K → Σ ⋯ ) \mathsf{K}_{\infty}=\operatorname{colim}(\mathsf{K}\xrightarrow{\Sigma}\mathsf{K}\xrightarrow{\Sigma}\cdots) K ∞ = colim ( K Σ K Σ ⋯ ) m ≃ − 1 s t m ~ m\simeq^{\mathrm{st}}_{-1}\widetilde{m} m ≃ − 1 st m m m m m ~ \widetilde{m} m K ∞ \mathsf{K}_{\infty} K ∞
Proof. The colimit of the sequence is a filtered colimit of categories, so its objects are the classes of pairs ( n , m ) (n,m) ( n , m ) ( n , m ) ∼ ( n + 1 , Σ m ) (n,m)\sim(n+1,\Sigma m) ( n , m ) ∼ ( n + 1 , Σ m ) Hom ( [ ( 0 , m ) ] , [ ( 0 , m ~ ) ] ) = colim p Hom K ( Σ p m , Σ p m ~ ) \operatorname{Hom}([(0,m)],[(0,\widetilde{m})])=\operatorname{colim}_p\operatorname{Hom}_{\mathsf{K}}(\Sigma^p m,\Sigma^p\widetilde{m}) Hom ([( 0 , m )] , [( 0 , m )]) = colim p Hom K ( Σ p m , Σ p m ) Σ \Sigma Σ K ∞ \mathsf{K}_{\infty} K ∞ Σ p m ⊴ Σ p m ~ \Sigma^p m\trianglelefteq\Sigma^p\widetilde{m} Σ p m ⊴ Σ p m p p p p p p p ′ p' p ′ Σ p ′ \Sigma^{p'} Σ p ′ Σ p \Sigma^p Σ p Proposition 8.6 ) puts both at the stage p + p ′ p+p' p + p ′ Σ p + p ′ m ≃ − 1 Σ p + p ′ m ~ \Sigma^{p+p'}m\simeq_{-1}\Sigma^{p+p'}\widetilde{m} Σ p + p ′ m ≃ − 1 Σ p + p ′ m Σ n m ≃ − 1 Σ n m ~ \Sigma^n m\simeq_{-1}\Sigma^n\widetilde{m} Σ n m ≃ − 1 Σ n m n n n
Stabilization does not leave the class of comparison data: it is a filtered colimit of data.
Proposition 8.8. Let Σ \Sigma Σ D \mathcal{D} D D ∞ \mathcal{D}^{\infty} D ∞ D → Σ D → Σ ⋯ \mathcal{D}\xrightarrow{\Sigma}\mathcal{D}\xrightarrow{\Sigma}\cdots D Σ D Σ ⋯ K ( D ∞ ) = K ∞ \mathsf{K}(\mathcal{D}^{\infty})=\mathsf{K}_{\infty} K ( D ∞ ) = K ∞ m m m D ∞ \mathcal{D}^{\infty} D ∞ m m m ≃ − 1 s t \simeq^{\mathrm{st}}_{-1} ≃ − 1 st − 1 -1 − 1 ≃ 0 s t \simeq^{\mathrm{st}}_{0} ≃ 0 st 0 0 0 π k ( D ∞ , [ m ] ) = colim n π k ( D , Σ n m ) \pi_k(\mathcal{D}^{\infty},[m])=\operatorname{colim}_n\pi_k(\mathcal{D},\Sigma^n m) π k ( D ∞ , [ m ]) = colim n π k ( D , Σ n m ) k ≥ 1 k\geq1 k ≥ 1
Proof. Objects, 1- and 2-morphisms of C ∞ \mathcal{C}^{\infty} C ∞ D ∞ \mathcal{D}^{\infty} D ∞ x ∼ Σ x x\sim\Sigma x x ∼ Σ x C ∞ , D ∞ \mathcal{C}^{\infty},\mathcal{D}^{\infty} C ∞ , D ∞ σ , τ i , U i ∞ = colim U i \sigma,\tau_i,U_i^{\infty}=\operatorname{colim}U_i σ , τ i , U i ∞ = colim U i W ∞ = colim W W^{\infty}=\operatorname{colim}W W ∞ = colim W Σ D σ = σ Σ C \Sigma_{\mathcal{D}}\sigma=\sigma\Sigma_{\mathcal{C}} Σ D σ = σ Σ C Σ D τ i = τ i Σ C \Sigma_{\mathcal{D}}\tau_i=\tau_i\Sigma_{\mathcal{C}} Σ D τ i = τ i Σ C Σ C U i ⊆ U i \Sigma_{\mathcal{C}}U_i\subseteq U_i Σ C U i ⊆ U i Σ D W ⊆ W \Sigma_{\mathcal{D}}W\subseteq W Σ D W ⊆ W W ∞ W^{\infty} W ∞ K ( D ∞ ) \mathsf{K}(\mathcal{D}^{\infty}) K ( D ∞ ) Hom ( [ m ] , [ m ~ ] ) = colim p Hom K ( Σ p m , Σ p m ~ ) \operatorname{Hom}([m],[\widetilde{m}])=\operatorname{colim}_p\operatorname{Hom}_{\mathsf{K}}(\Sigma^p m,\Sigma^p\widetilde{m}) Hom ([ m ] , [ m ]) = colim p Hom K ( Σ p m , Σ p m ) K ∞ \mathsf{K}_{\infty} K ∞ Proposition 8.7 . The nerve commutes with sequential colimits, and so do the π k \pi_k π k π 0 \pi_0 π 0
The higher stabilized levels are those of a mapping telescope.
Proposition 8.9. Let Σ \Sigma Σ D \mathcal{D} D K Σ = ∫ N K \mathsf{K}_{\Sigma}=\int_{\mathbb{N}}\mathsf{K} K Σ = ∫ N K N → C a t \mathbb{N}\to\mathsf{Cat} N → Cat n ≤ n ′ n\leq n' n ≤ n ′ Σ n ′ − n \Sigma^{n'-n} Σ n ′ − n
(a) m ≃ 0 s t m ~ m\simeq^{\mathrm{st}}_0\widetilde{m} m ≃ 0 st m ( 0 , m ) (0,m) ( 0 , m ) ( 0 , m ~ ) (0,\widetilde{m}) ( 0 , m ) K Σ \mathsf{K}_{\Sigma} K Σ
(b) ∣ N K Σ ∣ |\mathsf{N}\mathsf{K}_{\Sigma}| ∣ N K Σ ∣ Tel \operatorname{Tel} Tel ∣ N K ∣ → Σ ∣ N K ∣ → Σ ⋯ |\mathsf{N}\mathsf{K}|\xrightarrow{\Sigma}|\mathsf{N}\mathsf{K}|\xrightarrow{\Sigma}\cdots ∣ NK ∣ Σ ∣ NK ∣ Σ ⋯ π k ( Tel , ( 0 , m ) ) = colim n π k ( ∣ N K ∣ , Σ n m ) \pi_k(\operatorname{Tel},(0,m))=\operatorname{colim}_n\pi_k(|\mathsf{N}\mathsf{K}|,\Sigma^n m) π k ( Tel , ( 0 , m )) = colim n π k ( ∣ NK ∣ , Σ n m )
Proof. By Thomason’s theorem (Definition 5.7), ∣ N K Σ ∣ ≃ hocolim N ∣ N K ∣ |\mathsf{N}\mathsf{K}_{\Sigma}|\simeq\operatorname{hocolim}_{\mathbb{N}}|\mathsf{N}\mathsf{K}| ∣ N K Σ ∣ ≃ hocolim N ∣ NK ∣ k = 0 k=0 k = 0 π 0 K Σ = colim n π 0 K \pi_0K_\Sigma=\operatorname{colim}_n\pi_0K π 0 K Σ = colim n π 0 K m , m ~ m,\widetilde{m} m , m Σ n m \Sigma^n m Σ n m Σ n m ~ \Sigma^n\widetilde{m} Σ n m K K K n n n
For groupoid data the stabilized levels reduce to stabilized automorphism groups.
Corollary 8.10. Let Σ \Sigma Σ D \mathcal{D} D K = K ( D ) K=K(\mathcal{D}) K = K ( D ) m m m C C C k ≥ 2 k\geq2 k ≥ 2 m m m m m m colim n Aut K ( Σ n m ) \operatorname{colim}_n\operatorname{Aut}_K(\Sigma^n m) colim n Aut K ( Σ n m ) Σ \Sigma Σ
Proof. For a groupoid, ∣ N K ∣ |NK| ∣ N K ∣ π k ( ∣ N K ∣ , x ) = 0 \pi_k(|NK|,x)=0 π k ( ∣ N K ∣ , x ) = 0 k ≥ 2 k\geq2 k ≥ 2 π 1 ( ∣ N K ∣ , x ) = Aut K ( x ) \pi_1(|NK|,x)=\operatorname{Aut}_K(x) π 1 ( ∣ N K ∣ , x ) = Aut K ( x ) Proposition 8.9 (b).
Summary: tracing each relation to the normal form
The table records, for each relation treated, the chain of constructions leading from the normal form K ( D ) ⊆ ⋁ V H K(\mathcal{D})\subseteq\bigvee_V H K ( D ) ⊆ ⋁ V H Restriction means that the relation is read on the objects s ( m ) s(m) s ( m ) enrichment that a cost is attached to the 2-morphisms and lengths to the modifications; colimit that a filtered colimit of data is taken.
Table 1.
Two remarks fix what the table does not say. First, the entries trace the relations to the normal form, but not every equivalence relation on Mor ( C ) \operatorname{Mor}(C) Mor ( C ) σ \sigma σ σ \sigma σ § 10 . Second, enrichment is the only operation in the table that changes the kind of object one reads: limit relations are not levels of K ( D ) K(\mathcal{D}) K ( D ) K l i m K^{\mathrm{lim}} K lim
The axis of regularity
The normal form shows that the comparison category carries no constraint on its shape (Theorem 4.5 ). The constraints on the relations that occur on a fixed Mor ( C ) \operatorname{Mor}(C) Mor ( C ) σ , τ 1 , τ 2 \sigma,\tau_1,\tau_2 σ , τ 1 , τ 2 C C C σ \sigma σ
Seven classes
Definition 10.1. A comparison datum D \mathcal{D} D ρ i \rho_i ρ i
ρ 0 \rho_0 ρ 0 rigid ) D = C \mathcal{D}=C D = C σ = τ 1 = τ 2 = Id \sigma=\tau_1=\tau_2=\operatorname{Id} σ = τ 1 = τ 2 = Id D = Gr ( C ; U 1 , U 2 ) \mathcal{D}=\operatorname{Gr}(C;\mathcal{U}_1,\mathcal{U}_2) D = Gr ( C ; U 1 , U 2 ) U 1 , U 2 \mathcal{U}_1,\mathcal{U}_2 U 1 , U 2
ρ 1 \rho_1 ρ 1 diagonal ) W = 1 \mathcal{W}=1 W = 1 σ = τ 1 = τ 2 \sigma=\tau_1=\tau_2 σ = τ 1 = τ 2
ρ 2 \rho_2 ρ 2 completely functorial strict ) W = 1 \mathcal{W}=1 W = 1 σ , τ 1 , τ 2 \sigma,\tau_1,\tau_2 σ , τ 1 , τ 2
ρ 3 \rho_3 ρ 3 completely functorial ) σ , τ 1 , τ 2 \sigma,\tau_1,\tau_2 σ , τ 1 , τ 2 W \mathcal{W} W
ρ 4 \rho_4 ρ 4 pseudofunctorial ) τ 1 , τ 2 \tau_1,\tau_2 τ 1 , τ 2 σ \sigma σ σ ( m ′ ) σ ( m ) ⇒ σ ( m ′ m ) \sigma(m')\sigma(m)\Rightarrow\sigma(m'm) σ ( m ′ ) σ ( m ) ⇒ σ ( m ′ m ) 1 σ A ⇒ σ ( 1 A ) 1_{\sigma A}\Rightarrow\sigma(1_A) 1 σ A ⇒ σ ( 1 A ) W \mathcal{W} W
ρ 5 \rho_5 ρ 5 lax ) as ρ 4 \rho_4 ρ 4 σ \sigma σ
ρ 6 \rho_6 ρ 6 arbitrary ) σ \sigma σ
For ρ = ρ i \rho=\rho_i ρ = ρ i k ∈ { − 1 , 0 } k\in\{-1,0\} k ∈ { − 1 , 0 } F ρ i k ( C ) \mathcal{F}_{\rho_i}^k(C) F ρ i k ( C ) Mor ( C ) \operatorname{Mor}(C) Mor ( C ) ≃ k \simeq_k ≃ k ρ \rho ρ C C C F k s t r = F ρ 2 k \mathcal{F}_k^{\mathrm{str}}=\mathcal{F}_{\rho_2}^k F k str = F ρ 2 k functorial level of E E E λ ( E ) = min { k ∈ { − 1 , 0 } : E ∈ F ρ 2 k ( C ) } \lambda(E)=\min\{k\in\{-1,0\}:E\in\mathcal{F}_{\rho_2}^k(C)\} λ ( E ) = min { k ∈ { − 1 , 0 } : E ∈ F ρ 2 k ( C )} = ∞ =\infty = ∞
Proposition 10.2. F ρ 0 k ⊆ F ρ 1 k ⊆ F ρ 2 k ⊆ F ρ 3 k ⊆ F ρ 4 k ⊆ F ρ 5 k ⊆ F ρ 6 k \mathcal{F}_{\rho_0}^k\subseteq\mathcal{F}_{\rho_1}^k\subseteq\mathcal{F}_{\rho_2}^k\subseteq\mathcal{F}_{\rho_3}^k\subseteq\mathcal{F}_{\rho_4}^k\subseteq\mathcal{F}_{\rho_5}^k\subseteq\mathcal{F}_{\rho_6}^k F ρ 0 k ⊆ F ρ 1 k ⊆ F ρ 2 k ⊆ F ρ 3 k ⊆ F ρ 4 k ⊆ F ρ 5 k ⊆ F ρ 6 k k = − 1 , 0 k=-1,0 k = − 1 , 0
Proof. Each class is a special case of the next: Id \operatorname{Id} Id W = 1 \mathcal{W}=1 W = 1 W \mathcal{W} W W ∋ \mathcal{W}\ni W ∋
The questions are then: which relations occur at each step and level, which steps are separated, and what is lost when passing from a step to the next. We answer them for the first three steps completely or partially, and for the last by a normalization.
The rigid and diagonal classes are classified
For wide subcategories U 1 , U 2 \mathcal{U}_1,\mathcal{U}_2 U 1 , U 2 C C C m ⊴ U m ~ m\unlhd_{\mathcal{U}}\widetilde{m} m ⊴ U m m ~ = u 1 m u 2 \widetilde{m}=u_1mu_2 m = u 1 m u 2 u i ∈ U i u_i\in\mathcal{U}_i u i ∈ U i J C U 1 , U 2 J_C^{\mathcal{U}_1,\mathcal{U}_2} J C U 1 , U 2 m ⊴ U m ~ ⊴ U m m\unlhd_{\mathcal{U}}\widetilde{m}\unlhd_{\mathcal{U}}m m ⊴ U m ⊴ U m U 1 = U 2 = C \mathcal{U}_1=\mathcal{U}_2=C U 1 = U 2 = C J J J Proposition 6.4 ).
Theorem 10.3. Let D = Gr ( C ; U 1 , U 2 ) \mathcal{D}=\operatorname{Gr}(C;\mathcal{U}_1,\mathcal{U}_2) D = Gr ( C ; U 1 , U 2 ) K ( D ) = ∫ V Hom C K(\mathcal{D})=\int_V\operatorname{Hom}_C K ( D ) = ∫ V Hom C V = U 2 o p × U 1 V=\mathcal{U}_2^{\mathrm{op}}\times\mathcal{U}_1 V = U 2 op × U 1 Hom C \operatorname{Hom}_C Hom C ( u 2 , u 1 ) ⋅ p = u 1 p u 2 (u_2,u_1)\cdot p=u_1pu_2 ( u 2 , u 1 ) ⋅ p = u 1 p u 2 D \mathcal{D} D − 1 -1 − 1 J C U 1 , U 2 J_C^{\mathcal{U}_1,\mathcal{U}_2} J C U 1 , U 2 0 0 0 U 1 = 1 \mathcal{U}_1=1 U 1 = 1 U 2 = 1 \mathcal{U}_2=1 U 2 = 1
Proof. Apply Theorem 4.2 and Corollary 4.4 with the subcategories U i \mathcal{U}_i U i C C C H ( A , B ) = Hom C ( A , B ) H(A,B)=\operatorname{Hom}_C(A,B) H ( A , B ) = Hom C ( A , B ) s s s Ob ∫ V H \operatorname{Ob}\int_V H Ob ∫ V H m → m ~ m\to\widetilde{m} m → m ( u 1 , u 2 ) (u_1,u_2) ( u 1 , u 2 ) m ~ = u 1 m u 2 \widetilde{m}=u_1mu_2 m = u 1 m u 2
Theorem 10.4. Let ≈ \approx ≈ C C C q : C → C / ≈ q:C\to C/\mathord{\approx} q : C → C / ≈ W 1 , W 2 \mathcal{W}_1,\mathcal{W}_2 W 1 , W 2 C / ≈ C/\mathord{\approx} C / ≈ D ( ≈ ; W 1 , W 2 ) = ( C , C / ≈ , q , q , q ; q − 1 W 1 , q − 1 W 2 , 1 ) \mathcal{D}(\approx;\mathcal{W}_1,\mathcal{W}_2)=(C,C/\mathord{\approx},q,q,q;q^{-1}\mathcal{W}_1,q^{-1}\mathcal{W}_2,1) D ( ≈ ; W 1 , W 2 ) = ( C , C / ≈ , q , q , q ; q − 1 W 1 , q − 1 W 2 , 1 ) ρ 1 \rho_1 ρ 1 − 1 -1 − 1 q − 1 ( J C / ≈ W 1 , W 2 ) q^{-1}(J_{C/\approx}^{\mathcal{W}_1,\mathcal{W}_2}) q − 1 ( J C / ≈ W 1 , W 2 ) 0 0 0 q q q Gr ( C / ≈ ; W 1 , W 2 ) \operatorname{Gr}(C/\mathord{\approx};\mathcal{W}_1,\mathcal{W}_2) Gr ( C / ≈ ; W 1 , W 2 ) ρ 1 \rho_1 ρ 1 − 1 -1 − 1 0 0 0 D ( ≈ ; W 1 , W 2 ) \mathcal{D}(\approx;\mathcal{W}_1,\mathcal{W}_2) D ( ≈ ; W 1 , W 2 ) ≈ = ker σ \approx=\ker\sigma ≈= ker σ m ≈ m ′ m\approx m' m ≈ m ′ m , m ′ m,m' m , m ′ σ m = σ m ′ \sigma m=\sigma m' σ m = σ m ′ W i \mathcal{W}_i W i U i \mathcal{U}_i U i
F ρ 1 − 1 ( C ) = { q ≈ − 1 ( J C / ≈ W 1 , W 2 ) } . \mathcal{F}_{\rho_1}^{-1}(C)=\left\{q_{\approx}^{-1}\left(J_{C/\approx}^{\mathcal{W}_1,\mathcal{W}_2}\right)\right\}. F ρ 1 − 1 ( C ) = { q ≈ − 1 ( J C / ≈ W 1 , W 2 ) } . Proof. The kernel ≈ \approx ≈ σ = Q \sigma= Q σ = Q C → C / ≈ ↪ D \mathcal{C} \to\mathcal{C}/\mathord{\approx} \hookrightarrow\mathcal{D} C → C / ≈ ↪ D m → m ~ m \to\widetilde{m} m → m ( u 1 , u 2 ) (u_1,u_2) ( u 1 , u 2 ) Q m ~ = Q u 1 Q m Q u 2 Q\widetilde{m}=Qu_1\,Qm\,Qu_2 Q m = Q u 1 Q m Q u 2 m ~ ≈ u 1 m u 2 \widetilde{m}\approx u_1mu_2 m ≈ u 1 m u 2 m ~ ‾ = u ‾ 1 m ‾ u ‾ 2 \overline{\widetilde{m}}=\overline{u}_1\overline{m}\overline{u}_2 m = u 1 m u 2 C / ≈ \mathcal{C}/\mathord{\approx} C / ≈ u ‾ i \overline{u}_i u i W i W_i W i U i U_i U i u ‾ i \overline{u}_i u i u i ∈ U i u_i\in U_i u i ∈ U i m ⊴ m ~ m\unlhd\widetilde{m} m ⊴ m m ‾ ⊴ W m ~ ‾ \overline{m}\unlhd_W\overline{\widetilde{m}} m ⊴ W m C / ≈ \mathcal{C}/\mathord{\approx} C / ≈ − 1 -1 − 1 0 0 0 ( id , id ) (\operatorname{id},\operatorname{id}) ( id , id ) K K K K ( Gr ( C / ≈ ; W 1 , W 2 ) ) K(\operatorname{Gr}(\mathcal{C}/\mathord{\approx};W_1,W_2)) K ( Gr ( C / ≈ ; W 1 , W 2 )) q q q
Corollary 10.5. (a) Every congruence belongs to F ρ 0 − 1 \mathcal{F}^{-1}_{\rho_0} F ρ 0 − 1 W 1 = W 2 = 1 W_1=W_2=1 W 1 = W 2 = 1
(b) For a finite group G G G F ρ 0 − 1 ( B G ) = F ρ 1 − 1 ( B G ) = F ρ 0 0 ( B G ) = F ρ 1 0 ( B G ) \mathcal{F}^{-1}_{\rho_0}(\mathrm{BG})=\mathcal{F}^{-1}_{\rho_1}(\mathrm{BG})=\mathcal{F}^{0}_{\rho_0}(\mathrm{BG})=\mathcal{F}^{0}_{\rho_1}(\mathrm{BG}) F ρ 0 − 1 ( BG ) = F ρ 1 − 1 ( BG ) = F ρ 0 0 ( BG ) = F ρ 1 0 ( BG ) x ∼ y ⟺ y ∈ H 1 x H 2 x\sim y\Longleftrightarrow y\in H_1xH_2 x ∼ y ⟺ y ∈ H 1 x H 2 H 1 , H 2 ≤ G H_1,H_2\leq G H 1 , H 2 ≤ G
Proof. (a) is the statement for trivial W i W_i W i K K K ≃ − 1 = ≃ 0 \simeq_{-1}=\simeq_0 ≃ − 1 = ≃ 0 J U 1 , U 2 J_{U_1,U_2} J U 1 , U 2 D ( ≈ ; W 1 , W 2 ) \mathcal{D}(\approx;W_1,W_2) D ( ≈ ; W 1 , W 2 ) N N N ker q \ker q ker q H i = q − 1 ( W i ) ⊇ N H_i=q^{-1}(W_i)\supseteq N H i = q − 1 ( W i ) ⊇ N q − 1 ( H ‾ 1 y ˉ H ‾ 2 ) = H 1 y H 2 q^{-1}(\overline{H}_1\bar{y}\overline{H}_2)=H_1yH_2 q − 1 ( H 1 y ˉ H 2 ) = H 1 y H 2 H i ⊇ N H_i\supseteq N H i ⊇ N N N N ρ 1 \rho_1 ρ 1 ( H 1 , H 2 ) (H_1,H_2) ( H 1 , H 2 ) ρ 0 \rho_0 ρ 0 G G G H 1 y H 2 = y H 1 H 2 H_1yH_2=yH_1H_2 H 1 y H 2 = y H 1 H 2
Separations
Proposition 10.6. (a) F ρ 0 − 1 ⊊ F ρ 1 − 1 \mathcal{F}^{-1}_{\rho_0}\subsetneq\mathcal{F}^{-1}_{\rho_1} F ρ 0 − 1 ⊊ F ρ 1 − 1 ( N , + ) (\mathbb{N},+) ( N , + ) 2 2 2 ρ 1 \rho_1 ρ 1 ρ 0 \rho_0 ρ 0
(b) F ρ 1 − 1 ⊊ F ρ 2 − 1 \mathcal{F}^{-1}_{\rho_1}\subsetneq\mathcal{F}^{-1}_{\rho_2} F ρ 1 − 1 ⊊ F ρ 2 − 1 Z / 3 \mathbb{Z}/3 Z /3 { 0 , 1 } ∣ { 2 } \{0,1\}\mid\{2\} { 0 , 1 } ∣ { 2 } { 1 , 2 } ∣ { 0 } \{1,2\}\mid\{0\} { 1 , 2 } ∣ { 0 } ρ 2 \rho_2 ρ 2 Examples 11.12 , 11.13 ) but not cosets; and conjugacy of S 3 S_3 S 3 ρ 2 \rho_2 ρ 2 Theorem 10.17 ) but not a double coset relation.
(c) F ρ 2 − 1 ⊊ F ρ 6 − 1 \mathcal{F}^{-1}_{\rho_2}\subsetneq\mathcal{F}^{-1}_{\rho_6} F ρ 2 − 1 ⊊ F ρ 6 − 1 B M BM B M M = { 1 , a , 0 } M=\{1,a,0\} M = { 1 , a , 0 } E 0 E_0 E 0 ρ 6 \rho_6 ρ 6 Corollary 10.13 ) but not ρ 2 \rho_2 ρ 2
(d) F ρ 2 0 ⊈ F ρ 2 − 1 \mathcal{F}^{0}_{\rho_2}\not\subseteq\mathcal{F}^{-1}_{\rho_2} F ρ 2 0 ⊆ F ρ 2 − 1 { 1 , a , 0 } \{1,a,0\} { 1 , a , 0 } E 0 ∈ F ρ 2 0 E_0\in\mathcal{F}^{0}_{\rho_2} E 0 ∈ F ρ 2 0 E 0 ∉ F ρ 2 − 1 E_0\notin\mathcal{F}^{-1}_{\rho_2} E 0 ∈ / F ρ 2 − 1
Proof. (a) A ρ 0 \rho_0 ρ 0 ( N , + ) (\mathbb{N},+) ( N , + ) m ~ = m + u 1 + u 2 \widetilde{m}=m+u_1+u_2 m = m + u 1 + u 2 m = m ~ + v 1 + v 2 m=\widetilde{m}+v_1+v_2 m = m + v 1 + v 2 u 1 + u 2 + v 1 + v 2 = 0 u_1+u_2+v_1+v_2=0 u 1 + u 2 + v 1 + v 2 = 0 m ~ = m \widetilde{m}=m m = m { 0 , 1 } ∣ { 2 } \{0,1\}\mid\{2\} { 0 , 1 } ∣ { 2 } { 1 , 2 } ∣ { 0 } \{1,2\}\mid\{0\} { 1 , 2 } ∣ { 0 } ρ 2 \rho_2 ρ 2 Examples 11.12 and 11.13 , and conjugacy of S 3 S_3 S 3 ρ 2 \rho_2 ρ 2 Theorem 10.17 . By Corollary 10.5, ρ 1 \rho_1 ρ 1 Z / 3 \mathbb{Z}/3 Z /3 Z / 3 \mathbb{Z}/3 Z /3 S 3 S_3 S 3 { e } \{e\} { e } e e e H 1 H 2 = { e } H_1H_2=\{e\} H 1 H 2 = { e } E 0 E_0 E 0 ρ 6 \rho_6 ρ 6 Corollary 10.13 (B M BM B M ρ 2 \rho_2 ρ 2 E 0 ∈ F ρ 2 0 E_0\in\mathcal{F}^{0}_{\rho_2} E 0 ∈ F ρ 2 0 E 0 ∉ F ρ 2 − 1 E_0\notin\mathcal{F}^{-1}_{\rho_2} E 0 ∈ / F ρ 2 − 1 F ρ 2 − 1 ⊂ F ρ 2 0 \mathcal{F}^{-1}_{\rho_2}\subset\mathcal{F}^{0}_{\rho_2} F ρ 2 − 1 ⊂ F ρ 2 0 K K K
Abelian groups, and where the four families sit
Proposition 10.7. Let G G G F ρ 0 k ( B G ) = F ρ 1 k ( B G ) \mathcal{F}^{k}_{\rho_0}(\mathrm{BG})=\mathcal{F}^{k}_{\rho_1}(\mathrm{BG}) F ρ 0 k ( BG ) = F ρ 1 k ( BG ) k = − 1 k=-1 k = − 1 k = 0 k=0 k = 0
Proof. For ρ 0 \rho_0 ρ 0 U 1 , U 2 U_1,U_2 U 1 , U 2 P = U 1 + U 2 P=U_1+U_2 P = U 1 + U 2 m ⊴ m ~ m\unlhd\widetilde{m} m ⊴ m m ~ − m ∈ P \widetilde{m}-m\in P m − m ∈ P − 1 -1 − 1 m ~ − m ∈ P ∩ ( − P ) \widetilde{m}-m\in P\cap(-P) m − m ∈ P ∩ ( − P ) P P P H H H 0 0 0 m ~ − m ∈ P \widetilde{m}-m\in P m − m ∈ P m ~ − m ∈ ⟨ P ⟩ \widetilde{m}-m\in\langle P\rangle m − m ∈ ⟨ P ⟩ H H H U 1 = H U_1=H U 1 = H U 2 = 0 U_2=0 U 2 = 0 ρ 1 \rho_1 ρ 1 q − 1 q^{-1} q − 1 G / N G/N G / N
Regularity is relative to the category on which the relation is considered. The same relation can be rigid on one category and only completely functorial on another.
Proposition 10.8. (a) A transport datum with discrete F F F Theorem 5.5 , W = 1 \mathcal{W}=1 W = 1 Gr ( C F ; 1 , U ∪ { id ⋆ } ) \operatorname{Gr}(\mathcal{C}_F;1,\mathcal{U}\cup\{\operatorname{id}_{\star}\}) Gr ( C F ; 1 , U ∪ { id ⋆ }) C F \mathcal{C}_F C F F F F ρ 3 \rho_3 ρ 3 σ = Id \sigma=\operatorname{Id} σ = Id
(b) A limit datum (Definition 7.1) of the closure type has regularity ρ 3 \rho_3 ρ 3 σ = Id \sigma=\operatorname{Id} σ = Id ρ 0 \rho_0 ρ 0
(c) A strict stabilizer preserves each regularity class ρ 0 , … , ρ 3 \rho_0,\ldots,\rho_3 ρ 0 , … , ρ 3 D ∞ \mathcal{D}^{\infty} D ∞ D \mathcal{D} D
(d) Conjugacy of a group G G G G conj G^{\operatorname{conj}} G conj ρ 0 \rho_0 ρ 0 ( B G ) F (\mathbf{B}G)_F ( B G ) F ρ 2 \rho_2 ρ 2 ρ 1 \rho_1 ρ 1 Mor ( B G ) \operatorname{Mor}(\mathbf{B}G) Mor ( B G ) G G G
Proof. (a) By construction σ = τ i = Id \sigma=\tau_i=\operatorname{Id} σ = τ i = Id C F \mathcal{C}_F C F D F \mathcal{D}_F D F F F F W = 1 \mathcal{W}=1 W = 1 Proposition 8.8 . (d) Conjugacy is level − 1 -1 − 1 G G G G conj G^{\operatorname{conj}} G conj ( B G ) F (\mathbf{B}G)_F ( B G ) F F ( ∗ ) = G F(\ast)=G F ( ∗ ) = G ρ 2 \rho_2 ρ 2 Mor ( B G ) \operatorname{Mor}(\mathbf{B}G) Mor ( B G ) Theorem 10.17 . It is not ρ 1 \rho_1 ρ 1 ρ 1 \rho_1 ρ 1 H 1 x H 2 H_1xH_2 H 1 x H 2 e e e H 1 H 2 H_1H_2 H 1 H 2 e e e { e } \{e\} { e } H 1 = H 2 = { e } H_1=H_2=\{e\} H 1 = H 2 = { e } G G G
So the four families of the introduction all sit at σ = Id \sigma=\operatorname{Id} σ = Id non-identity σ \sigma σ Example 11.12 .
A calculus of regularity classes
Proposition 10.9. For ρ ∈ { ρ 0 , ρ 1 , ρ 2 } \rho\in\{\rho_0,\rho_1,\rho_2\} ρ ∈ { ρ 0 , ρ 1 , ρ 2 } k ∈ { − 1 , 0 } k\in\{-1,0\} k ∈ { − 1 , 0 }
(a) Products. E ∈ F ρ k ( C ) E\in\mathcal{F}^{k}_{\rho}(\mathcal{C}) E ∈ F ρ k ( C ) E ′ ∈ F ρ k ( C ′ ) E'\in\mathcal{F}^{k}_{\rho}(\mathcal{C}') E ′ ∈ F ρ k ( C ′ ) E × E ′ ∈ F ρ k ( C × C ′ ) E\times E'\in\mathcal{F}^{k}_{\rho}(\mathcal{C}\times\mathcal{C}') E × E ′ ∈ F ρ k ( C × C ′ )
(b) Opposite. E ∈ F ρ k ( C ) E\in\mathcal{F}^{k}_{\rho}(\mathcal{C}) E ∈ F ρ k ( C ) E ∈ F ρ k ( C op ) E\in\mathcal{F}^{k}_{\rho}(\mathcal{C}^{\operatorname{op}}) E ∈ F ρ k ( C op )
(c) Full pullbacks. For ρ ∈ { ρ 1 , ρ 2 } \rho\in\{\rho_1,\rho_2\} ρ ∈ { ρ 1 , ρ 2 } Q : C → C ′ Q:\mathcal{C}\to\mathcal{C}' Q : C → C ′ E ∈ F ρ k ( C ′ ) E\in\mathcal{F}^{k}_{\rho}(\mathcal{C}') E ∈ F ρ k ( C ′ ) Q − 1 E ∈ F ρ k ( C ) Q^{-1}E\in\mathcal{F}^{k}_{\rho}(\mathcal{C}) Q − 1 E ∈ F ρ k ( C ) ρ 0 \rho_0 ρ 0 Q − 1 E ∈ F ρ 0 k ( C ) Q^{-1}E\in\mathcal{F}^{k}_{\rho_0}(\mathcal{C}) Q − 1 E ∈ F ρ 0 k ( C )
(d) Disjoint unions. If C = C 1 ⊔ C 2 \mathcal{C}=\mathcal{C}_1\sqcup\mathcal{C}_2 C = C 1 ⊔ C 2 E = E 1 ⊔ E 2 E=E_1\sqcup E_2 E = E 1 ⊔ E 2 E i ∈ F ρ k ( C i ) E_i\in\mathcal{F}^{k}_{\rho}(\mathcal{C}_i) E i ∈ F ρ k ( C i ) E ∈ F ρ k ( C ) E\in\mathcal{F}^{k}_{\rho}(\mathcal{C}) E ∈ F ρ k ( C )
Proof. (a) Product datum, with U i × U i ′ \mathcal{U}_i\times\mathcal{U}'_i U i × U i ′ K ( D × D ′ ) = K ( D ) × K ( D ′ ) \mathcal{K}(\mathcal{D}\times\mathcal{D}')=\mathcal{K}(\mathcal{D})\times\mathcal{K}(\mathcal{D}') K ( D × D ′ ) = K ( D ) × K ( D ′ ) U 1 ↔ U 2 \mathcal{U}_1\leftrightarrow\mathcal{U}_2 U 1 ↔ U 2 τ 1 ↔ τ 2 \tau_1\leftrightarrow\tau_2 τ 1 ↔ τ 2 ( σ Q , τ i Q ) (\sigma Q,\tau_iQ) ( σ Q , τ i Q ) U i \mathcal{U}_i U i Q − 1 U i Q^{-1}\mathcal{U}_i Q − 1 U i Q Q Q C ′ \mathcal{C}' C ′ ρ 0 \rho_0 ρ 0 σ = Q \sigma=Q σ = Q
Level 1 of completely functorial data
For ρ 2 \rho_2 ρ 2
Proposition 10.10. Let G G G D \mathcal{D} D ρ 2 \rho_2 ρ 2 B G \mathbf{B}G B G D = B Γ \mathcal{D}=\mathbf{B}\Gamma D = B Γ Γ \Gamma Γ U 1 = U 2 = G \mathcal{U}_1=\mathcal{U}_2=G U 1 = U 2 = G m m m σ ( m ) = 1 \sigma(m)=1 σ ( m ) = 1 π 1 ( ∣ N K ∣ , m ) ≅ { ( g 1 , g 2 ) ∈ G × G : τ 1 ( g 1 ) = τ 2 ( g 2 ) − 1 } \pi_1(|\mathcal{N}K|,m)\cong\{(g_1,g_2)\in G\times G:\tau_1(g_1)=\tau_2(g_2)^{-1}\} π 1 ( ∣ N K ∣ , m ) ≅ {( g 1 , g 2 ) ∈ G × G : τ 1 ( g 1 ) = τ 2 ( g 2 ) − 1 } ( g 1 ′ , g 2 ′ ) ( g 1 , g 2 ) = ( g 1 ′ g 1 , g 2 g 2 ′ ) (g'_1,g'_2)(g_1,g_2)=(g'_1g_1,g_2g'_2) ( g 1 ′ , g 2 ′ ) ( g 1 , g 2 ) = ( g 1 ′ g 1 , g 2 g 2 ′ ) τ 1 \tau_1 τ 1 τ 2 − 1 \tau_2^{-1} τ 2 − 1 Γ \Gamma Γ { ( g , g − 1 ) } ≅ G \{(g,g^{-1})\}\cong G {( g , g − 1 )} ≅ G m = 1 m=1 m = 1 Gr ( B G ) \operatorname{Gr}(\mathbf{B}G) Gr ( B G ) { ( g , g − 1 ) } ≅ G \{(g,g^{-1})\}\cong G {( g , g − 1 )} ≅ G
Proof. Since G G G W = 1 \mathcal{W}=1 W = 1 K \mathcal{K} K π 1 ( ∣ N K ∣ , m ) ≅ Aut K ( m ) \pi_1(|\mathcal{N}K|,m)\cong\operatorname{Aut}_{\mathcal{K}}(m) π 1 ( ∣ N K ∣ , m ) ≅ Aut K ( m ) m m m ( u 1 , u 2 ) (u_1,u_2) ( u 1 , u 2 ) τ 1 ( u 1 ) σ ( m ) τ 2 ( u 2 ) = σ ( m ) \tau_1(u_1)\sigma(m)\tau_2(u_2)=\sigma(m) τ 1 ( u 1 ) σ ( m ) τ 2 ( u 2 ) = σ ( m ) ( u 1 ′ , u 2 ′ ) ( u 1 , u 2 ) = ( u 1 ′ u 1 , u 2 u 2 ′ ) (u'_1,u'_2)(u_1,u_2)=(u'_1u_1,u_2u'_2) ( u 1 ′ , u 2 ′ ) ( u 1 , u 2 ) = ( u 1 ′ u 1 , u 2 u 2 ′ ) σ ( m ) = 1 \sigma(m)=1 σ ( m ) = 1 τ 1 ( u 1 ) τ 2 ( u 2 ) = 1 \tau_1(u_1)\tau_2(u_2)=1 τ 1 ( u 1 ) τ 2 ( u 2 ) = 1 Theorem 10.17 with φ = id \varphi=\operatorname{id} φ = id σ ( 1 ) = ( 1 , 1 ) \sigma(1)=(1,1) σ ( 1 ) = ( 1 , 1 ) τ 1 ( u ) = τ 2 ( u ) = ( u , u ) \tau_1(u)=\tau_2(u)=(u,u) τ 1 ( u ) = τ 2 ( u ) = ( u , u ) u 1 u 2 = 1 u_1u_2=1 u 1 u 2 = 1
Arbitrary (6): a normalization
Definition 10.11. A category C C C strongly connected if for any two objects A , B A,B A , B A → B A\to B A → B B → A B\to A B → A Vect F \operatorname{Vect}_{\mathrm{F}} Vect F B M BM B M
Proposition 10.12. Let E E E equivalence relation on Mor ( C ) \operatorname{Mor}(C) Mor ( C ) R c h R_{\mathrm{ch}} R ch ( m , m ~ ) (m,\widetilde{m}) ( m , m ) m : A → B m:A\to B m : A → B m ~ : A ~ → B ~ \widetilde{m}:\widetilde{A}\to\widetilde{B} m : A → B B → B ~ B\to\widetilde{B} B → B B ~ → B \widetilde{B}\to B B → B A ~ → A \widetilde{A}\to A A → A A → A ~ A\to\widetilde{A} A → A ( U 1 , U 2 , W ) = ( C , C , 1 ) (\mathcal{U}_1,\mathcal{U}_2,\mathcal{W})=(C,C,1) ( U 1 , U 2 , W ) = ( C , C , 1 ) ≃ − 1 = E ∩ R c h \simeq_{-1}=E\cap R_{\mathrm{ch}} ≃ − 1 = E ∩ R ch
Proof. Write [ m ] [m] [ m ] E E E M = { 1 , 0 } ⊔ { [ m ] } M=\{1,0\}\sqcup\{[m]\} M = { 1 , 0 } ⊔ {[ m ]} x y = 0 xy=0 x y = 0 x , y ≠ 1 x,y\ne1 x , y = 1 1 1 1 1 1 1 0 0 0 1 1 1 D D D C C C 1 1 1 M M M 2 2 2 τ 1 = τ 2 \tau_1=\tau_2 τ 1 = τ 2 1 1 1 σ ( m ) = [ m ] \sigma(m)=[m] σ ( m ) = [ m ] Θ ( m ) u 1 , u 2 = [ m ] \Theta^{u_1,u_2}_{(m)}=[m] Θ ( m ) u 1 , u 2 = [ m ] m → m ~ m\to\widetilde{m} m → m B → B ~ B\to\widetilde{B} B → B A ~ → A \widetilde{A}\to A A → A [ m ] = [ m ~ ] [m]=[\widetilde{m}] [ m ] = [ m ]
Corollary 10.13. If C C C Mor ( C ) \operatorname{Mor}(C) Mor ( C ) − 1 -1 − 1 ( U 1 , U 2 , W ) = ( C , C , 1 ) (\mathcal{U}_1,\mathcal{U}_2,\mathcal{W})=(C,C,1) ( U 1 , U 2 , W ) = ( C , C , 1 )
This is a normalization of the framework and not a classification: if σ \sigma σ σ \sigma σ
Completely functorial data (2)
We write F k s t r = F ρ 2 k \mathcal{F}^{\mathrm{str}}_k=\mathcal{F}^{k}_{\rho_2} F k str = F ρ 2 k
Theorem 10.14. Let C C C
(a) J J J R R R L L L H H H Proposition 6.8 ), the transport relations of Theorem 5.5 , and every congruence E E E C C C F − 1 s t r \mathcal{F}^{\mathrm{str}}_{-1} F − 1 str Y Y Y − 1 -1 − 1
(b) Let D \mathcal{D} D m ≃ − 1 m ~ m\simeq_{-1}\widetilde{m} m ≃ − 1 m s ( m ) s(m) s ( m ) s ( m ~ ) s(\widetilde{m}) s ( m ) ∫ V H \int_{\mathcal{V}}H ∫ V H − 1 -1 − 1 s s s σ \sigma σ s s s C C C σ \sigma σ s ( a m b ) = ( A ′ , B ′ , σ ( a ) σ ( m ) σ ( b ) ) s(amb)=(A',B',\sigma(a)\sigma(m)\sigma(b)) s ( amb ) = ( A ′ , B ′ , σ ( a ) σ ( m ) σ ( b )) b : A ′ → A b:A'\to A b : A ′ → A m : A → B m:A\to B m : A → B a : B → B ′ a:B\to B' a : B → B ′ σ \sigma σ
(c) Let Q : C → C ′ Q:C\to C' Q : C → C ′ D Q = ( C , C ′ , Q , Q , Q ; C , C , 1 ) \mathcal{D}_Q=(C,C',Q,Q,Q;C,C,1) D Q = ( C , C ′ , Q , Q , Q ; C , C , 1 ) − 1 -1 − 1 D Q \mathcal{D}_Q D Q Q − 1 ( J C ′ ) Q^{-1}(J_{C'}) Q − 1 ( J C ′ ) J C ′ J_{C'} J C ′ C ′ C' C ′
Proof. (a) Green and transport data have σ = Id \sigma=\operatorname{Id} σ = Id GrH ( M ) \operatorname{GrH}(M) GrH ( M ) Definition 6.7 ). For a congruence E E E Q : C → C / E Q:C\to C/E Q : C → C / E σ = τ i = Q \sigma=\tau_i=Q σ = τ i = Q U 1 = U 2 = 1 \mathcal{U}_1=\mathcal{U}_2=1 U 1 = U 2 = 1 W = 1 \mathcal{W}=1 W = 1 m → m ~ m\to\widetilde{m} m → m u i = id u_i=\operatorname{id} u i = id Q m = Q m ~ Qm=Q\widetilde{m} Q m = Q m Y Y Y E E E G G G Y Y Y Corollary 5.11 (b) applies. (b) Theorem 4.2 for the first assertion; the equivariance is the functoriality of σ \sigma σ Q Q Q C ′ C' C ′ Q ( u ) Q(u) Q ( u ) m ⪯ m ~ m\preceq\widetilde{m} m ⪯ m Q m ~ ∈ C ′ Q ( m ) C ′ Q\widetilde{m}\in C'Q(m)C' Q m ∈ C ′ Q ( m ) C ′
An obstruction at level − 1 -1 − 1
Call a monoid M M M nil monoid if it is nontrivial, has a zero element 0 0 0 1 1 1 { 1 , a , 0 } \{1,a,0\} { 1 , a , 0 } a 2 = 0 a^2=0 a 2 = 0 { 1 } ∪ I \{1\}\cup I { 1 } ∪ I I I I 1 1 1 0 = 1 0=1 0 = 1
Theorem 10.15. Let M M M E ∈ F − 1 s t r ( B M ) E\in\mathcal{F}^{\mathrm{str}}_{-1}(\mathbf{BM}) E ∈ F − 1 str ( BM ) W = 1 \mathcal{W}=1 W = 1 1 1 1 { 1 } \{1\} { 1 } E E E E 0 E_0 E 0 { 1 , a } , { 0 } \{1,a\},\{0\} { 1 , a } , { 0 } { 1 , a , 0 } \{1,a,0\} { 1 , a , 0 } F − 1 s t r ( B M ) \mathcal{F}^{\mathrm{str}}_{-1}(\mathbf{BM}) F − 1 str ( BM )
Proof. Let D \mathcal{D} D 1 ≃ − 1 a 1\simeq_{-1}a 1 ≃ − 1 a a ≠ 1 a\ne1 a = 1 σ ( M ) , τ i ( M ) \sigma(M),\tau_i(M) σ ( M ) , τ i ( M ) N = End D ( σ ( ∗ ) ) N=\operatorname{End}_{\mathcal{D}}(\sigma(*)) N = End D ( σ ( ∗ )) σ , τ i : M → N \sigma,\tau_i:M\to N σ , τ i : M → N 1 → a 1\to a 1 → a u 1 , u 2 u_1,u_2 u 1 , u 2 τ 1 ( u 1 ) τ 2 ( u 2 ) = σ ( a ) \tau_1(u_1)\tau_2(u_2)=\sigma(a) τ 1 ( u 1 ) τ 2 ( u 2 ) = σ ( a ) a → 1 a\to1 a → 1 v 1 , v 2 v_1,v_2 v 1 , v 2 τ 1 ( v 1 ) σ ( a ) τ 2 ( v 2 ) = 1 \tau_1(v_1)\sigma(a)\tau_2(v_2)=1 τ 1 ( v 1 ) σ ( a ) τ 2 ( v 2 ) = 1 τ 1 ( p ) τ 2 ( q ) = 1 \tau_1(p)\tau_2(q)=1 τ 1 ( p ) τ 2 ( q ) = 1 p = v 1 u 1 p=v_1u_1 p = v 1 u 1 q = u 2 v 2 q=u_2v_2 q = u 2 v 2
( A ) (A) ( A ) p ≠ 1 p\ne1 p = 1 τ 1 ≡ 1 \tau_1\equiv1 τ 1 ≡ 1 p n = 0 p^n=0 p n = 0 τ 1 ( p ) \tau_1(p) τ 1 ( p ) τ 2 ( q ) \tau_2(q) τ 2 ( q ) τ 1 ( p ) n = τ 1 ( 0 ) = : e \tau_1(p)^n=\tau_1(0)=:e τ 1 ( p ) n = τ 1 ( 0 ) =: e e e e e y = 1 ey=1 ey = 1 e = e e y = e y = 1 e=eey=ey=1 e = eey = ey = 1 x x x τ 1 ( x ) = τ 1 ( x ) τ 1 ( 0 ) = τ 1 ( x 0 ) = τ 1 ( 0 ) = 1 \tau_1(x)=\tau_1(x)\tau_1(0)=\tau_1(x0)=\tau_1(0)=1 τ 1 ( x ) = τ 1 ( x ) τ 1 ( 0 ) = τ 1 ( x 0 ) = τ 1 ( 0 ) = 1
( B ) (B) ( B ) q ≠ 1 q\ne1 q = 1 τ 2 ≡ 1 \tau_2\equiv1 τ 2 ≡ 1 τ 2 ( q ) \tau_2(q) τ 2 ( q ) f = τ 2 ( q ) n = τ 2 ( 0 ) f=\tau_2(q)^n=\tau_2(0) f = τ 2 ( q ) n = τ 2 ( 0 ) x f = 1 xf=1 x f = 1 f = x f f = x f = 1 f=xff=xf=1 f = x f f = x f = 1 τ 2 ( x ) = τ 2 ( x 0 ) = 1 \tau_2(x)=\tau_2(x0)=1 τ 2 ( x ) = τ 2 ( x 0 ) = 1 x x x
( C ) (C) ( C ) p = 1 p=1 p = 1 u 1 = v 1 = 1 u_1=v_1=1 u 1 = v 1 = 1 q = 1 q=1 q = 1 u 2 = v 2 = 1 u_2=v_2=1 u 2 = v 2 = 1 v 1 u 1 = 1 v_1u_1=1 v 1 u 1 = 1 e = u 1 v 1 e=u_1v_1 e = u 1 v 1 e = 1 e=1 e = 1 u 1 u_1 u 1 u 1 = 1 = v 1 u_1=1=v_1 u 1 = 1 = v 1 e e e e = e n = 0 e=e^n=0 e = e n = 0 u 1 = u 1 v 1 u 1 = 0 u_1=u_1v_1u_1=0 u 1 = u 1 v 1 u 1 = 0 1 = v 1 u 1 = 0 1=v_1u_1=0 1 = v 1 u 1 = 0
By (A) and (C), τ 1 ( u 1 ) = 1 \tau_1(u_1)=1 τ 1 ( u 1 ) = 1 τ 2 ( u 2 ) = 1 \tau_2(u_2)=1 τ 2 ( u 2 ) = 1 σ ( a ) = τ 1 ( u 1 ) τ 2 ( u 2 ) = 1 \sigma(a)=\tau_1(u_1)\tau_2(u_2)=1 σ ( a ) = τ 1 ( u 1 ) τ 2 ( u 2 ) = 1 a n = 0 a^n=0 a n = 0 n n n σ ( 0 ) = 1 \sigma(0)=1 σ ( 0 ) = 1 x x x σ ( x ) = σ ( x ) σ ( 0 ) = σ ( x 0 ) = σ ( 0 ) = 1 \sigma(x)=\sigma(x)\sigma(0)=\sigma(x0)=\sigma(0)=1 σ ( x ) = σ ( x ) σ ( 0 ) = σ ( x 0 ) = σ ( 0 ) = 1 σ ≡ 1 \sigma\equiv1 σ ≡ 1 m , m ~ m,\widetilde m m , m ( id , id ) (\operatorname{id},\operatorname{id}) ( id , id ) m → m ~ m\to\widetilde m m → m E E E
An extension to invertible 2-morphisms is not asserted here: replacing equalities by invertible cells requires a separate argument, since isomorphic 1-morphisms need not be equal as elements of the endomorphism monoid N N N 0 0 0 E 0 E_0 E 0 a a a 1 1 1 a 2 a^2 a 2 1 2 1^2 1 2 − 1 -1 − 1
Level 0 is more flexible than level − 1 -1 − 1
Level 0 only asks for a zigzag, and a single arrow 1 → a 1\to a 1 → a 1 1 1 a a a
Proposition 10.16. For M = { 1 , a , 0 } M=\{1,a,0\} M = { 1 , a , 0 } a 2 = 0 a^2=0 a 2 = 0 E 0 E_0 E 0 { 1 , a } , { 0 } \{1,a\},\{0\} { 1 , a } , { 0 } E 0 ∈ F 0 s t r ( B M ) E_0\in\mathcal{F}^{\mathrm{str}}_0(\mathbf{BM}) E 0 ∈ F 0 str ( BM ) λ ( E 0 ) = 0 \lambda(E_0)=0 λ ( E 0 ) = 0 F 0 s t r ( B M ) ⊈ F − 1 s t r ( B M ) \mathcal{F}^{\mathrm{str}}_0(\mathbf{BM})\not\subseteq\mathcal{F}^{\mathrm{str}}_{-1}(\mathbf{BM}) F 0 str ( BM ) ⊆ F − 1 str ( BM ) M M M
Proof. Let N N N X , Y X,Y X , Y X X X = X X XXX=XX X X X = X X Y Y Y = Y Y YYY=YY Y Y Y = Y Y X Y X Y X Y = X Y X Y XYXYXY=XYXY X Y X Y X Y = X Y X Y X X X Y X Y X Y XXX YXYXY X X X Y X Y X Y X Y X Y X Y Y Y XYXYXYYY X Y X Y X Y Y Y X Y X Y X XYXYX X Y X Y X X Y X Y Y XYXYY X Y X Y Y X Y X Y X Y X Y X Y XYXYXYXYXY X Y X Y X Y X Y X Y X Y X Y XYXY X Y X Y N N N M → N M\to N M → N σ ( 1 ) = τ i ( 1 ) = 1 \sigma(1)=\tau_i(1)=1 σ ( 1 ) = τ i ( 1 ) = 1 σ ( a ) = X Y \sigma(a)=XY σ ( a ) = X Y σ ( 0 ) = X Y X Y \sigma(0)=XYX Y σ ( 0 ) = X Y X Y τ 1 ( a ) = X \tau_1(a)=X τ 1 ( a ) = X τ 1 ( 0 ) = X X \tau_1(0)=XX τ 1 ( 0 ) = X X τ 2 ( a ) = Y \tau_2(a)=Y τ 2 ( a ) = Y τ 2 ( 0 ) = Y Y \tau_2(0)=YY τ 2 ( 0 ) = Y Y a 2 = 0 a^2=0 a 2 = 0 a 0 = 0 a = 0 ⋅ 0 = 0 a0=0a=0\cdot0=0 a 0 = 0 a = 0 ⋅ 0 = 0 σ \sigma σ ( X Y ) 3 (XY)^3 ( X Y ) 3 ( X Y ) 4 (XY)^4 ( X Y ) 4 ( X Y ) 2 (XY)^2 ( X Y ) 2 τ 1 \tau_1 τ 1 X 3 = X 4 = X 2 X^3=X^4=X^2 X 3 = X 4 = X 2 τ 2 \tau_2 τ 2 D = ( B M , B M , σ , τ 1 , τ 2 ; B M , B M , 1 ) , \mathcal{D}=(\mathbf{BM},\mathbf{BM},\sigma,\tau_1,\tau_2;\mathbf{BM},\mathbf{BM},1), D = ( BM , BM , σ , τ 1 , τ 2 ; BM , BM , 1 ) , m m m ( u 1 , u 2 ) (u_1,u_2) ( u 1 , u 2 ) τ 1 ( u 1 ) σ ( m ) τ 2 ( u 2 ) ∈ σ ( M ) = { 1 , X Y , X Y X Y } \tau_1(u_1)\sigma(m)\tau_2(u_2) \in\sigma(M)=\{1,XY,XYXY\} τ 1 ( u 1 ) σ ( m ) τ 2 ( u 2 ) ∈ σ ( M ) = { 1 , X Y , X Y X Y } m = 1 m=1 m = 1 X i Y j X^iY^j X i Y j 0 ≤ i , j ≤ 2 0\le i,j\le2 0 ≤ i , j ≤ 2 1 1 1 X Y XY X Y ( u 1 , u 2 ) = ( a , a ) (u_1,u_2)=(a,a) ( u 1 , u 2 ) = ( a , a ) σ ( M ) \sigma(M) σ ( M ) X Y X Y XYXY X Y X Y m = a m=a m = a X i + 1 Y j + 1 X^{i+1}Y^{j+1} X i + 1 Y j + 1 X Y XY X Y σ ( M ) \sigma(M) σ ( M ) m = 0 m=0 m = 0 X i + 1 Y X Y j + 1 X^{i+1}YXY^{j+1} X i + 1 Y X Y j + 1 X Y X Y XYXY X Y X Y K ( D ) \mathrm{K}(\mathcal{D}) K ( D ) 1 → a 1\to a 1 → a { 1 , a } \{1,a\} { 1 , a } { 0 } \{0\} { 0 } ≃ 0 = E 0 \simeq_0=E_0 ≃ 0 = E 0
Commutativity of the image would give τ 1 ( 0 ) τ 2 ( 0 ) = σ ( 0 ) \tau_1(0)\tau_2(0)=\sigma(0) τ 1 ( 0 ) τ 2 ( 0 ) = σ ( 0 ) 1 → 0 1\to0 1 → 0 σ ( 0 ) = ( X Y ) 2 ≠ X 2 Y 2 \sigma(0)=(XY)^2\ne X^2Y^2 σ ( 0 ) = ( X Y ) 2 = X 2 Y 2 N N N
Twisted conjugacy: constrained Green data
Level − 1 -1 − 1 J J J D \mathcal{D} D
Theorem 10.17. Let M , Γ M,\Gamma M , Γ β 1 , β 2 : M → Γ \beta_1,\beta_2:M\to\Gamma β 1 , β 2 : M → Γ D = ( B M , B ( M × Γ ) , σ , τ 1 , τ 2 ; B M , B M , 1 ) \mathcal{D}=(BM,B(M\times\Gamma),\sigma,\tau_1,\tau_2;BM,BM,1) D = ( B M , B ( M × Γ ) , σ , τ 1 , τ 2 ; B M , B M , 1 ) σ ( m ) = ( m , 1 ) \sigma(m)=(m,1) σ ( m ) = ( m , 1 ) τ i ( u ) = ( u , β i ( u ) ) \tau_i(u)=(u,\beta_i(u)) τ i ( u ) = ( u , β i ( u )) D \mathcal{D} D m → m ~ m\to\widetilde m m → m ( u 1 , u 2 ) (u_1,u_2) ( u 1 , u 2 ) m ~ = u 1 m u 2 \widetilde m=u_1mu_2 m = u 1 m u 2 β 1 ( u 1 ) β 2 ( u 2 ) = 1 \beta_1(u_1)\beta_2(u_2)=1 β 1 ( u 1 ) β 2 ( u 2 ) = 1 M = G M=G M = G φ \varphi φ Γ = G \Gamma=G Γ = G β 1 = id \beta_1=\operatorname{id} β 1 = id β 2 = φ \beta_2=\varphi β 2 = φ ≃ − 1 \simeq_{-1} ≃ − 1 φ \varphi φ m ~ = φ ( g ) − 1 m g \widetilde m=\varphi(g)^{-1}mg m = φ ( g ) − 1 m g φ = id \varphi=\operatorname{id} φ = id
Proof. τ 1 ( u 1 ) σ ( m ) τ 2 ( u 2 ) = ( u 1 m u 2 , β 1 ( u 1 ) β 2 ( u 2 ) ) \tau_1(u_1)\sigma(m)\tau_2(u_2)=(u_1mu_2,\beta_1(u_1)\beta_2(u_2)) τ 1 ( u 1 ) σ ( m ) τ 2 ( u 2 ) = ( u 1 m u 2 , β 1 ( u 1 ) β 2 ( u 2 )) σ ( m ~ ) = ( m ~ , 1 ) \sigma(\widetilde m)=(\widetilde m,1) σ ( m ) = ( m , 1 ) β 1 ( u 1 ) β 2 ( u 2 ) = u 1 φ ( u 2 ) = 1 \beta_1(u_1)\beta_2(u_2)=u_1\varphi(u_2)=1 β 1 ( u 1 ) β 2 ( u 2 ) = u 1 φ ( u 2 ) = 1 u 1 = φ ( u 2 ) − 1 u_1=\varphi(u_2)^{-1} u 1 = φ ( u 2 ) − 1 m ~ = φ ( g ) − 1 m g \widetilde m=\varphi(g)^{-1}mg m = φ ( g ) − 1 m g g = u 2 g=u_2 g = u 2 ( φ ( g ) , g − 1 ) (\varphi(g),g^{-1}) ( φ ( g ) , g − 1 ) − 1 -1 − 1
For a nonabelian group, conjugacy is not a congruence (t ∼ t ′ t\sim t' t ∼ t ′ S 3 S_3 S 3 t t = e ≁ t ′ t ′ tt=e\not\sim t't' tt = e ∼ t ′ t ′ Q − 1 ( J C ′ ) Q^{-1}(J_{\mathcal C'}) Q − 1 ( J C ′ ) J J J Theorem 10.14 does not contain it, and the class F − 1 s t r \mathcal F^{\mathrm{str}}_{-1} F − 1 str
Closure properties and finite groups
Proposition 10.18. (a) If E ∈ F − 1 s t r ( C ′ ) E\in\mathcal F^{\mathrm{str}}_{-1}(\mathcal C') E ∈ F − 1 str ( C ′ ) Q : C → C ′ Q:\mathcal C\to\mathcal C' Q : C → C ′ Q − 1 ( E ) ∈ F − 1 s t r ( C ) Q^{-1}(E)\in\mathcal F^{\mathrm{str}}_{-1}(\mathcal C) Q − 1 ( E ) ∈ F − 1 str ( C ) E ∈ F − 1 s t r ( C ) E\in\mathcal F^{\mathrm{str}}_{-1}(\mathcal C) E ∈ F − 1 str ( C ) E ′ ∈ F − 1 s t r ( C ′ ) E'\in\mathcal F^{\mathrm{str}}_{-1}(\mathcal C') E ′ ∈ F − 1 str ( C ′ ) E × E ′ ∈ F − 1 s t r ( C × C ′ ) E\times E'\in\mathcal F^{\mathrm{str}}_{-1}(\mathcal C\times\mathcal C') E × E ′ ∈ F − 1 str ( C × C ′ ) K ( D ) \mathrm{K}(\mathcal D) K ( D ) ≃ − 1 = ≃ 0 \simeq_{-1}=\simeq_0 ≃ − 1 = ≃ 0 F − 1 s t r ( B G ) = F 0 s t r ( B G ) \mathcal F^{\mathrm{str}}_{-1}(BG)=\mathcal F^{\mathrm{str}}_0(BG) F − 1 str ( B G ) = F 0 str ( B G ) G G G
Proof. (a) Compose a datum for E E E Q Q Q σ Q , τ i Q \sigma Q,\tau_iQ σ Q , τ i Q U i \mathcal U_i U i Q − 1 U i Q^{-1}\mathcal U_i Q − 1 U i Q Q Q u u u C ′ \mathcal C' C ′ U i × U i ′ \mathcal U_i\times\mathcal U'_i U i × U i ′ G G G W = 1 \mathcal W=1 W = 1 U i ⊆ G \mathcal U_i\subseteq G U i ⊆ G ( u 1 , u 2 ) (u_1,u_2) ( u 1 , u 2 ) ( u 1 − 1 , u 2 − 1 ) (u_1^{-1},u_2^{-1}) ( u 1 − 1 , u 2 − 1 ) τ 1 ( u 1 − 1 ) σ ( m ~ ) τ 2 ( u 2 − 1 ) = σ ( m ) \tau_1(u_1^{-1})\sigma(\widetilde m)\tau_2(u_2^{-1})=\sigma(m) τ 1 ( u 1 − 1 ) σ ( m ) τ 2 ( u 2 − 1 ) = σ ( m )
By (c), on a finite group the two questions of whether a relation is level 0 or level − 1 -1 − 1 U 1 × U 2 \mathcal U_1\times\mathcal U_2 U 1 × U 2 Γ \Gamma Γ τ 1 , τ 2 \tau_1,\tau_2 τ 1 , τ 2
An exhaustive computer search over homomorphisms Z / 4 → S 5 \mathbb Z/4\to S_5 Z /4 → S 5 U i ≤ Z / 4 \mathcal U_i\le\mathbb Z/4 U i ≤ Z /4 Z / 4 \mathbb Z/4 Z /4 − 1 -1 − 1 F − 1 s t r \mathcal F^{\mathrm{str}}_{-1} F − 1 str
Levels k ≥ 1 k \ge1 k ≥ 1
By Corollary 5.6, every connected simplicial set T T T K ( D ) \mathrm{K}(\mathcal{D}) K ( D ) k ≥ 1 k \ge1 k ≥ 1 − 1 -1 − 1 0 0 0 λ ( E ) \lambda(E) λ ( E ) E E E k ≥ 1 k \ge1 k ≥ 1 0 0 0 E 0 E_0 E 0 − 1 -1 − 1 J J J
Examples and counterexamples
The examples below illustrate the notions above. Examples 11.12 and 11.13 exhibit the relations of Proposition 10.6(b) and are used in its proof; the others are only illustrations, although some are cited in statements (Proposition 3.11 , Remark 3.7, Corollary 7.6).
Example 11.1. The levels are defined for an arbitrary category K \mathrm{K} K Theorem 5.5 every small category is, up to opposite, a full subcategory of some K ( D ) \mathrm{K}(\mathcal{D}) K ( D ) K \mathrm{K} K x , y , z x,y,z x , y , z x → y x \to y x → y z → y z \to y z → y x x x z z z x ̸ ≃ − 1 z x \not\simeq_{-1} z x ≃ − 1 z x ≃ 0 z x \simeq_0 z x ≃ 0 z y y y K \mathrm{K} K 0 , 1 0,1 0 , 1 a , b : 0 → 1 a,b:0 \to1 a , b : 0 → 1 0 ̸ ≃ − 1 1 0 \not\simeq_{-1} 1 0 ≃ − 1 1 1 → 0 1 \to0 1 → 0 0 ≃ 0 1 0 \simeq_0 1 0 ≃ 0 1 π 0 \pi_0 π 0 ∣ N K ∣ |N\mathrm{K}| ∣ N K ∣ Z \mathbb{Z} Z Aut K ( 0 ) \operatorname{Aut}_{\mathrm{K}}(0) Aut K ( 0 )
Example 11.2. Let C = p t \mathcal{C}=\mathrm{pt} C = pt D \mathcal{D} D Z \mathbb{Z} Z σ = τ i = Id \sigma=\tau_i=\operatorname{Id} σ = τ i = Id W W W K ( D ) = B Z \mathrm{K}(\mathcal{D})=B\mathbb{Z} K ( D ) = B Z − 1 -1 − 1 0 0 0 Z \mathbb{Z} Z D \mathcal{D} D
Example 11.3. Let ( Y ′ , ≾ ) (Y',\precsim) ( Y ′ , ≾ ) f : Y → Y ′ f:Y\to Y' f : Y → Y ′ C = p t \mathcal{C}=\mathrm{pt} C = pt F ( ∗ ) = Y F(*)=Y F ( ∗ ) = Y s ≾ s ~ ⟺ f ( s ) ≾ f ( s ~ ) s\precsim\tilde{s}\Longleftrightarrow f(s)\precsim f(\tilde{s}) s ≾ s ~ ⟺ f ( s ) ≾ f ( s ~ ) U = C \mathcal{U}=\mathcal{C} U = C W = { ≃ } W=\{\simeq\} W = { ≃ } s ≃ − 1 s ~ ⟺ f ( s ) ≾ f ( s ~ ) ≾ f ( s ) s\simeq_{-1}\tilde{s}\Longleftrightarrow f(s)\precsim f(\tilde{s})\precsim f(s) s ≃ − 1 s ~ ⟺ f ( s ) ≾ f ( s ~ ) ≾ f ( s ) ≃ f \simeq_f ≃ f ≾ \precsim ≾ f = det f=\det f = det ≾ \precsim ≾ K \mathcal{K} K K \mathcal{K} K
Indeed, since C = p t \mathcal{C}=\mathrm{pt} C = pt s → s ~ s\to\tilde{s} s → s ~ J U F J_UF J U F ( id , φ ) (\operatorname{id},\varphi) ( id , φ ) φ : s → s ~ \varphi:s\to\tilde{s} φ : s → s ~ F ( ∗ ) = ( Y , ≾ ) F(*)=(Y,\precsim) F ( ∗ ) = ( Y , ≾ ) W W W s ≾ s ~ ≾ s s\precsim\tilde{s}\precsim s s ≾ s ~ ≾ s f ( s ) ≾ f ( s ~ ) ≾ f ( s ) f(s)\precsim f(\tilde{s})\precsim f(s) f ( s ) ≾ f ( s ~ ) ≾ f ( s ) s ≃ − 1 s ~ s\simeq_{-1}\tilde{s} s ≃ − 1 s ~ f ( s ) ≾ f ( s ~ ) f(s)\precsim f(\tilde{s}) f ( s ) ≾ f ( s ~ ) f ( s ~ ) ≾ f ( s ) f(\tilde{s})\precsim f(s) f ( s ~ ) ≾ f ( s ) ≾ \precsim ≾ f ( s ) f(s) f ( s ) f ( s ~ ) f(\tilde{s}) f ( s ~ ) ≃ f \simeq_f ≃ f f f f
Example 11.4. Let C \mathcal{C} C F \mathbb{F} F Gr × ( C ) \operatorname{Gr}^{\times}(\mathcal{C}) Gr × ( C ) m : A → B m:A\to B m : A → B r r r dim A = n \dim A=n dim A = n dim B = p \dim B=p dim B = p ≃ − 1 = ≃ 0 \simeq_{-1}=\simeq_0 ≃ − 1 = ≃ 0 A → B A\to B A → B ( u 1 , u 2 ) ↦ ( u 1 , u 2 − 1 ) (u_1,u_2)\mapsto(u_1,u_2^{-1}) ( u 1 , u 2 ) ↦ ( u 1 , u 2 − 1 ) { ( g , h ) ∈ GL ( B ) × GL ( A ) : g m = m h } \{(g,h)\in\operatorname{GL}(B)\times\operatorname{GL}(A):gm=mh\} {( g , h ) ∈ GL ( B ) × GL ( A ) : g m = mh } m = ( I r 0 0 0 ) m=\begin{pmatrix}I_r&0\\0&0\end{pmatrix} m = ( I r 0 0 0 ) g m = m h gm=mh g m = mh g = ( a b 0 d ) g=\begin{pmatrix}a&b\\0&d\end{pmatrix} g = ( a 0 b d ) h = ( a 0 c e ) h=\begin{pmatrix}a&0\\c&e\end{pmatrix} h = ( a c 0 e ) a ∈ GL r ( F ) a\in\operatorname{GL}_r(\mathbb{F}) a ∈ GL r ( F ) d , e d,e d , e m m m GL r ( F ) \operatorname{GL}_r(\mathbb{F}) GL r ( F ) GL ( B ) \operatorname{GL}(B) GL ( B ) GL ( A ) \operatorname{GL}(A) GL ( A ) m m m r r r
Example 11.5. Let B 2 = { ( i , j ) : i , j ∈ { 1 , 2 } } ∪ { 0 } B_2=\{(i,j):i,j\in\{1,2\}\}\cup\{0\} B 2 = {( i , j ) : i , j ∈ { 1 , 2 }} ∪ { 0 } ( i , j ) ( k , l ) = ( i , l ) (i,j)(k,l)=(i,l) ( i , j ) ( k , l ) = ( i , l ) j = k j=k j = k 0 0 0 B 2 1 B_2^1 B 2 1 B 2 1 B_2^1 B 2 1 ( 1 , 2 ) = ( 1 , 1 ) ( 1 , 2 ) (1,2)=(1,1)(1,2) ( 1 , 2 ) = ( 1 , 1 ) ( 1 , 2 ) ( 1 , 1 ) = ( 1 , 2 ) ( 2 , 1 ) (1,1)=(1,2)(2,1) ( 1 , 1 ) = ( 1 , 2 ) ( 2 , 1 ) ( 1 , 1 ) , ( 1 , 2 ) (1,1),(1,2) ( 1 , 1 ) , ( 1 , 2 ) ( 1 , 1 ) J ( 1 , 2 ) (1,1)\mathrel{\mathcal{J}}(1,2) ( 1 , 1 ) J ( 1 , 2 ) ( 2 , 2 ) (2,2) ( 2 , 2 ) ( 1 , 1 ) ( 2 , 2 ) = 0 (1,1)(2,2)=0 ( 1 , 1 ) ( 2 , 2 ) = 0 1 ≠ 2 1\ne2 1 = 2 ( 1 , 2 ) ( 2 , 2 ) = ( 1 , 2 ) (1,2)(2,2)=(1,2) ( 1 , 2 ) ( 2 , 2 ) = ( 1 , 2 ) 0 0 0 { 0 } \{0\} { 0 } ( 1 , 2 ) (1,2) ( 1 , 2 ) ( 1 , 2 ) ≠ 0 (1,2)\ne0 ( 1 , 2 ) = 0 0 0 0 ( 1 , 2 ) (1,2) ( 1 , 2 ) J \mathcal{J} J J \mathcal{J} J § 10 ).
Example 11.6. Let X X X C = p t \mathcal{C}=\mathrm{pt} C = pt F ( ∗ ) F(*) F ( ∗ ) X X X d H d_H d H ≃ − 1 l i m = { d H = 0 } \mathrel{\simeq}^{\mathrm{lim}}_{-1}=\{d_H=0\} ≃ − 1 lim = { d H = 0 } ≃ − 1 f i n \mathrel{\simeq}^{\mathrm{fin}}_{-1} ≃ − 1 fin
Example 11.7. Let H H H G = U ( H ) G=U(H) G = U ( H ) Bdd ( H ) \operatorname{Bdd}(H) Bdd ( H ) x ↦ u ∗ x u x\mapsto u^*xu x ↦ u ∗ xu ≃ − 1 f i n \mathrel{\simeq}^{\mathrm{fin}}_{-1} ≃ − 1 fin d ≤ ∥ x − x ~ ∥ d\le\lVert x-\widetilde{x}\rVert d ≤ ∥ x − x ∥ ≃ − 1 l i m \mathrel{\simeq}^{\mathrm{lim}}_{-1} ≃ − 1 lim approximate unitary equivalence : { u ∗ x u } = { u ∗ x ~ u } \{u^*xu\}=\{u^*\widetilde{x}u\} { u ∗ xu } = { u ∗ x u } u n u_n u n u n ∗ x u n → x ~ u_n^*xu_n\to\widetilde{x} u n ∗ x u n → x ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) x e n = e n / ( n + 1 ) xe_n=e_n/(n+1) x e n = e n / ( n + 1 ) x ~ e 0 = 0 \widetilde{x}e_0=0 x e 0 = 0 x ~ e n = e n / n \widetilde{x}e_n=e_n/n x e n = e n / n n ≥ 1 n\ge1 n ≥ 1 N ≥ 1 N\ge1 N ≥ 1 u N u_N u N e 0 ↦ e N e_0\mapsto e_N e 0 ↦ e N e k ↦ e k − 1 e_k\mapsto e_{k-1} e k ↦ e k − 1 1 ≤ k ≤ N 1\le k\le N 1 ≤ k ≤ N e k ↦ e k e_k\mapsto e_k e k ↦ e k k > N k>N k > N ∥ u N ∗ x u N − x ~ ∥ ≤ 1 / ( N + 1 ) → 0 \lVert u_N^*xu_N-\widetilde{x}\rVert\le1/(N+1)\to0 ∥ u N ∗ x u N − x ∥ ≤ 1/ ( N + 1 ) → 0 ker x = 0 ≠ ker x ~ \ker x=0\ne\ker\widetilde{x} ker x = 0 = ker x C ∗ C^* C ∗ [11 ] .
Example 11.8. Let G G G W W W X ⊆ W X\subseteq W X ⊆ W G G G g g g − 1 -1 − 1 G x ‾ = G y ‾ \overline{Gx}=\overline{Gy} G x = G y O ( x ) O(x) O ( x ) [7 ] . If y ∈ G x ‾ y\in\overline{Gx} y ∈ G x O ( y ) = O ( x ) O(y)=O(x) O ( y ) = O ( x ) x → z x\to z x → z z ∈ O ( x ) z\in O(x) z ∈ O ( x ) K l i m K^{\mathrm{lim}} K lim x ↦ O ( x ) x\mapsto O(x) x ↦ O ( x ) X → X / / G X\to X/\!/G X → X / / G 0 0 0 S S S k ≥ 1 k\ge1 k ≥ 1 G m G_m G m C 2 \mathbb{C}^2 C 2 ( 1 , − 1 ) (1,-1) ( 1 , − 1 ) − 1 -1 − 1 0 0 0 { x y = 0 } \{xy=0\} { x y = 0 } x y = c ≠ 0 xy=c\ne0 x y = c = 0
Example 11.9. (a) Stable isomorphism. C = \mathcal{C}= C = id X ≃ id Y ⟺ X ≃ Y \operatorname{id}_X\simeq\operatorname{id}_Y\Longleftrightarrow X\simeq Y id X ≃ id Y ⟺ X ≃ Y Σ P = − ⊕ P \Sigma_P=-\oplus P Σ P = − ⊕ P T = \mathcal{T}= T = id X ≃ T id Y ⟺ X ⊕ P ≃ Y ⊕ P \operatorname{id}_X\simeq_{\mathcal{T}}\operatorname{id}_Y\Longleftrightarrow X\oplus P\simeq Y\oplus P id X ≃ T id Y ⟺ X ⊕ P ≃ Y ⊕ P P P P K 0 ( R ) K_0(R) K 0 ( R ) Witt equivalence. Nondegenerate quadratic spaces with isometries, Σ n = − ⊥ Hyp ∘ \Sigma_n=-\perp\operatorname{Hyp}^{\circ} Σ n = − ⊥ Hyp ∘ T = N × N \mathcal{T}=\mathbb{N}\times\mathbb{N} T = N × N q ⊥ Hyp ∘ ≃ q ′ ⊥ Hyp ∘ q\perp\operatorname{Hyp}^{\circ}\simeq q'\perp\operatorname{Hyp}^{\circ} q ⊥ Hyp ∘ ≃ q ′ ⊥ Hyp ∘ Stable homotopy type of finite complexes. C = \mathcal{C}= C = Σ n = S n \Sigma_n=S^n Σ n = S n n n n T = \mathcal{T}= T = X ≃ T Y ⟺ S n X ≃ S n Y X\simeq_{\mathcal{T}}Y\Longleftrightarrow S^nX\simeq S^nY X ≃ T Y ⟺ S n X ≃ S n Y n n n X X X Y Y Y K 0 ( R ) K_0(R) K 0 ( R ) [8 ] ), and stable homotopy theory, where suspension inverts the instability of unstable homotopy types (Freudenthal’s theorem).
Example 11.10. In Example 11.9(c) suspension is a strict stabilizer of the invertible datum, so the stable homotopy types are the components of the telescope T e l \mathrm{Tel} Tel k ≥ 1 k \ge1 k ≥ 1 k k k T \mathcal{T} T S \mathcal{S} S
Example 11.11. In the invertible versions of Example 11.9, with D = C \mathcal{D}=\mathcal{C} D = C K \mathcal{K} K Aut K ( id x ) ≅ Aut C ( x ) \operatorname{Aut}_{\mathcal{K}}(\operatorname{id}_x)\cong\operatorname{Aut}_{\mathcal{C}}(x) Aut K ( id x ) ≅ Aut C ( x ) ( u 1 , u 2 ) (u_1,u_2) ( u 1 , u 2 ) id x \operatorname{id}_x id x u 2 = u 1 − 1 u_2=u_1^{-1} u 2 = u 1 − 1 id x \operatorname{id}_x id x colim n Aut ( Σ n x ) \operatorname*{colim}_{n}\operatorname{Aut}(\Sigma^n x) colim n Aut ( Σ n x )
(a) For Σ = − ⊕ R \Sigma=-\oplus R Σ = − ⊕ R R R R x = R k x=R^k x = R k GL ( R ) = colim n GL k + n ( R ) \operatorname{GL}(R)=\operatorname*{colim}_{n}\operatorname{GL}_{k+n}(R) GL ( R ) = colim n GL k + n ( R ) K 1 ( R ) K_1(R) K 1 ( R )
(b) For Σ = − ⊥ Hyp \Sigma=-\perp\operatorname{Hyp} Σ = − ⊥ Hyp colim n O ( q ⊥ Hyp n ) \operatorname*{colim}_{n}O(q\perp\operatorname{Hyp}^n) colim n O ( q ⊥ Hyp n )
(c) For Σ = S \Sigma=S Σ = S colim n Aut ( S n X ) \operatorname*{colim}_{n}\operatorname{Aut}(S^nX) colim n Aut ( S n X ) X X X
Thus the relations of Example 11.9 sit at the stabilized levels − 1 -1 − 1 0 0 0 GL ( R ) \operatorname{GL}(R) GL ( R ) O ∞ O_\infty O ∞
The last two examples concern the regularity axis (Section 10).
Example 11.12. Let C = B Z / 3 \mathcal{C}=B\mathbb{Z}/3 C = B Z /3 D = B A 4 \mathcal{D}=BA_4 D = B A 4 σ , τ 1 , τ 2 : Z / 3 → A 4 \sigma,\tau_1,\tau_2:\mathbb{Z}/3\to A_4 σ , τ 1 , τ 2 : Z /3 → A 4 c = ( 234 ) c=(234) c = ( 234 ) a = ( 123 ) a=(123) a = ( 123 ) b = ( 134 ) b=(134) b = ( 134 ) U 1 = U 2 = C U_1=U_2=\mathcal{C} U 1 = U 2 = C W = 1 W=1 W = 1 − 1 -1 − 1 m , m ~ m,\widetilde{m} m , m c m ~ ∈ ⟨ a ⟩ c m ⟨ b ⟩ c^{\widetilde{m}}\in\langle a\rangle c^m\langle b\rangle c m ∈ ⟨ a ⟩ c m ⟨ b ⟩ c = a b c=ab c = ab ⟨ a ⟩ ⟨ b ⟩ \langle a\rangle\langle b\rangle ⟨ a ⟩ ⟨ b ⟩ e e e c c c c 2 = ( 243 ) c^2=(243) c 2 = ( 243 ) { 0 , 1 } \{0,1\} { 0 , 1 } { 2 } \{2\} { 2 } Z / 3 \mathbb{Z}/3 Z /3 Z / 3 \mathbb{Z}/3 Z /3 Z / 3 \mathbb{Z}/3 Z /3 σ \sigma σ A 4 A_4 A 4
Example 11.13. Every equivalence relation on Z / 3 \mathbb{Z}/3 Z /3 − 1 -1 − 1 U 1 = U 2 = Z / 3 U_1=U_2=\mathbb{Z}/3 U 1 = U 2 = Z /3 σ = Id \sigma=\operatorname{Id} σ = Id τ i \tau_i τ i Θ u 1 , u 2 ( m ) = m \Theta_{u_1,u_2}(m)=m Θ u 1 , u 2 ( m ) = m { 0 , 1 } ∣ { 2 } \{0,1\}\mid\{2\} { 0 , 1 } ∣ { 2 } { 0 , 2 } ∣ { 1 } \{0,2\}\mid\{1\} { 0 , 2 } ∣ { 1 } m ↦ − m m\mapsto-m m ↦ − m { 1 , 2 } ∣ { 0 } \{1,2\}\mid\{0\} { 1 , 2 } ∣ { 0 } Γ = S 4 \Gamma=S_4 Γ = S 4 σ \sigma σ ↦ c = ( 123 ) \mapsto c=(123) ↦ c = ( 123 ) τ 1 = τ 2 \tau_1=\tau_2 τ 1 = τ 2 ↦ a = ( 012 ) \mapsto a=(012) ↦ a = ( 012 ) ⟨ a ⟩ ⟨ a ⟩ = ⟨ a ⟩ \langle a\rangle\langle a\rangle=\langle a\rangle ⟨ a ⟩ ⟨ a ⟩ = ⟨ a ⟩ c c c c 2 c^2 c 2 c 2 ∈ ⟨ a ⟩ c ⟨ a ⟩ c^2\in\langle a\rangle c\langle a\rangle c 2 ∈ ⟨ a ⟩ c ⟨ a ⟩ { 0 } \{0\} { 0 } { 1 , 2 } \{1,2\} { 1 , 2 }
Conclusion
Every comparison category is a Grothendieck construction restricted along a map s s s σ , τ 1 , τ 2 \sigma,\tau_1,\tau_2 σ , τ 1 , τ 2 σ \sigma σ − 1 -1 − 1 − 1 -1 − 1 ρ 2 , … , ρ 5 \rho_2,\ldots,\rho_5 ρ 2 , … , ρ 5 ( ∞ , 2 ) (\infty,2) ( ∞ , 2 )
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This article was generated by a generative artificial intelligence system (Claude Sonnet 5.5, Anthropic, October 2026) from an initial draft written by the author. The system restructured and rewrote the text, proposed and formulated the definitions, statements and proofs of the new results, carried out the supporting computations, and produced the LaTeX source. The research questions, the initial draft and the scientific direction were provided by the author, who directed the generation through successive instructions and revisions. The author takes full responsibility for the content, and the AI system is not listed as an author.
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References
Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco
Email address: nizar.elidrissi@uit.ac.ma
