Introduction
First proved independently by Dold and Kan in 1958, the Dold–Kan correspondence [5 , 15 ] gives an equivalence between simplicial k k k
Fun ( Δ o p , k Mod ) ≃ Ch ≥ 0 ( k Mod ) . \operatorname{Fun}(\Delta^{\mathrm{op}}, k\operatorname{Mod}) \simeq\operatorname{Ch}_{\geq0}(k\operatorname{Mod}). Fun ( Δ op , k Mod ) ≃ Ch ≥ 0 ( k Mod ) . It has since inspired numerous generalizations to other indexing categories and to higher categorical settings: Pirashvili establishes a Dold–Kan-type theorem for abelian Γ \Gamma Γ [25 ] , while Słomińska studies such correspondences from the perspective of Morita equivalences of functor categories [26 ] . Lack and Street construct Dold–Kan correspondences for categories which satisfy certain factorization conditions [19 ] .
Walde develops a homotopy-coherent extension [27 ] of the Lack–Street framework. Related developments include the Dold–Kan correspondence for stable ∞ \infty ∞ [21 ] . Dyckerhoff categorifies the Dold–Kan correspondence using 2-simplicial stable ∞ \infty ∞ [9 ] , and Heine extends this approach to the Dwyer–Kan correspondence using 2-duplicial stable ∞ \infty ∞ [14 ] .
2020 Mathematics Subject Classification . Primary: 18G31; Secondary: 16E40, 16D90.
Returning to the classical setting, Lyubashenko describes the Dold–Kan correspondence using a family of idempotents in the linearized simplex category [23 ] . This family is related by order-reversal to the Dold–Puppe projectors [6 ] , which are used by Getzler to give a direct proof of the Dwyer–Kan theorem for paracyclic modules [13 ] . In this article, we combine Lyubashenko’s and Getzler’s insights to develop a functorial framework for constructing Dold–Kan-type correspondences.
Main results
Here we briefly recall the necessary definitions and then state the main results of the article. Let Δ \Delta Δ Δ max , Δ min ⊆ Δ \Delta_{\max}, \Delta_{\min} \subseteq\Delta Δ m a x , Δ m i n ⊆ Δ k k k k Mod k\operatorname{Mod} k Mod k Mod k\operatorname{Mod} k Mod C \mathcal{C} C k [ C ] k[\mathcal{C}] k [ C ] A \mathcal{A} A A \mathcal{A} A C \mathcal{C} C A → C \mathcal{A} \to\mathcal{C} A → C A \mathcal{A} A Ring A \operatorname{Ring}_{\mathcal{A}} Ring A A ↓ k Cat \mathcal{A} \downarrow k\operatorname{Cat} A ↓ k Cat
For each k [ Δ max ] k[\Delta_{\max}] k [ Δ m a x ] C \mathcal{C} C N ( C ) N(\mathcal{C}) N ( C ) N : Ring k [ Δ max ] → Ring N ( Δ max ) N \colon\operatorname{Ring}_{k[\Delta_{\max}]} \to\operatorname{Ring}_{N(\Delta_{\max})} N : Ring k [ Δ m a x ] → Ring N ( Δ m a x )
Theorem 4.11 (Abstract Dold–Kan correspondence).
(1) The categories C \mathcal{C} C N ( C ) N(\mathcal{C}) N ( C ) Fun k ( C op , A ) \operatorname{Fun}_{k}(\mathcal{C}^{\operatorname{op}},\mathcal{A}) Fun k ( C op , A ) Fun k ( N ( C ) op , A ) \operatorname{Fun}_{k}(N(\mathcal{C})^{\operatorname{op}},\mathcal{A}) Fun k ( N ( C ) op , A ) A \mathcal{A} A
(2) The functor N : Ring k [ Δ max ] → Ring N ( k [ Δ max ] ) N \colon\operatorname{Ring}_{k[\Delta_{\max}]} \to\operatorname{Ring}_{N(k[\Delta_{\max}])} N : Ring k [ Δ m a x ] → Ring N ( k [ Δ m a x ])
As an application, we recover the Dwyer–Kan correspondences for duplicial, paracyclic and N N N § 6 .
More generally, Theorem 4.11 yields a Dold–Kan-type correspondence for every crossed simplicial group Δ G \Delta G Δ G N ( Δ G ) ≔ N ( k [ Δ G ] ) N(\Delta G) \coloneqq N(k[\Delta G]) N ( Δ G ) : = N ( k [ Δ G ])
Δ op ⟶ ( Δ G ) op → Δ G ( − , 0 ) Set \Delta^{\operatorname{op}} \longrightarrow(\Delta G)^{\operatorname{op}} \xrightarrow{\Delta G(-,0)} \operatorname{Set} Δ op ⟶ ( Δ G ) op Δ G ( − , 0 ) Set which we denote by G G G Theorem 5.10 we prove a rank formula for the hom-modules based on the nondegenerate simplices of G G G Δ G \Delta G Δ G
Theorem 5.20 . The following are equivalent for a crossed simplicial group Δ G \Delta G Δ G
(1) There exists an equivalence ( Δ G ) op ≃ Δ G (\Delta G)^{\operatorname{op}} \simeq\Delta G ( Δ G ) op ≃ Δ G
(2) There exists a linear equivalence N ( Δ G ) op ≃ N ( Δ G ) N(\Delta G)^{\operatorname{op}} \simeq N(\Delta G) N ( Δ G ) op ≃ N ( Δ G )
(3) The simplicial dimension of G G G
Moreover, every equivalence ( Δ G ) op ≃ Δ G (\Delta G)^{\operatorname{op}} \simeq\Delta G ( Δ G ) op ≃ Δ G N ( Δ G ) op ≃ N ( Δ G ) N(\Delta G)^{\operatorname{op}} \simeq N(\Delta G) N ( Δ G ) op ≃ N ( Δ G ) Lemma 5.15 and Theorem 5.18 .
The proof of Theorem 5.20 makes use of the following constructions we develop in § 7 .
We construct normalization-compatible self-dualities of the paracyclic and paradih edral categories using order adjoints on the integers.
Using a quotient presentation, we extend this construction to all crossed simplicial groups of simplicial dimension one.
Acknowledgements. The author would like to thank Ulrich Krähmer for many useful discussions and valuable comments on the initial draft of this article, Ivan Bartulović for encouraging him to pursue the topic and Tony Zorman for enlightening discussions on promonads.
Preliminaries
Simplicial constructions
Notation 2.1. For every natural number n ∈ N n \in\mathbb{N} n ∈ N [ n ] ≔ { 0 , 1 , … , n } [n] \coloneqq\{0,1,\ldots,n\} [ n ] : = { 0 , 1 , … , n }
Definition 2.2. We define a category Δ \Delta Δ
The objects are the natural numbers n ∈ N n \in\mathbb{N} n ∈ N
A morphism f ∈ Δ ( m , n ) f \in\Delta(m,n) f ∈ Δ ( m , n ) f : [ m ] ⟶ [ n ] f\colon[m] \longrightarrow[n] f : [ m ] ⟶ [ n ]
The following morphisms of Δ \Delta Δ
δ i n ∈ Δ ( n − 1 , n ) \delta_i^n \in\Delta(n-1,n) δ i n ∈ Δ ( n − 1 , n ) i ∈ [ n ] i \in[n] i ∈ [ n ]
σ j n ∈ Δ ( n , n − 1 ) \sigma_j^n \in\Delta(n,n-1) σ j n ∈ Δ ( n , n − 1 ) j ∈ [ n − 1 ] j \in[n-1] j ∈ [ n − 1 ]
We will refer to the δ i n \delta_i^n δ i n cofaces and to the σ j n \sigma_j^n σ j n codegeneracies .
Definition 2.3. We denote by Rev : Δ → Δ \operatorname{Rev}\colon\Delta\to\Delta Rev : Δ → Δ Δ \Delta Δ f ∈ Δ ( m , n ) f \in\Delta(m,n) f ∈ Δ ( m , n ) Rev ( f ) ( i ) = n − f ( m − i ) \operatorname{Rev}(f)(i)=n-f(m-i) Rev ( f ) ( i ) = n − f ( m − i ) i ∈ [ m ] i \in[m] i ∈ [ m ]
Definition 2.5. A simplicial object in a category C \mathcal{C} C X : Δ o p → C X\colon\Delta^{\mathrm{op}}\to\mathcal{C} X : Δ op → C
Notation 2.6. Given a simplicial object X : Δ o p → C X\colon\Delta^{\mathrm{op}}\to\mathcal{C} X : Δ op → C X n ≔ X ( n ) X_n\coloneqq X(n) X n : = X ( n ) d i n ≔ X ( δ i n ) : X n → X n − 1 d_i^n\coloneqq X(\delta_i^n)\colon X_n\to X_{n-1} d i n : = X ( δ i n ) : X n → X n − 1 s j n ≔ X ( σ j n ) : X n − 1 → X n s_j^n\coloneqq X(\sigma_j^n)\colon X_{n-1}\to X_n s j n : = X ( σ j n ) : X n − 1 → X n
Notation 2.7. The upper indices of δ i n \delta_i^n δ i n σ j n \sigma_j^n σ j n
Definition 2.9. Let X : Δ o p → S e t X\colon\Delta^{\mathrm{op}}\to\mathsf{Set} X : Δ op → Set x ∈ X n x\in X_n x ∈ X n degenerate , if x = s j ( y ) x=s_j(y) x = s j ( y ) y ∈ X n − 1 y\in X_{n-1} y ∈ X n − 1 j ∈ [ n − 1 ] j\in[n-1] j ∈ [ n − 1 ] x x x nondegenerate . We denote the nondegenerate elements by X n n d ≔ X n ∖ ⋃ j ∈ [ n − 1 ] s j ( X n − 1 ) X_n^{\mathrm{nd}}\coloneqq X_n\setminus\bigcup_{j\in[n-1]}s_j(X_{n-1}) X n nd : = X n ∖ ⋃ j ∈ [ n − 1 ] s j ( X n − 1 ) X X X dim X ≔ sup ( { − 1 } ∪ { n ∈ N : X n n d ≠ ∅ } ) \dim X\coloneqq\sup\left(\{-1\}\cup\{n\in\mathbb{N}:X_n^{\mathrm{nd}}\neq\varnothing\}\right) dim X : = sup ( { − 1 } ∪ { n ∈ N : X n nd = ∅ } )
Proposition 2.10. Let X : Δ o p → S e t X \colon\Delta^{\mathrm{op}} \to\mathbf{Set} X : Δ op → Set x ∈ X n x \in X_n x ∈ X n
x = X ( φ ) ( x ′ ) x = X(\varphi)(x') x = X ( φ ) ( x ′ ) for a surjective φ ∈ Δ ( n , r ) \varphi\in\Delta(n,r) φ ∈ Δ ( n , r ) x ′ ∈ X r n d x' \in X_r^{\mathrm{nd}} x ′ ∈ X r nd
Proof. Choose r ∈ N r \in\mathbb{N} r ∈ N x = X ( φ ) ( x ′ ) x = X(\varphi)(x') x = X ( φ ) ( x ′ ) φ ∈ Δ ( n , r ) \varphi\in\Delta(n,r) φ ∈ Δ ( n , r ) x ′ ∈ X r x' \in X_r x ′ ∈ X r r r r φ \varphi φ φ \varphi φ x ′ ∈ X r n d x' \in X_r^{\mathrm{nd}} x ′ ∈ X r nd [22 Proposition 0014] . □ \square □
Definition 2.11. We refer to the decomposition x = X ( φ ) ( x ′ ) x = X(\varphi)(x') x = X ( φ ) ( x ′ ) Proposition 2.10 as the Eilenberg–Zilber decomposition or EZ-decomposition of x x x x ′ x' x ′ x x x
Proposition 2.13. Regard Δ \Delta Δ
A morphism φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) φ ( m ) = n \varphi(m) = n φ ( m ) = n
A morphism φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) φ ( 0 ) = 0 \varphi(0) = 0 φ ( 0 ) = 0
Proof. Let φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) ψ ∈ Δ ( n , m ) \psi\in\Delta(n,m) ψ ∈ Δ ( n , m ) φ ⊣ ψ \varphi\dashv\psi φ ⊣ ψ φ ( x ) ≤ y ⇔ x ≤ ψ ( y ) \varphi(x) \le y \Leftrightarrow x \le\psi(y) φ ( x ) ≤ y ⇔ x ≤ ψ ( y ) x ∈ [ m ] x \in[m] x ∈ [ m ] y ∈ [ n ] y \in[n] y ∈ [ n ] S ( y ) = { x ∈ [ m ] : φ ( x ) ≤ y } S(y) = \{x \in[m] : \varphi(x) \le y\} S ( y ) = { x ∈ [ m ] : φ ( x ) ≤ y } ψ ( y ) = max S ( y ) \psi(y) = \max S(y) ψ ( y ) = max S ( y ) y ∈ [ n ] y \in[n] y ∈ [ n ] φ ( 0 ) = 0 \varphi(0) = 0 φ ( 0 ) = 0 φ ( 0 ) = 0 \varphi(0) = 0 φ ( 0 ) = 0 S ( y ) S(y) S ( y ) y ↦ max S ( y ) y \mapsto\max S(y) y ↦ max S ( y ) φ \varphi φ □ \square □
We will adopt and extend some of the notation found in [22 Subsection 04S6] .
Definition 2.14. Let Δ min ⊆ Δ \Delta_{\min} \subseteq\Delta Δ m i n ⊆ Δ Δ max ⊆ Δ \Delta_{\max} \subseteq\Delta Δ m a x ⊆ Δ
Corollary 2.15. The functor ( Δ min ) o p → Δ max (\Delta_{\min})^{\mathrm{op}} \to\Delta_{\max} ( Δ m i n ) op → Δ m a x Rev ( Δ max ) = Δ min \operatorname{Rev}(\Delta_{\max}) = \Delta_{\min} Rev ( Δ m a x ) = Δ m i n
Definition 2.17. The concatenation functor C 0 : Δ → Δ min C_0 \colon\Delta\to\Delta_{\min} C 0 : Δ → Δ m i n Δ min ↪ Δ \Delta_{\min} \hookrightarrow\Delta Δ m i n ↪ Δ C 0 ( n ) = n + 1 C_0(n) = n + 1 C 0 ( n ) = n + 1 φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) C 0 ( φ ) ∈ Δ min ( m + 1 , n + 1 ) C_0(\varphi) \in\Delta_{\min}(m + 1,n + 1) C 0 ( φ ) ∈ Δ m i n ( m + 1 , n + 1 )
C 0 ( φ ) ( x ) = { 0 if x = 0 , φ ( x − 1 ) + 1 else . C_0(\varphi)(x) =
\begin{cases}
0 & \text{if } x = 0,\\
\varphi(x - 1) + 1 & \text{else}.
\end{cases} C 0 ( φ ) ( x ) = { 0 φ ( x − 1 ) + 1 if x = 0 , else . Similarly, the concatenation functor C 0 : Δ → Δ max C^{0} : \Delta\to\Delta_{\max} C 0 : Δ → Δ m a x Δ max ↪ Δ \Delta_{\max} \hookrightarrow\Delta Δ m a x ↪ Δ C 0 ( n ) = n + 1 C^{0}(n) = n + 1 C 0 ( n ) = n + 1 φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) C 0 ( φ ) ∈ Δ max ( m + 1 , n + 1 ) C^{0}(\varphi) \in\Delta_{\max}(m+1,n+1) C 0 ( φ ) ∈ Δ m a x ( m + 1 , n + 1 )
C 0 ( φ ) ( x ) = { n + 1 if x = m + 1 , φ ( x ) else. C^{0}(\varphi)(x) =
\begin{cases}
n+1 & \text{if } x=m+1,\\
\varphi(x) & \text{else.}
\end{cases} C 0 ( φ ) ( x ) = { n + 1 φ ( x ) if x = m + 1 , else. Definition 2.19. Let X : Δ o p → C X : \Delta^{\mathrm{op}} \to\mathcal{C} X : Δ op → C décalage of X X X Dec 0 ( X ) ≔ X ∘ ( ι ∘ C 0 ) o p \operatorname{Dec}_{0}(X) \coloneqq X \circ(\iota\circ C_{0})^{\mathrm{op}} Dec 0 ( X ) : = X ∘ ( ι ∘ C 0 ) op Dec 0 ( X ) ≔ X ∘ ( ι ∘ C 0 ) o p \operatorname{Dec}^{0}(X) \coloneqq X \circ(\iota\circ C^{0})^{\mathrm{op}} Dec 0 ( X ) : = X ∘ ( ι ∘ C 0 ) op d min : Dec 0 ( X ) ⇒ X d_{\min} : \operatorname{Dec}_{0}(X) \Rightarrow X d m i n : Dec 0 ( X ) ⇒ X d max : Dec 0 ( X ) ⇒ X d_{\max} : \operatorname{Dec}^{0}(X) \Rightarrow X d m a x : Dec 0 ( X ) ⇒ X
Linear categories and modules
For the remainder of the article we fix a (nonzero) commutative unital ring k k k k M o d k\mathrm{Mod} k Mod [3 Sections 1 and 6] and [16 ] ).
Definition 2.20. A linear category A \mathcal{A} A k M o d k\mathrm{Mod} k Mod F : A → B F : \mathcal{A} \to\mathcal{B} F : A → B k M o d k\mathrm{Mod} k Mod A \mathcal{A} A Fun k ( A , B ) \operatorname{Fun}_{k}(\mathcal{A},\mathcal{B}) Fun k ( A , B ) A → B \mathcal{A} \to\mathcal{B} A → B
If X X X k [ X ] k[X] k [ X ] k k k X X X
Definition 2.21. The linearization of a category C \mathcal{C} C k [ C ] k[\mathcal{C}] k [ C ] C \mathcal{C} C k k k k [ C ] ( x , y ) = k [ C ( x , y ) ] k[\mathcal{C}](x,y) = k[\mathcal{C}(x,y)] k [ C ] ( x , y ) = k [ C ( x , y )] C \mathcal{C} C
Following the perspective introduced by Mitchell [24 ] , we regard linear categories as k k k
Definition 2.22. A (two-sided) ideal I I I C \mathcal{C} C k k k I ( x , y ) ⊆ C ( x , y ) I(x,y) \subseteq\mathcal{C}(x,y) I ( x , y ) ⊆ C ( x , y ) I ( y , z ) ∘ C ( x , y ) ⊆ I ( x , z ) I(y,z) \circ\mathcal{C}(x,y) \subseteq I(x,z) I ( y , z ) ∘ C ( x , y ) ⊆ I ( x , z ) C ( y , z ) ∘ I ( x , y ) ⊆ I ( x , z ) \mathcal{C}(y,z) \circ I(x,y) \subseteq I(x,z) C ( y , z ) ∘ I ( x , y ) ⊆ I ( x , z ) x , y , z ∈ C x,y,z \in\mathcal{C} x , y , z ∈ C
Composition in C \mathcal{C} C quotient category C / I \mathcal{C}/I C / I C → C / I \mathcal{C} \to\mathcal{C}/I C → C / I
Definition 2.23. The tensor product A ⊗ k B \mathcal{A} \otimes_{k} \mathcal{B} A ⊗ k B
Objects of A ⊗ k B \mathcal{A} \otimes_{k} \mathcal{B} A ⊗ k B ( a , b ) (a,b) ( a , b ) a ∈ A a \in\mathcal{A} a ∈ A b ∈ B b \in\mathcal{B} b ∈ B
Hom-modules are ( A ⊗ k B ) ( ( a , b ) , ( a ′ , b ′ ) ) = A ( a , a ′ ) ⊗ k B ( b , b ′ ) (\mathcal{A} \otimes_{k} \mathcal{B})((a,b),(a',b')) = \mathcal{A}(a,a') \otimes_{k} \mathcal{B}(b,b') ( A ⊗ k B ) (( a , b ) , ( a ′ , b ′ )) = A ( a , a ′ ) ⊗ k B ( b , b ′ )
Composition is defined in the obvious way, using the tensor product on k M o d k\mathrm{Mod} k Mod
Definition 2.25. Let A , B \mathcal{A}, \mathcal{B} A , B
A right A \mathcal{A} A A op → k Mod \mathcal{A}^{\operatorname{op}} \to\Bbbk\operatorname{Mod} A op → k Mod
A left B \mathcal{B} B B → k Mod \mathcal{B} \to\Bbbk\operatorname{Mod} B → k Mod
An ( A , B ) (\mathcal{A},\mathcal{B}) ( A , B ) A ⊗ k B op → k Mod \mathcal{A} \otimes_{\Bbbk} \mathcal{B}^{\operatorname{op}} \to\Bbbk\operatorname{Mod} A ⊗ k B op → k Mod
The corresponding functor categories are labeled Mod A \operatorname{Mod}_{\mathcal{A}} Mod A B Mod {}_{\mathcal{B}}\operatorname{Mod} B Mod A Mod B {}_{\mathcal{A}}\operatorname{Mod}_{\mathcal{B}} A Mod B a ∈ A a \in\mathcal{A} a ∈ A
A a : = A ( a , − ) : A → k Mod and A a : = A ( − , a ) : A op → k Mod \mathcal{A}_{a} := \mathcal{A}(a,-): \mathcal{A} \to\Bbbk\operatorname{Mod}
\qquad\text{and}\qquad
\mathcal{A}^{a} := \mathcal{A}(-,a): \mathcal{A}^{\operatorname{op}} \to\Bbbk\operatorname{Mod} A a := A ( a , − ) : A → k Mod and A a := A ( − , a ) : A op → k Mod and identify the category A \mathcal{A} A
A ⊗ k A op ≃ A op ⊗ k A → A ( − , − ) k Mod . \mathcal{A} \otimes_{\Bbbk} \mathcal{A}^{\operatorname{op}} \simeq\mathcal{A}^{\operatorname{op}} \otimes_{\Bbbk} \mathcal{A} \xrightarrow{\mathcal{A}(-,-)} \Bbbk\operatorname{Mod}. A ⊗ k A op ≃ A op ⊗ k A A ( − , − ) k Mod . Definition 2.26. Given a small linear category A \mathcal{A} A y : A → Mod A \mathord{\mathbf{y}}: \mathcal{A} \to\operatorname{Mod}_{\mathcal{A}} y : A → Mod A
Definition 2.27. Let A , B , C \mathcal{A},\mathcal{B},\mathcal{C} A , B , C tensor product of bimodules M ∈ A Mod B M \in{}_{\mathcal{A}}\operatorname{Mod}_{\mathcal{B}} M ∈ A Mod B N ∈ B Mod C N \in{}_{\mathcal{B}}\operatorname{Mod}_{\mathcal{C}} N ∈ B Mod C
M ⊗ B N : = ∫ b ∈ B M ( − , b ) ⊗ k N ( b , − ) ∈ A Mod C . M \otimes_{\mathcal{B}} N := \int^{b \in\mathcal{B}} M(-,b) \otimes_{\Bbbk} N(b,-) \in{}_{\mathcal{A}}\operatorname{Mod}_{\mathcal{C}}. M ⊗ B N := ∫ b ∈ B M ( − , b ) ⊗ k N ( b , − ) ∈ A Mod C . We assume familiarity with the calculus of (co)ends and refer to [16 ] and [Lor21] for a detailed treatment of the subject.
We remark that our definition of the tensor product is (up to variance convention) the composition of linear profunctors; see e.g. [Lor21, Section 5].
Generalized Morita theory
The results of this section are special cases of more general results on modules and profunctors of enriched bicategories (see e.g. [GS16]) and are included to keep the article self-contained. For the remainder of this section, let A \mathcal{A} A B \mathcal{B} B
Definition 3.1. We say that A \mathcal{A} A B \mathcal{B} B Morita equivalent , if there exists a linear equivalence Mod A ≃ Mod B \operatorname{Mod}_{\mathcal{A}} \simeq\operatorname{Mod}_{\mathcal{B}} Mod A ≃ Mod B
Adjunctions arising from bimodules
Just as in the classical setting, we may study Morita equivalences by means of bimodules.
Proposition 3.3. Every bimodule X ∈ A M o d B X \in{}_{\mathcal A}\mathrm{Mod}_{\mathcal B} X ∈ A Mod B
( − ) ⊗ A X : M o d A ⇄ M o d B : M o d B ( X , − ) . (-) \otimes_{\mathcal A} X : \mathrm{Mod}_{\mathcal A} \rightleftarrows\mathrm{Mod}_{\mathcal B} : \mathrm{Mod}_{\mathcal B}(X,-). ( − ) ⊗ A X : Mod A ⇄ Mod B : Mod B ( X , − ) . Proof. Let M ∈ M o d A M \in\mathrm{Mod}_{\mathcal A} M ∈ Mod A N ∈ M o d B N \in\mathrm{Mod}_{\mathcal B} N ∈ Mod B
M o d B ( M ⊗ A X , N ) ≅ ∫ a ∫ b k M o d ( M ( a ) ⊗ k X ( a , b ) , N ( b ) ) ≅ ∫ a ∫ b k M o d ( M ( a ) , k M o d ( X ( a , b ) , N ( b ) ) ) ≅ M o d A ( M , M o d B ( X , N ) ) . \begin{aligned}
\mathrm{Mod}_{\mathcal B}(M \otimes_{\mathcal A} X,N)
&\cong\int_{a}\int_{b} k\mathrm{Mod}(M(a) \otimes_k X(a,b),N(b)) \\
&\cong\int_{a}\int_{b} k\mathrm{Mod}(M(a),k\mathrm{Mod}(X(a,b),N(b))) \\
&\cong\mathrm{Mod}_{\mathcal A}(M,\mathrm{Mod}_{\mathcal B}(X,N)).
\end{aligned} Mod B ( M ⊗ A X , N ) ≅ ∫ a ∫ b k Mod ( M ( a ) ⊗ k X ( a , b ) , N ( b )) ≅ ∫ a ∫ b k Mod ( M ( a ) , k Mod ( X ( a , b ) , N ( b ))) ≅ Mod A ( M , Mod B ( X , N )) . Proposition 3.4. If F : M o d A → M o d B F:\mathrm{Mod}_{\mathcal A}\to\mathrm{Mod}_{\mathcal B} F : Mod A → Mod B F ≅ ( − ) ⊗ A X F\cong(-)\otimes_{\mathcal A}X F ≅ ( − ) ⊗ A X X X X ( F ∘ ♮ ) b (F\circ\mathord{\natural})^{\mathrm b} ( F ∘ ♮ ) b
Proof. Consider the following natural isomorphisms in M ∈ M o d A M\in\mathrm{Mod}_{\mathcal A} M ∈ Mod A
F ( M ) ≅ F ( ∫ a M ( a ) ⊗ k ♮ ( a ) ) ≅ ∫ a M ( a ) ⊗ k F ( ♮ ( a ) ) ≅ M ⊗ A X . F(M)\cong F\left(\int^{a}M(a)\otimes_k\mathord{\natural}(a)\right)\cong\int^{a}M(a)\otimes_kF(\mathord{\natural}(a))\cong M\otimes_{\mathcal A}X. F ( M ) ≅ F ( ∫ a M ( a ) ⊗ k ♮ ( a ) ) ≅ ∫ a M ( a ) ⊗ k F ( ♮ ( a )) ≅ M ⊗ A X . We obtain the (co)extension of scalars along a functor as a special case of this adjunction.
Definition 3.5. A functor F : A → B F:\mathcal A\to\mathcal B F : A → B B ( − , F ( − ) ) ∈ A M o d B \mathcal B(-,F(-))\in{}_{\mathcal A}\mathrm{Mod}_{\mathcal B} B ( − , F ( − )) ∈ A Mod B
F ! : M o d A ⇄ M o d B : F ∗ . F_!:\mathrm{Mod}_{\mathcal A}\rightleftarrows\mathrm{Mod}_{\mathcal B}:F^*. F ! : Mod A ⇄ Mod B : F ∗ . The other bimodule B ( F ( − ) , − ) ∈ B M o d A \mathcal B(F(-),-)\in{}_{\mathcal B}\mathrm{Mod}_{\mathcal A} B ( F ( − ) , − ) ∈ B Mod A
F ∗ : M o d B ⇄ M o d A : F ∗ . F^*:\mathrm{Mod}_{\mathcal B}\rightleftarrows\mathrm{Mod}_{\mathcal A}:F_*. F ∗ : Mod B ⇄ Mod A : F ∗ . Note that F ∗ F^* F ∗ F F F
Tiny modules and generators
Definition 3.7. A module M ∈ M o d A M\in\mathrm{Mod}_{\mathcal A} M ∈ Mod A tiny if M o d A ( M , − ) \mathrm{Mod}_{\mathcal A}(M,-) Mod A ( M , − ) ( M i ) i ∈ I ⊆ M o d A (M_i)_{i\in I}\subseteq\mathrm{Mod}_{\mathcal A} ( M i ) i ∈ I ⊆ Mod A tiny generator if they jointly reflect zero objects, i.e. if M o d A ( M i , X ) = 0 \mathrm{Mod}_{\mathcal A}(M_i,X)=0 Mod A ( M i , X ) = 0 i ∈ I i\in I i ∈ I X = 0 X=0 X = 0
Proposition 3.9. A module in M o d A \mathrm{Mod}_{\mathcal A} Mod A
Proof. Observe that all representables are tiny. Indeed, colimits of diagrams in M o d A \mathrm{Mod}_{\mathcal A} Mod A M ∈ M o d A M\in\mathrm{Mod}_{\mathcal A} M ∈ Mod A
is a retract of a tiny module T ∈ Mod A T \in\operatorname{Mod}_{\mathcal A} T ∈ Mod A Mod A ( M , − ) \operatorname{Mod}_{\mathcal A}(M,-) Mod A ( M , − ) Mod A ( T , − ) \operatorname{Mod}_{\mathcal A}(T,-) Mod A ( T , − ) M M M
Given a tiny module P ∈ Mod A P \in\operatorname{Mod}_{\mathcal A} P ∈ Mod A P P P f : A a → P f:\mathcal A^a \to P f : A a → P a ∈ A a \in\mathcal A a ∈ A π : ⨁ f A a → P \pi:\bigoplus_f \mathcal A^a \to P π : ⨁ f A a → P π \pi π coker π = 0 \operatorname{coker}\pi=0 coker π = 0 A a \mathcal A^a A a A a → coker π \mathcal A^a \to\operatorname{coker}\pi A a → coker π P ↠ coker π P \twoheadrightarrow\operatorname{coker}\pi P ↠ coker π f : A a → P f:\mathcal A^a \to P f : A a → P f f f Mod A ( A a , coker π ) = 0 \operatorname{Mod}_{\mathcal A}(\mathcal A^a,\operatorname{coker}\pi)=0 Mod A ( A a , coker π ) = 0 a ∈ A a \in\mathcal A a ∈ A coker π = 0 \operatorname{coker}\pi=0 coker π = 0 P P P π \pi π Mod A ( P , ⨁ f A a ) ≅ ⨁ f Mod A ( P , A a ) \operatorname{Mod}_{\mathcal A}(P,\bigoplus_f\mathcal A^a)\cong\bigoplus_f\operatorname{Mod}_{\mathcal A}(P,\mathcal A^a) Mod A ( P , ⨁ f A a ) ≅ ⨁ f Mod A ( P , A a ) □ \square □
Porism 3.10. In the proof above we may replace ( A a ) a ∈ A (\mathcal A^a)_{a\in\mathcal A} ( A a ) a ∈ A ( M i ) i ∈ I (M_i)_{i\in I} ( M i ) i ∈ I M i M_i M i
Remark 3.11. If A \mathcal A A k k k M ∈ Mod A M\in\operatorname{Mod}_{\mathcal A} M ∈ Mod A Mod A ( M , − ) \operatorname{Mod}_{\mathcal A}(M,-) Mod A ( M , − ) Mod A \operatorname{Mod}_{\mathcal A} Mod A M M M M M M
Proposition 3.12. Let X ∈ A Mod B X\in{}_{\mathcal A}\operatorname{Mod}_{\mathcal B} X ∈ A Mod B
( − ) ⊗ A X : Mod A → Mod B (-)\otimes_{\mathcal A}X:\operatorname{Mod}_{\mathcal A}\to\operatorname{Mod}_{\mathcal B} ( − ) ⊗ A X : Mod A → Mod B
X ♯ : A → Mod B X^\sharp:\mathcal A\to\operatorname{Mod}_{\mathcal B} X ♯ : A → Mod B ( X ♯ ( a ) ) a ∈ A ⊆ Mod B (X^\sharp(a))_{a\in\mathcal A}\subseteq\operatorname{Mod}_{\mathcal B} ( X ♯ ( a ) ) a ∈ A ⊆ Mod B
Proof. Suppose (1) holds. We have seen in Remark 3.6 that the family ( X ♯ ( a ) ) a ∈ A (X^\sharp(a))_{a\in\mathcal A} ( X ♯ ( a ) ) a ∈ A X ♯ X^\sharp X ♯ A ( a , a ′ ) ≅ Mod A ( A a , A a ′ ) ≅ Mod B ( A a ⊗ A X , A a ′ ⊗ A X ) ≅ Mod B ( X ♯ ( a ) , X ♯ ( a ′ ) ) \mathcal A(a,a')\cong\operatorname{Mod}_{\mathcal A}(\mathcal A^a,\mathcal A^{a'})\cong\operatorname{Mod}_{\mathcal B}(\mathcal A^a\otimes_{\mathcal A}X,\mathcal A^{a'}\otimes_{\mathcal A}X)\cong\operatorname{Mod}_{\mathcal B}(X^\sharp(a),X^\sharp(a')) A ( a , a ′ ) ≅ Mod A ( A a , A a ′ ) ≅ Mod B ( A a ⊗ A X , A a ′ ⊗ A X ) ≅ Mod B ( X ♯ ( a ) , X ♯ ( a ′ )) a , a ′ ∈ A a,a'\in\mathcal A a , a ′ ∈ A Mod B ( X , − ) \operatorname{Mod}_{\mathcal B}(X,-) Mod B ( X , − ) M ∈ Mod A M\in\operatorname{Mod}_{\mathcal A} M ∈ Mod A Mod B ( X , M ⊗ A X ) ≅ ∫ a M ( a ) ⊗ k Mod B ( X , X ♯ ( a ) ) ≅ ∫ a M ( a ) ⊗ k A a ≅ M \operatorname{Mod}_{\mathcal B}(X,M\otimes_{\mathcal A}X)\cong\int^{a}M(a)\otimes_k\operatorname{Mod}_{\mathcal B}(X,X^\sharp(a))\cong\int^{a}M(a)\otimes_k\mathcal A^a\cong M Mod B ( X , M ⊗ A X ) ≅ ∫ a M ( a ) ⊗ k Mod B ( X , X ♯ ( a )) ≅ ∫ a M ( a ) ⊗ k A a ≅ M ( − ) ⊗ A X ⊣ Mod B ( X , − ) (-)\otimes_{\mathcal A}X\dashv\operatorname{Mod}_{\mathcal B}(X,-) ( − ) ⊗ A X ⊣ Mod B ( X , − ) Mod B ( X , − ) \operatorname{Mod}_{\mathcal B}(X,-) Mod B ( X , − ) Mod B ( X , M ) = 0 \operatorname{Mod}_{\mathcal B}(X,M)=0 Mod B ( X , M ) = 0 Mod B ( X ♯ ( a ) , M ) = 0 \operatorname{Mod}_{\mathcal B}(X^\sharp(a),M)=0 Mod B ( X ♯ ( a ) , M ) = 0 a ∈ A a\in\mathcal A a ∈ A M = 0 M=0 M = 0 □ \square □
:::
Corollary 3.13. Let P = ( P i ) i ∈ I ⊆ Mod A P=(P_i)_{i\in I}\subseteq\operatorname{Mod}_{\mathcal A} P = ( P i ) i ∈ I ⊆ Mod A End A ( P ) \operatorname{End}_{\mathcal A}(P) End A ( P ) Mod A \operatorname{Mod}_{\mathcal A} Mod A P i P_i P i A \mathcal A A ( End A ( P ) → Mod A ) b (\operatorname{End}_{\mathcal A}(P)\to\operatorname{Mod}_{\mathcal A})^b ( End A ( P ) → Mod A ) b
Transport of ring structure
If R R R k k k R R R R R R [1 ] or [2 Section 2.1] ). The monoid structures on an R R R A A A k k k R → A R\to A R → A
Definition 3.14. An A \mathcal A A u : A → C u:\mathcal A\to\mathcal C u : A → C A \mathcal A A Ring A \operatorname{Ring}_{\mathcal A} Ring A A ↓ k Cat \mathcal A\downarrow k\operatorname{Cat} A ↓ k Cat C \mathcal{C} C A \mathcal{A} A u u u A \mathcal{A} A u ( A ) ⊆ C u(\mathcal{A}) \subseteq\mathcal{C} u ( A ) ⊆ C
Construction 3.16. Given a functor Φ : Mod A → Mod B \Phi:\operatorname{Mod}_{\mathcal{A}} \to\operatorname{Mod}_{\mathcal{B}} Φ : Mod A → Mod B T Φ : Ring B → Ring A T_{\Phi}:\operatorname{Ring}_{\mathcal{B}} \to\operatorname{Ring}_{\mathcal{A}} T Φ : Ring B → Ring A P a = Φ ( A a ) ∈ Mod B P_a = \Phi(\mathcal{A}^a) \in\operatorname{Mod}_{\mathcal{B}} P a = Φ ( A a ) ∈ Mod B B \mathcal{B} B u : B → C u:\mathcal{B} \to\mathcal{C} u : B → C C u \mathcal{C}_u C u
Ob ( C u ) = Ob ( A ) \operatorname{Ob}(\mathcal{C}_u) = \operatorname{Ob}(\mathcal{A}) Ob ( C u ) = Ob ( A )
C u ( x , y ) = Mod C ( u ! ( P x ) , u ! ( P y ) ) \mathcal{C}_u(x,y) = \operatorname{Mod}_{\mathcal{C}}(u_!(P_x),u_!(P_y)) C u ( x , y ) = Mod C ( u ! ( P x ) , u ! ( P y )) Mod C \operatorname{Mod}_{\mathcal{C}} Mod C
This category comes with a functor u Φ : A → C u u_{\Phi}:\mathcal{A} \to\mathcal{C}_u u Φ : A → C u u ! ∘ Φ ∘ ↓ A u_! \circ\Phi\circ\mathord{\downarrow}_{\mathcal{A}} u ! ∘ Φ ∘ ↓ A T Φ ( C ) = C u T_{\Phi}(\mathcal{C}) = \mathcal{C}_u T Φ ( C ) = C u v : B → D v:\mathcal{B} \to\mathcal{D} v : B → D B \mathcal{B} B F : C → D F:\mathcal{C} \to\mathcal{D} F : C → D B \mathcal{B} B F F F F Φ : C u → C v F_{\Phi}:\mathcal{C}_u \to\mathcal{C}_v F Φ : C u → C v α ↦ θ ∘ F ! ( α ) ∘ θ − 1 \alpha\mapsto\theta\circ F_!(\alpha) \circ\theta^{-1} α ↦ θ ∘ F ! ( α ) ∘ θ − 1 θ : F ! ∘ u ! ⇒ v ! \theta:F_! \circ u_! \Rightarrow v_! θ : F ! ∘ u ! ⇒ v ! F ∘ u = v F \circ u = v F ∘ u = v T Φ ( F ) = F Φ T_{\Phi}(F) = F_{\Phi} T Φ ( F ) = F Φ
The following result is essentially the observation that the Yoneda embedding is dense.
Lemma 3.17. Suppose F , G : Mod A → C F,G:\operatorname{Mod}_{\mathcal{A}} \to\mathcal{C} F , G : Mod A → C C \mathcal{C} C
(1) F F F G G G
(2) F ∘ ↓ F \circ\mathord{\downarrow} F ∘ ↓ G ∘ ↓ G \circ\mathord{\downarrow} G ∘ ↓
Proof. It suffices to show that (2) implies (1). The co-Yoneda lemma gives the following isomorphisms that are natural in M ∈ Mod A M \in\operatorname{Mod}_{\mathcal{A}} M ∈ Mod A
F ( M ) ≅ F ( ∫ a M ( a ) ⊗ k A a ) ≅ ∫ a M ( a ) ⊗ k F ( A a ) ≅ ∫ a M ( a ) ⊗ k G ( A a ) ≅ G ( M ) . F(M) \cong F\left(\int^{a} M(a) \otimes_k \mathcal{A}^a\right) \cong\int^{a} M(a) \otimes_k F(\mathcal{A}^a) \cong\int^{a} M(a) \otimes_k G(\mathcal{A}^a) \cong G(M). F ( M ) ≅ F ( ∫ a M ( a ) ⊗ k A a ) ≅ ∫ a M ( a ) ⊗ k F ( A a ) ≅ ∫ a M ( a ) ⊗ k G ( A a ) ≅ G ( M ) . We will now show that Morita equivalences induce equivalences on ring structures. In bimodule language, this amounts to observing that conjugation by an invertible bimodule is a strong monoidal equivalence.
Theorem 3.18. If Φ : Mod A → Mod B \Phi:\operatorname{Mod}_{\mathcal{A}} \to\operatorname{Mod}_{\mathcal{B}} Φ : Mod A → Mod B T Φ : Ring B → Ring A T_{\Phi}:\operatorname{Ring}_{\mathcal{B}} \to\operatorname{Ring}_{\mathcal{A}} T Φ : Ring B → Ring A
Proof. Choose an adjoint quasi-inverse Ψ : Mod B → Mod A \Psi:\operatorname{Mod}_{\mathcal{B}} \to\operatorname{Mod}_{\mathcal{A}} Ψ : Mod B → Mod A T Ψ T_{\Psi} T Ψ T Φ T_{\Phi} T Φ u : B → C u:\mathcal{B} \to\mathcal{C} u : B → C B \mathcal{B} B P a = Φ ( A a ) P_a = \Phi(\mathcal{A}^a) P a = Φ ( A a ) a ∈ A a \in\mathcal{A} a ∈ A
J u : C u → Mod C , a ↦ u ! ( P a ) J_u:\mathcal{C}_u \to\operatorname{Mod}_{\mathcal{C}}, \qquad a \mapsto u_!(P_a) J u : C u → Mod C , a ↦ u ! ( P a ) the canonical functor which is the identity on morphisms. From the adjunction
Mod C ( u ! ( P a ) , N ) ≅ Mod B ( P a , u ∗ ( N ) ) \operatorname{Mod}_{\mathcal{C}}(u_!(P_a),N) \cong\operatorname{Mod}_{\mathcal{B}}(P_a,u^*(N)) Mod C ( u ! ( P a ) , N ) ≅ Mod B ( P a , u ∗ ( N )) we deduce that the family u ! ( P a ) u_!(P_a) u ! ( P a ) Mod C \operatorname{Mod}_{\mathcal{C}} Mod C L u : = ( − ) ⊗ C u J u b : Mod C u → Mod C L_u := (-) \otimes_{\mathcal{C}_u} J_u^b:\operatorname{Mod}_{\mathcal{C}_u} \to\operatorname{Mod}_{\mathcal{C}} L u := ( − ) ⊗ C u J u b : Mod C u → Mod C
Now let v = u Φ : A → C u v = u_{\Phi}:\mathcal{A} \to\mathcal{C}_u v = u Φ : A → C u L u ∘ v ! ≅ u ! ∘ Φ L_u \circ v_! \cong u_! \circ\Phi L u ∘ v ! ≅ u ! ∘ Φ
L u ( v ! ( A a ) ) ≅ L u ( C u a ) ≅ J u ( a ) = u ! ( P a ) = u ! ( Φ ( A a ) ) . L_u(v_!(\mathcal{A}^a)) \cong L_u(\mathcal{C}_u^a) \cong J_u(a) = u_!(P_a) = u_!(\Phi(\mathcal{A}^a)). L u ( v ! ( A a )) ≅ L u ( C u a ) ≅ J u ( a ) = u ! ( P a ) = u ! ( Φ ( A a )) . Hence, by Lemma 3.17 this extends to L u ∘ v ! ≅ u ! ∘ Φ L_u \circ v_! \cong u_! \circ\Phi L u ∘ v ! ≅ u ! ∘ Φ ( C u ) v (\mathcal{C}_u)_v ( C u ) v v ! ( Ψ ( B b ) ) v!(\Psi(\mathcal{B}^b)) v ! ( Ψ ( B b )) b ∈ B b \in\mathcal{B} b ∈ B
L u ∘ v ! ∘ Φ ∘ ≅ B ≅ u ! ∘ Φ ∘ Ψ ∘ ≅ B ≅ u ! ∘ ≅ B ≅ ≅ C ∘ u L_u \circ v_! \circ\Phi\circ\mathop{\cong}_{\mathcal{B}} \cong u_! \circ\Phi\circ\Psi\circ\mathop{\cong}_{\mathcal{B}} \cong u_! \circ\cong_{\mathcal{B}} \cong\cong_{\mathcal{C}} \circ u L u ∘ v ! ∘ Φ ∘ ≅ B ≅ u ! ∘ Φ ∘ Ψ ∘ ≅ B ≅ u ! ∘ ≅ B ≅ ≅ C ∘ u and the fact that L u L_u L u
C ⟶ ( C u ) v , u ( b ) ⟼ b . \mathcal{C} \longrightarrow(\mathcal{C}_u)_v,\qquad u(b) \longmapsto b. C ⟶ ( C u ) v , u ( b ) ⟼ b . Since u u u B \mathcal{B} B u u u T Ψ ∘ T Φ ≅ id T_{\Psi} \circ T_{\Phi} \cong\operatorname{id} T Ψ ∘ T Φ ≅ id Φ \Phi Φ Ψ \Psi Ψ
Change of coefficients
Our notion of Morita equivalence relies on categories of the form Fun k ( A o p , k Mod ) \operatorname{Fun}_k(\mathcal{A}^{\mathrm{op}},k\operatorname{Mod}) Fun k ( A op , k Mod ) k Mod k\operatorname{Mod} k Mod
Definition 3.19. Let A \mathcal{A} A A ^ \widehat{\mathcal{A}} A e e e im e ∈ Mod A \operatorname{im} e \in\operatorname{Mod}_{\mathcal{A}} im e ∈ Mod A A ^ \widehat{\mathcal{A}} A Mod A \operatorname{Mod}_{\mathcal{A}} Mod A im e \operatorname{im} e im e e e e
Proposition 3.20. Let C \mathcal{C} C k k k A ↪ A ^ \mathcal{A} \hookrightarrow\widehat{\mathcal{A}} A ↪ A k k k Fun k ( A ^ o p , C ) ≃ Fun k ( A o p , C ) \operatorname{Fun}_k(\widehat{\mathcal{A}}^{\mathrm{op}},\mathcal{C}) \simeq\operatorname{Fun}_k(\mathcal{A}^{\mathrm{op}},\mathcal{C}) Fun k ( A op , C ) ≃ Fun k ( A op , C )
Proof. This amounts to observing that a k k k C \mathcal{C} C C \mathcal{C} C [4 ] for more details. □
Corollary 3.21. Suppose A \mathcal{A} A B \mathcal{B} B k k k C \mathcal{C} C Fun k ( A o p , C ) ≃ Fun k ( B o p , C ) \operatorname{Fun}_k(\mathcal{A}^{\mathrm{op}},\mathcal{C}) \simeq\operatorname{Fun}_k(\mathcal{B}^{\mathrm{op}},\mathcal{C}) Fun k ( A op , C ) ≃ Fun k ( B op , C )
Proof. Let F : Mod A → Mod B F:\operatorname{Mod}_{\mathcal{A}} \to\operatorname{Mod}_{\mathcal{B}} F : Mod A → Mod B F F F A ^ ≃ B ^ \widehat{\mathcal{A}} \simeq\widehat{\mathcal{B}} A ≃ B
Fun k ( A o p , C ) ≃ Fun k ( A ^ o p , C ) ≃ Fun k ( B ^ o p , C ) ≃ Fun k ( B o p , C ) . \operatorname{Fun}_k(\mathcal{A}^{\mathrm{op}},\mathcal{C}) \simeq\operatorname{Fun}_k(\widehat{\mathcal{A}}^{\mathrm{op}},\mathcal{C}) \simeq\operatorname{Fun}_k(\widehat{\mathcal{B}}^{\mathrm{op}},\mathcal{C}) \simeq\operatorname{Fun}_k(\mathcal{B}^{\mathrm{op}},\mathcal{C}). Fun k ( A op , C ) ≃ Fun k ( A op , C ) ≃ Fun k ( B op , C ) ≃ Fun k ( B op , C ) . □
Morita normalization
The Dold–Puppe projectors
The basis of our normalization approach is a family of idempotent projectors which can be constructed from the counits of the lower and upper décalage comonads.
Construction 4.1. Let X : Δ o p → k Mod X:\Delta^{\mathrm{op}} \to k\operatorname{Mod} X : Δ op → k Mod K ( X ) = ker ( d min : Dec 0 ( X ) ⇒ X ) K(X)=\ker(d_{\min}:\operatorname{Dec}_0(X)\Rightarrow X) K ( X ) = ker ( d m i n : Dec 0 ( X ) ⇒ X ) n n n
0 ⟶ K ( X ) n ⟶ Dec 0 ( X ) n → ( d min ) n X n ⟶ 0 0 \longrightarrow K(X)_n \longrightarrow\operatorname{Dec}_0(X)_n \xrightarrow{(d_{\min})_n} X_n \longrightarrow0 0 ⟶ K ( X ) n ⟶ Dec 0 ( X ) n ( d m i n ) n X n ⟶ 0 which is right split by s 0 s_0 s 0 X n + 1 = Dec 0 ( X ) n → K ( X ) n X_{n+1}=\operatorname{Dec}_0(X)_n \to K(X)_n X n + 1 = Dec 0 ( X ) n → K ( X ) n id − s 0 ∘ d 0 \operatorname{id}-s_0\circ d_0 id − s 0 ∘ d 0
K 0 ( X ) n ⇄ K 1 ( X ) n − 1 ⇄ K 2 ( X ) n − 2 ⇄ ⋯ ⇄ K n − 1 ( X ) 1 ⇄ K n ( X ) 0 K^0(X)_n \rightleftarrows K^1(X)_{n-1} \rightleftarrows K^2(X)_{n-2} \rightleftarrows\cdots\rightleftarrows K^{n-1}(X)_1 \rightleftarrows K^n(X)_0 K 0 ( X ) n ⇄ K 1 ( X ) n − 1 ⇄ K 2 ( X ) n − 2 ⇄ ⋯ ⇄ K n − 1 ( X ) 1 ⇄ K n ( X ) 0 which have the same form. Since Dec 0 \operatorname{Dec}_0 Dec 0 K K K K 0 ( X ) n = X n K^0(X)_n=X_n K 0 ( X ) n = X n
( id − s n − 1 ∘ d n − 1 ) ∘ ⋯ ∘ ( id − s 1 ∘ d 1 ) ∘ ( id − s 0 ∘ d 0 ) . (\operatorname{id}-s_{n-1}\circ d_{n-1})\circ\cdots\circ(\operatorname{id}-s_1\circ d_1)\circ(\operatorname{id}-s_0\circ d_0). ( id − s n − 1 ∘ d n − 1 ) ∘ ⋯ ∘ ( id − s 1 ∘ d 1 ) ∘ ( id − s 0 ∘ d 0 ) . The analogous construction with Dec 0 ( X ) \operatorname{Dec}^{0}(X) Dec 0 ( X ) ker ( d max : Dec 0 ( X ) ⇒ X ) \ker(d_{\max}:\operatorname{Dec}^{0}(X)\Rightarrow X) ker ( d m a x : Dec 0 ( X ) ⇒ X )
( id − s 0 ∘ d 1 ) ∘ ( id − s 1 ∘ d 2 ) ∘ ⋯ ∘ ( id − s n − 1 ∘ d n ) . (\operatorname{id}-s_{0}\circ d_{1})\circ(\operatorname{id}-s_{1}\circ d_{2})\circ\cdots\circ(\operatorname{id}-s_{n-1}\circ d_{n}). ( id − s 0 ∘ d 1 ) ∘ ( id − s 1 ∘ d 2 ) ∘ ⋯ ∘ ( id − s n − 1 ∘ d n ) . Under the Yoneda lemma, these projectors correspond to certain morphisms in k [ Δ ] k[\Delta] k [ Δ ]
Definition 4.2. For n ∈ N n\in\mathbb{N} n ∈ N
p n : = ( id − δ 0 ∘ σ 0 ) ∘ ( id − δ 1 ∘ σ 1 ) ∘ ⋯ ∘ ( id − δ n − 1 ∘ σ n − 1 ) ∈ k [ Δ max ] p_{n}:=(\operatorname{id}-\delta_{0}\circ\sigma_{0})\circ(\operatorname{id}-\delta_{1}\circ\sigma_{1})\circ\cdots\circ(\operatorname{id}-\delta_{n-1}\circ\sigma_{n-1})\in k[\Delta_{\max}] p n := ( id − δ 0 ∘ σ 0 ) ∘ ( id − δ 1 ∘ σ 1 ) ∘ ⋯ ∘ ( id − δ n − 1 ∘ σ n − 1 ) ∈ k [ Δ m a x ] q n : = ( id − δ n ∘ σ n − 1 ) ∘ ( id − δ n − 1 ∘ σ n − 2 ) ∘ ⋯ ∘ ( id − δ 1 ∘ σ 0 ) ∈ k [ Δ min ] . q_{n}:=(\operatorname{id}-\delta_{n}\circ\sigma_{n-1})\circ(\operatorname{id}-\delta_{n-1}\circ\sigma_{n-2})\circ\cdots\circ(\operatorname{id}-\delta_{1}\circ\sigma_{0})\in k[\Delta_{\min}]. q n := ( id − δ n ∘ σ n − 1 ) ∘ ( id − δ n − 1 ∘ σ n − 2 ) ∘ ⋯ ∘ ( id − δ 1 ∘ σ 0 ) ∈ k [ Δ m i n ] . By convention, p 0 p_{0} p 0 q 0 q_{0} q 0 id 0 \operatorname{id}_{0} id 0
Dold–Kan correspondences
Throughout this section, we will fix an object
( u : k [ Δ max ] → C ) ∈ Ring k [ Δ max ] . (u:k[\Delta_{\max}]\to\mathcal{C})\in\operatorname{Ring}_{k[\Delta_{\max}]}. ( u : k [ Δ m a x ] → C ) ∈ Ring k [ Δ m a x ] . All results carry over to Ring k [ Δ min ] \operatorname{Ring}_{k[\Delta_{\min}]} Ring k [ Δ m i n ]
( − ) ∘ k [ Rev ] : Ring k [ Δ min ] → Ring k [ Δ max ] . (-)\circ k[\operatorname{Rev}]:\operatorname{Ring}_{k[\Delta_{\min}]}\to\operatorname{Ring}_{k[\Delta_{\max}]}. ( − ) ∘ k [ Rev ] : Ring k [ Δ m i n ] → Ring k [ Δ m a x ] . Under this identification, we replace p n p_{n} p n Rev ( p n ) = q n \operatorname{Rev}(p_{n})=q_{n} Rev ( p n ) = q n
Definition 4.6. Let X ∈ Mod C X\in\operatorname{Mod}_{\mathcal{C}} X ∈ Mod C degenerate submodule in degree n ∈ N n\in\mathbb{N} n ∈ N k k k D n ( X ) : = ∑ j < n im s j n ⊆ X n D_{n}(X):=\sum_{j<n}\operatorname{im}s^{n}_{j}\subseteq X_{n} D n ( X ) := ∑ j < n im s j n ⊆ X n ( D n : Mod C → k Mod ) n ∈ N (D_{n}:\operatorname{Mod}_{\mathcal{C}}\to k\operatorname{Mod})_{n\in\mathbb{N}} ( D n : Mod C → k Mod ) n ∈ N
Proposition 4.7. The morphisms id − p n \operatorname{id}-p_{n} id − p n are idempotent and exhibit D n D_{n} D n as a retract of the evaluation functor ( − ) n (-)_{n} ( − ) n
Proof. By Remark 4.5 we may write p n − i d = ∑ i < n b i ∘ σ i p_n-\mathrm{id}=\sum_{i<n} b_i\circ\sigma_i p n − id = ∑ i < n b i ∘ σ i Remark 2.8 , we see that σ i ∘ p n = 0 \sigma_i\circ p_n=0 σ i ∘ p n = 0 i < n i<n i < n ( p n − i d ) ∘ p n = 0 (p_n-\mathrm{id})\circ p_n=0 ( p n − id ) ∘ p n = 0 D n ( X ) = ker X ( p n ) D_n(X)=\ker X(p_n) D n ( X ) = ker X ( p n ) X ∈ M o d C X\in\mathbf{Mod}_{\mathcal{C}} X ∈ Mod C
Proposition 4.8. The morphism p n p_n p n C n ≃ P n ⊕ Q n \mathcal{C}_n\simeq P_n\oplus Q_n C n ≃ P n ⊕ Q n M o d C \mathbf{Mod}_{\mathcal{C}} Mod C
P n ≔ im C ( p n ) = ⋂ i = 0 n − 1 ker C ( σ i n ) and Q n ≔ ker C ( p n ) = ∑ i = 0 n − 1 im C ( δ i n ) . P_n\coloneqq\operatorname{im}_{\mathcal{C}}(p_n)=\bigcap_{i=0}^{n-1}\ker_{\mathcal{C}}(\sigma_i^n)
\qquad\text{and}\qquad
Q_n\coloneqq\ker_{\mathcal{C}}(p_n)=\sum_{i=0}^{n-1}\operatorname{im}_{\mathcal{C}}(\delta_i^n). P n : = im C ( p n ) = i = 0 ⋂ n − 1 ker C ( σ i n ) and Q n : = ker C ( p n ) = i = 0 ∑ n − 1 im C ( δ i n ) . Proof. The proof of Proposition 4.7 already shows im C ( p n ) ⊆ ⋂ i < n ker C ( σ i ) \operatorname{im}_{\mathcal{C}}(p_n)\subseteq\bigcap_{i<n}\ker_{\mathcal{C}}(\sigma_i) im C ( p n ) ⊆ ⋂ i < n ker C ( σ i ) σ i ∘ x = 0 \sigma_i\circ x=0 σ i ∘ x = 0 i < n i<n i < n p n ∘ x = x p_n\circ x=x p n ∘ x = x im C ( p n ) = ⋂ i < n ker C ( σ i ) \operatorname{im}_{\mathcal{C}}(p_n)=\bigcap_{i<n}\ker_{\mathcal{C}}(\sigma_i) im C ( p n ) = ⋂ i < n ker C ( σ i ) ker p n \ker p_n ker p n Remark 4.5 we may write p n − i d = ∑ i < n δ i ∘ a i p_n-\mathrm{id}=\sum_{i<n}\delta_i\circ a_i p n − id = ∑ i < n δ i ∘ a i Remark 2.8 we deduce p n ∘ δ i = 0 p_n\circ\delta_i=0 p n ∘ δ i = 0 i < n i<n i < n ker C ( p n ) = im C ( p n − i d ) \ker_{\mathcal{C}}(p_n)=\operatorname{im}_{\mathcal{C}}(p_n-\mathrm{id}) ker C ( p n ) = im C ( p n − id )
Lemma 4.9. The family ( P n ) n ∈ N (P_n)_{n\in\mathbb{N}} ( P n ) n ∈ N M o d C \mathbf{Mod}_{\mathcal{C}} Mod C
Proof. Each P n P_n P n Proposition 3.9 ). For X ∈ M o d C X\in\mathbf{Mod}_{\mathcal{C}} X ∈ Mod C M o d C ( P n , X ) ≅ X n / D n ( X ) \mathbf{Mod}_{\mathcal{C}}(P_n,X)\cong X_n/D_n(X) Mod C ( P n , X ) ≅ X n / D n ( X ) X ≠ 0 X\ne0 X = 0 n ∈ N n\in\mathbb{N} n ∈ N X n ≠ 0 X_n\ne0 X n = 0 D n ( X ) = 0 D_n(X)=0 D n ( X ) = 0 M o d C ( P n , X ) ≠ 0 \mathbf{Mod}_{\mathcal{C}}(P_n,X)\ne0 Mod C ( P n , X ) = 0 ( P n ) (P_n) ( P n )
Definition 4.10. The normalization of C \mathcal{C} C N ( C ) \mathcal{N}(\mathcal{C}) N ( C )
Ob N ( C ) = N \operatorname{Ob}\mathcal{N}(\mathcal{C})=\mathbb{N} Ob N ( C ) = N
N ( C ) ( m , n ) = M o d C ( P u ( m ) , P u ( n ) ) \mathcal{N}(\mathcal{C})(m,n)=\mathbf{Mod}_{\mathcal{C}}(P_{u(m)},P_{u(n)}) N ( C ) ( m , n ) = Mod C ( P u ( m ) , P u ( n ) ) M o d C \mathbf{Mod}_{\mathcal{C}} Mod C
If C = k [ D ] \mathcal{C}=k[\mathcal{D}] C = k [ D ] D \mathcal{D} D N ( D ) ≔ N ( C ) \mathcal{N}(\mathcal{D})\coloneqq\mathcal{N}(\mathcal{C}) N ( D ) : = N ( C ) u = i d k [ Δ max ] u=\mathrm{id}_{k[\Delta_{\max}]} u = id k [ Δ m a x ] J : N ( Δ max ) ↪ M o d k [ Δ max ] J:\mathcal{N}(\Delta_{\max})\hookrightarrow\mathbf{Mod}_{k[\Delta_{\max}]} J : N ( Δ m a x ) ↪ Mod k [ Δ m a x ] Φ = ( − ) ⊗ N ( Δ max ) J b \Phi=(-)\otimes_{\mathcal{N}(\Delta_{\max})}J^{\mathrm{b}} Φ = ( − ) ⊗ N ( Δ m a x ) J b
N ≔ T Φ : R i n g k [ Δ max ] ⟶ R i n g N ( Δ max ) . \mathcal{N}\coloneqq T_{\Phi}:\mathbf{Ring}_{k[\Delta_{\max}]}\longrightarrow\mathbf{Ring}_{\mathcal{N}(\Delta_{\max})}. N : = T Φ : Ring k [ Δ m a x ] ⟶ Ring N ( Δ m a x ) . Extension of scalars sends the image of p n p_n p n u ( p n ) u(p_n) u ( p n ) N ( C ) \mathcal{N}(\mathcal{C}) N ( C )
Theorem 4.11 (Abstract Dold–Kan correspondence).
The categories C \mathcal{C} C N ( C ) \mathcal{N}(\mathcal{C}) N ( C ) Fun k ( C o p , A ) \operatorname{Fun}_k(\mathcal{C}^{\mathrm{op}},\mathcal{A}) Fun k ( C op , A ) Fun k ( N ( C ) o p , A ) \operatorname{Fun}_k(\mathcal{N}(\mathcal{C})^{\mathrm{op}},\mathcal{A}) Fun k ( N ( C ) op , A ) A \mathcal{A} A
The functor N : R i n g k [ Δ max ] ⟶ R i n g N ( Δ max ) \mathcal{N}:\mathbf{Ring}_{k[\Delta_{\max}]}\longrightarrow\mathbf{Ring}_{\mathcal{N}(\Delta_{\max})} N : Ring k [ Δ m a x ] ⟶ Ring N ( Δ m a x )
Proof. (1) follows from Proposition 3.12 and Lemma 4.9 applied to the bimodule
( N ( C ) ⟶ M o d C ) b ∈ N ( C ) M o d C , (\mathcal{N}(\mathcal{C})\longrightarrow\mathbf{Mod}_{\mathcal{C}})^{\mathrm{b}}\in{}_{\mathcal{N}(\mathcal{C})}\mathbf{Mod}_{\mathcal{C}}, ( N ( C ) ⟶ Mod C ) b ∈ N ( C ) Mod C , and then applying Corollary 3.21 . (2) follows by Theorem 3.18 .
Construction 4.12. We give a description of N ( C ) \mathcal{N}(\mathcal{C}) N ( C ) C ⟶ M o d C \mathcal{C}\longrightarrow\mathbf{Mod}_{\mathcal{C}} C ⟶ Mod C k k k
p n ∘ C ( m , n ) ∘ p m ⟶ ∼ N ( C ) ( m , n ) . p_n\circ\mathcal{C}(m,n)\circ p_m\overset{\sim}{\longrightarrow}\mathcal{N}(\mathcal{C})(m,n). p n ∘ C ( m , n ) ∘ p m ⟶ ∼ N ( C ) ( m , n ) . Denote by L ≔ p n ∘ ( − ) L\coloneqq p_n\circ(-) L : = p n ∘ ( − ) R ≔ ( − ) ∘ p m R\coloneqq(-)\circ p_m R : = ( − ) ∘ p m C ( m , n ) \mathcal{C}(m,n) C ( m , n ) Proposition 4.8 shows ker L = ∑ i = 0 n − 1 δ i ∘ C ( m , n − 1 ) \ker L=\sum_{i=0}^{n-1}\delta_i\circ\mathcal{C}(m,n-1) ker L = ∑ i = 0 n − 1 δ i ∘ C ( m , n − 1 ) Proposition 4.7 shows ker R = ∑ j = 0 m − 1 C ( m − 1 , n ) ∘ σ j \ker R=\sum_{j=0}^{m-1}\mathcal{C}(m-1,n)\circ\sigma_j ker R = ∑ j = 0 m − 1 C ( m − 1 , n ) ∘ σ j R ∘ L = L ∘ R R\circ L=L\circ R R ∘ L = L ∘ R
N ( C ) ( m , n ) ≅ C ( m , n ) / ( ker L + ker R ) . \mathcal{N}(\mathcal{C})(m,n)\cong\mathcal{C}(m,n)/(\ker L+\ker R). N ( C ) ( m , n ) ≅ C ( m , n ) / ( ker L + ker R ) . Under this identification, composition of the classes of f ∈ C ( m , r ) f \in\mathcal{C}(m,r) f ∈ C ( m , r ) g ∈ C ( r , n ) g \in\mathcal{C}(r,n) g ∈ C ( r , n ) [ g ] ∘ [ f ] = [ g ∘ p r ∘ f ] [g] \circ[f] = [g \circ p_r \circ f] [ g ] ∘ [ f ] = [ g ∘ p r ∘ f ]
Even more is true if u : k [ Δ max ] → C u\colon k[\Delta_{\max}] \to\mathcal{C} u : k [ Δ m a x ] → C Δ max → D \Delta_{\max} \to\mathcal{D} Δ m a x → D ker L \ker L ker L ker R \ker R ker R D ( m , n ) \mathcal{D}(m,n) D ( m , n ) N ( D ) ( m , n ) \mathcal{N}(\mathcal{D})(m,n) N ( D ) ( m , n ) f ∈ D ( m , n ) f \in\mathcal{D}(m,n) f ∈ D ( m , n ) [ f ] = 0 [f] = 0 [ f ] = 0
f = δ i ∘ f ′ f = \delta_i \circ f' f = δ i ∘ f ′ i < n i < n i < n
f = f ′ ∘ σ j f = f' \circ\sigma_j f = f ′ ∘ σ j j < m j < m j < m
Construction 4.13. We give a description of the equivalence M o d C → M o d N ( C ) \mathbf{Mod}_{\mathcal{C}} \to\mathbf{Mod}_{\mathcal{N}(\mathcal{C})} Mod C → Mod N ( C ) ( N ( C ) → M o d C ) b (\mathcal{N}(\mathcal{C}) \to\mathbf{Mod}_{\mathcal{C}})^{\mathrm{b}} ( N ( C ) → Mod C ) b X ∈ M o d C X \in\mathbf{Mod}_{\mathcal{C}} X ∈ Mod C N ( X ) ∈ M o d N ( C ) \mathrm{N}(X) \in\mathbf{Mod}_{\mathcal{N}(\mathcal{C})} N ( X ) ∈ Mod N ( C )
N ( X ) n = M o d C ( P n , X ) ≅ X n / D n ( X ) . \mathrm{N}(X)_n = \mathbf{Mod}_{\mathcal{C}}(P_n,X) \cong X_n/D_n(X). N ( X ) n = Mod C ( P n , X ) ≅ X n / D n ( X ) . Under this identification, the right action of N ( C ) \mathcal{N}(\mathcal{C}) N ( C ) x ∈ X n x \in X_n x ∈ X n f ∈ C ( m , n ) f \in\mathcal{C}(m,n) f ∈ C ( m , n ) [ x ] ◃ [ f ] = [ x ◃ ( p n ∘ f ) ] [x] \triangleleft[f] = [x \triangleleft(p_n \circ f)] [ x ] ◃ [ f ] = [ x ◃ ( p n ∘ f )]
Example 4.14 (The trivial case). We consider u = id k [ Δ max ] u = \operatorname{id}_{k[\Delta_{\max}]} u = id k [ Δ m a x ] φ ∈ Δ max ( m , n ) \varphi\in\Delta_{\max}(m,n) φ ∈ Δ m a x ( m , n ) [ φ ] ≠ 0 [\varphi] \ne0 [ φ ] = 0 [ n − 1 ] ⊆ φ ( [ m ] ) [n-1] \subseteq\varphi([m]) [ n − 1 ] ⊆ φ ([ m ]) φ \varphi φ N ( Δ max ) \mathcal{N}(\Delta_{\max}) N ( Δ m a x ) M o d N ( Δ max ) \mathbf{Mod}_{\mathcal{N}(\Delta_{\max})} Mod N ( Δ m a x ) N \mathbb{N} N k k k
Example 4.15 (The classical case). We consider the inclusion u = ( k [ Δ max ] ↪ k [ Δ ] ) u = (k[\Delta_{\max}] \hookrightarrow k[\Delta]) u = ( k [ Δ m a x ] ↪ k [ Δ ]) φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) [ φ ] ≠ 0 [\varphi] \ne0 [ φ ] = 0 [ n − 1 ] ⊆ φ ( [ m ] ) [n-1] \subseteq\varphi([m]) [ n − 1 ] ⊆ φ ([ m ]) φ \varphi φ
N ( Δ ) ( m , n ) ≅ { ⟨ [ id ] ⟩ if m = n , ⟨ [ δ n ] ⟩ if m = n − 1 , 0 otherwise. \mathcal{N}(\Delta)(m,n) \cong
\begin{cases}
\langle[\operatorname{id}] \rangle& \text{if } m=n,\\
\langle[\delta_n] \rangle& \text{if } m=n-1,\\
0 & \text{otherwise.}
\end{cases} N ( Δ ) ( m , n ) ≅ ⎩ ⎨ ⎧ ⟨[ id ]⟩ ⟨[ δ n ]⟩ 0 if m = n , if m = n − 1 , otherwise. In view of Construction 4.13, we compute p n ∘ δ n p_n \circ\delta_n p n ∘ δ n (1) yield
p n ∘ δ n = ( ∑ i = 0 n ( − 1 ) i δ n − i ) ∘ p n − 1 . p_n \circ\delta_n = \left(\sum_{i=0}^{n}(-1)^i\delta_{n-i}\right)\circ p_{n-1}. p n ∘ δ n = ( i = 0 ∑ n ( − 1 ) i δ n − i ) ∘ p n − 1 . If δ n \delta_n δ n ( − 1 ) n δ n (-1)^n\delta_n ( − 1 ) n δ n [ x ] ◃ [ ( − 1 ) n δ n ] = [ x ◃ ∑ i = 0 n ( − 1 ) i δ i ] [x] \triangleleft[(-1)^n\delta_n] = [x \triangleleft\sum_{i=0}^{n}(-1)^i\delta_i] [ x ] ◃ [( − 1 ) n δ n ] = [ x ◃ ∑ i = 0 n ( − 1 ) i δ i ] N ( Δ ) \mathcal{N}(\Delta) N ( Δ ) C h ≥ 0 \mathbf{Ch}_{\geq0} Ch ≥ 0 ( ∂ n : n → n + 1 ) n ∈ N (\partial^n\colon n \to n+1)_{n\in\mathbb{N}} ( ∂ n : n → n + 1 ) n ∈ N ∂ n + 1 ∘ ∂ n = 0 \partial^{n+1}\circ\partial^n = 0 ∂ n + 1 ∘ ∂ n = 0 C h ≥ 0 \mathbf{Ch}_{\geq0} Ch ≥ 0 k M o d k\mathbf{Mod} k Mod
Fun ( Δ o p , k M o d ) ≃ M o d k [ Δ ] ≃ M o d C h ≥ 0 ≃ C h ≥ 0 ( k M o d ) . \operatorname{Fun}(\Delta^{\mathrm{op}},k\mathbf{Mod}) \simeq\mathbf{Mod}_{k[\Delta]} \simeq\mathbf{Mod}_{\mathbf{Ch}_{\geq0}} \simeq\mathbf{Ch}_{\geq0}(k\mathbf{Mod}). Fun ( Δ op , k Mod ) ≃ Mod k [ Δ ] ≃ Mod Ch ≥ 0 ≃ Ch ≥ 0 ( k Mod ) . Interaction with standard constructions
For this section we fix a functor u : k [ Δ max ] → C u\colon k[\Delta_{\max}] \to\mathcal{C} u : k [ Δ m a x ] → C u u u
Remark 4.16 (Base change and duality). Suppose we have a commutative diagram in k C a t k\mathbf{Cat} k Cat
C → F D u ↓ ↓ v k [ Δ max ] → Ψ A \begin{CD}
\mathcal{C} @>F>> \mathcal{D}\\
@VuVV @VVvV\\
k[\Delta_{\max}] @>\Psi>> \mathcal{A}
\end{CD} C u ↓ ⏐ k [ Δ m a x ] F Ψ D ↓ ⏐ v A where all functors are bijective on objects. Then v ∘ Ψ v \circ\Psi v ∘ Ψ k [ Δ max ] k[\Delta_{\max}] k [ Δ m a x ] N ( F ) : N ( C ) → N ( D ) \mathcal{N}(F): \mathcal{N}(\mathcal{C}) \to\mathcal{N}(\mathcal{D}) N ( F ) : N ( C ) → N ( D )
C o p → F C u o p ↑ ↑ u k [ Δ max ] o p → Ψ k [ Δ max ] \begin{CD}
\mathcal{C}^{\mathrm{op}} @>F>> \mathcal{C} \\
@A u^{\mathrm{op}} AA @AA u A \\
k[\Delta_{\max}]^{\mathrm{op}} @>\Psi>> k[\Delta_{\max}]
\end{CD} C op u op ⏐ ↑ k [ Δ m a x ] op F Ψ C ⏐ ↑ u k [ Δ m a x ] where F F F Ψ \Psi Ψ N ( C ) o p ≅ N ( C ) \mathcal{N}(\mathcal{C})^{\mathrm{op}} \cong\mathcal{N}(\mathcal{C}) N ( C ) op ≅ N ( C )
Remark 4.17 (Quotients). Let I ⊆ C I \subseteq\mathcal{C} I ⊆ C
I ‾ ( m , n ) = u ( p n ) ∘ I ( m , n ) ∘ u ( p m ) . \overline{I}(m,n)=u(p_n)\circ I(m,n)\circ u(p_m). I ( m , n ) = u ( p n ) ∘ I ( m , n ) ∘ u ( p m ) . Then I ‾ ⊆ N ( C ) \overline{I}\subseteq\mathcal{N}(\mathcal{C}) I ⊆ N ( C ) k [ Δ max ] → C → C / I k[\Delta_{\max}] \to\mathcal{C} \to\mathcal{C}/I k [ Δ m a x ] → C → C / I N ( C ) / I ‾ → N ( C / I ) \mathcal{N}(\mathcal{C})/\overline{I}\to\mathcal{N}(\mathcal{C}/I) N ( C ) / I → N ( C / I )
Remark 4.18 (Localization). Let η : id C ⇒ id C \eta:\operatorname{id}_{\mathcal{C}}\Rightarrow\operatorname{id}_{\mathcal{C}} η : id C ⇒ id C η ‾ : id N ( C ) ⇒ id N ( C ) \overline{\eta}:\operatorname{id}_{\mathcal{N}(\mathcal{C})}\Rightarrow\operatorname{id}_{\mathcal{N}(\mathcal{C})} η : id N ( C ) ⇒ id N ( C ) η ‾ n = u ( p n ) ∘ η n ∘ u ( p n ) \overline{\eta}_n=u(p_n)\circ\eta_n\circ u(p_n) η n = u ( p n ) ∘ η n ∘ u ( p n ) k [ Δ max ] → C → C [ η − 1 ] k[\Delta_{\max}]\to\mathcal{C}\to\mathcal{C}[\eta^{-1}] k [ Δ m a x ] → C → C [ η − 1 ] N ( C ) [ η ‾ − 1 ] → N ( C [ η − 1 ] ) \mathcal{N}(\mathcal{C})[\overline{\eta}^{-1}]\to\mathcal{N}(\mathcal{C}[\eta^{-1}]) N ( C ) [ η − 1 ] → N ( C [ η − 1 ])
Crossed simplicial groups
Crossed simplicial groups furnish Δ \Delta Δ Λ \Lambda Λ Definition 7.13 . We refer the reader to [18 , 11 , 8 ] for more details on the subject.
Preliminaries
Definition 5.1. A crossed simplicial group is a functor i : Δ → Δ G i:\Delta\to\Delta G i : Δ → Δ G f ∈ Δ G ( i ( m ) , i ( n ) ) f\in\Delta G(i(m),i(n)) f ∈ Δ G ( i ( m ) , i ( n )) f = i ( φ ) ∘ g f=i(\varphi)\circ g f = i ( φ ) ∘ g φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) g ∈ Aut Δ G ( i ( m ) ) g\in\operatorname{Aut}_{\Delta G}(i(m)) g ∈ Aut Δ G ( i ( m )) φ ∘ g \varphi\circ g φ ∘ g normal form of f f f Δ ↓ Cat \Delta\downarrow\operatorname{Cat} Δ ↓ Cat
Convention 5.2. By abuse of notation, we will usually assume that i i i Δ \Delta Δ i ( Δ ) ⊆ Δ G i(\Delta)\subseteq\Delta G i ( Δ ) ⊆ Δ G
Definition 5.3. Given a crossed simplicial group Δ G \Delta G Δ G G G G
Δ o p ⟶ ( Δ G ) o p → Δ G ( − , 0 ) Set . \Delta^{\mathrm{op}}\longrightarrow(\Delta G)^{\mathrm{op}}\xrightarrow{\Delta G(-,0)}\operatorname{Set}. Δ op ⟶ ( Δ G ) op Δ G ( − , 0 ) Set . The normal form of Δ G \Delta G Δ G G n = Δ G ( n , 0 ) ≅ Aut Δ G ( n ) G_n=\Delta G(n,0)\cong\operatorname{Aut}_{\Delta G}(n) G n = Δ G ( n , 0 ) ≅ Aut Δ G ( n ) h = g ◃ φ h=g\triangleleft\varphi h = g ◃ φ ω n ∘ h = ω m ∘ g ∘ φ \omega_n\circ h=\omega_m\circ g\circ\varphi ω n ∘ h = ω m ∘ g ∘ φ ω i ∈ Δ ( i , 0 ) \omega_i\in\Delta(i,0) ω i ∈ Δ ( i , 0 )
Definition 5.4. Let f : Δ G → Δ H f:\Delta G\to\Delta H f : Δ G → Δ H i : Δ → Δ G i:\Delta\to\Delta G i : Δ → Δ G j : Δ → Δ H j:\Delta\to\Delta H j : Δ → Δ H f f f
Δ K → Δ ↓ ↓ j Δ G → f Δ H \begin{CD}
\Delta K @>>> \Delta\\
@VVV @VV j V \\
\Delta G @>f>> \Delta H
\end{CD} Δ K ↓ ⏐ Δ G f Δ ↓ ⏐ j Δ H equipped with the universal functor Δ → Δ K \Delta\to\Delta K Δ → Δ K Δ ← i d Δ → i Δ G \Delta\xleftarrow{\mathrm{id}} \Delta\xrightarrow{i} \Delta G Δ id Δ i Δ G Δ K → Δ G \Delta K \to\Delta G Δ K → Δ G
Definition 5.5. A morphism of crossed simplicial groups p : Δ G → Δ H p\colon\Delta G \to\Delta H p : Δ G → Δ H extension , if it is full. Let Δ K \Delta K Δ K p p p
Δ K ⟶ Δ G → p Δ H \Delta K \longrightarrow\Delta G \xrightarrow{p} \Delta H Δ K ⟶ Δ G p Δ H induces a degreewise short exact sequence of groups
1 ⟶ K n ⟶ G n ⟶ H n ⟶ 1. 1 \longrightarrow K_n \longrightarrow G_n \longrightarrow H_n \longrightarrow1. 1 ⟶ K n ⟶ G n ⟶ H n ⟶ 1. Theorem 5.6. For every crossed simplicial group Δ G \Delta G Δ G Δ G → Δ H \Delta G \to\Delta H Δ G → Δ H Δ K \Delta K Δ K K K K Δ H \Delta H Δ H [8 Table 2] . This extension is unique up to isomorphism.
Proof. [8 Theorem I.7] . □
Proposition 5.7. If g ∘ σ i = h ∘ σ j g \circ\sigma_i = h \circ\sigma_j g ∘ σ i = h ∘ σ j i = j i = j i = j g = h g = h g = h g , h g,h g , h
Proof. From the cosimplicial identities σ i ∘ δ i = i d = σ i ∘ δ i + 1 \sigma_i \circ\delta_i = \mathrm{id} = \sigma_i \circ\delta_{i+1} σ i ∘ δ i = id = σ i ∘ δ i + 1
g ∘ σ i = h ∘ σ j ⟹ h ∘ σ j ∘ δ i = h ∘ σ j ∘ δ i + 1 ⟹ σ j ∘ δ i = σ j ∘ δ i + 1 . g \circ\sigma_i = h \circ\sigma_j \Longrightarrow h \circ\sigma_j \circ\delta_i = h \circ\sigma_j \circ\delta_{i+1} \Longrightarrow\sigma_j \circ\delta_i = \sigma_j \circ\delta_{i+1}. g ∘ σ i = h ∘ σ j ⟹ h ∘ σ j ∘ δ i = h ∘ σ j ∘ δ i + 1 ⟹ σ j ∘ δ i = σ j ∘ δ i + 1 . Evaluating these maps on i i i σ j ( i + 1 ) = σ j ( i ) \sigma_j(i+1) = \sigma_j(i) σ j ( i + 1 ) = σ j ( i ) i = j i = j i = j g = h g = h g = h
Definition 5.8. For g ∈ Aut Δ G ( n ) g \in\operatorname{Aut}_{\Delta G}(n) g ∈ Aut Δ G ( n ) j ∈ [ n − 1 ] j \in[n-1] j ∈ [ n − 1 ] degeneracy direction of g g g σ j ∘ g \sigma_j \circ g σ j ∘ g Δ G ( − , n − 1 ) ∣ Δ o p \Delta G(-,n-1)|_{\Delta^{\mathrm{op}}} Δ G ( − , n − 1 ) ∣ Δ op g g g D ( g ) ⊆ [ n − 1 ] D(g) \subseteq[n-1] D ( g ) ⊆ [ n − 1 ]
Proposition 5.9. Suppose g ∈ Aut Δ G ( n ) g \in\operatorname{Aut}_{\Delta G}(n) g ∈ Aut Δ G ( n ) g = h ◃ φ g = h \triangleleft\varphi g = h ◃ φ G G G h ∈ G r n d h \in G_r^{\mathrm{nd}} h ∈ G r nd g g g n − r n-r n − r
Proof. Let S ⊆ [ n − 1 ] S \subseteq[n-1] S ⊆ [ n − 1 ] j j j φ ( j ) = φ ( j + 1 ) \varphi(j) = \varphi(j+1) φ ( j ) = φ ( j + 1 ) ∣ S ∣ = n − r |S| = n-r ∣ S ∣ = n − r φ \varphi φ j ∈ S j \in S j ∈ S φ = φ j ∘ σ j \varphi= \varphi_j \circ\sigma_j φ = φ j ∘ σ j φ j ∈ Δ ( n − 1 , r ) \varphi_j \in\Delta(n-1,r) φ j ∈ Δ ( n − 1 , r ) h j = h ◃ φ j h_j = h \triangleleft\varphi_j h j = h ◃ φ j h j ∘ σ j = σ π ( j ) ∘ g h_j \circ\sigma_j = \sigma_{\pi(j)} \circ g h j ∘ σ j = σ π ( j ) ∘ g π ( j ) ∈ [ n − 1 ] \pi(j) \in[n-1] π ( j ) ∈ [ n − 1 ] j ↦ π ( j ) j \mapsto\pi(j) j ↦ π ( j ) S → D ( g ) S \to D(g) S → D ( g ) k ∈ D ( g ) k \in D(g) k ∈ D ( g ) σ k ∘ g = f ∘ σ j \sigma_k \circ g = f \circ\sigma_j σ k ∘ g = f ∘ σ j f f f g = f ◃ σ j g = f \triangleleft\sigma_j g = f ◃ σ j j ∈ S j \in S j ∈ S π ( j ) = k \pi(j) = k π ( j ) = k
Structure of the normalized category.
Theorem 5.10. Let g ∈ G m g \in G_m g ∈ G m r r r φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) [ φ ∘ g ] ≠ 0 [\varphi\circ g] \ne0 [ φ ∘ g ] = 0 N ( Δ G ) \mathcal{N}(\Delta G) N ( Δ G )
( r + 1 m − n + 1 ) \binom{r+1}{m-n+1} ( m − n + 1 r + 1 ) which we define to be zero when the lower index is outside the usual range. In particular, N ( Δ G ) ( m , n ) \mathcal{N}(\Delta G)(m,n) N ( Δ G ) ( m , n ) k k k
∑ r = 0 m ∣ G r n d ∣ ⋅ ( m r ) ⋅ ( r + 1 m − n + 1 ) . \sum_{r=0}^{m} |G_r^{\mathrm{nd}}| \cdot\binom{m}{r} \cdot\binom{r+1}{m-n+1}. r = 0 ∑ m ∣ G r nd ∣ ⋅ ( r m ) ⋅ ( m − n + 1 r + 1 ) . Proof. Let φ ∘ g ∈ Δ G ( m , n ) \varphi\circ g \in\Delta G(m,n) φ ∘ g ∈ Δ G ( m , n ) [ φ ∘ g ] ≠ 0 [\varphi\circ g] \ne0 [ φ ∘ g ] = 0 N ( Δ G ) \mathcal{N}(\Delta G) N ( Δ G ) [ n − 1 ] ⊆ im φ [n-1] \subseteq\operatorname{im}\varphi [ n − 1 ] ⊆ im φ φ ( j ) < φ ( j + 1 ) \varphi(j) < \varphi(j+1) φ ( j ) < φ ( j + 1 ) j ∈ D ( g ) j \in D(g) j ∈ D ( g ) g g g r r r s = ∣ D ( g ) ∣ = m − r s = \lvert D(g)\rvert= m-r s = ∣ D ( g )∣ = m − r φ \varphi φ
surjective onto [ n ] [n] [ n ] n n n
surjective onto [ n − 1 ] [n-1] [ n − 1 ] n − 1 n-1 n − 1
Since φ ( 0 ) = 0 \varphi(0) = 0 φ ( 0 ) = 0 φ \varphi φ s s s D ( g ) D(g) D ( g ) r r r
( r n − s ) + ( r n − 1 − s ) = ( r + 1 n − s ) = ( r + 1 m − n + 1 ) . \binom{r}{n-s}+\binom{r}{n-1-s}=\binom{r+1}{n-s}=\binom{r+1}{m-n+1}. ( n − s r ) + ( n − 1 − s r ) = ( n − s r + 1 ) = ( m − n + 1 r + 1 ) . Here, the first summand comes from case (1) and the second summand from case (2). If n = 0 n = 0 n = 0 h ∈ G r n d h \in G_r^{\mathrm{nd}} h ∈ G r nd g ∈ G m g \in G_m g ∈ G m ψ ∈ Δ ( m , r ) \psi\in\Delta(m,r) ψ ∈ Δ ( m , r ) ( m r ) \binom{m}{r} ( r m ) Δ ( m , r ) \Delta(m,r) Δ ( m , r ) □ \square □
Corollary 5.11. Let d m = max { r ≤ m : G r n d ≠ ∅ } d_m = \max\{r \le m : G_r^{\mathrm{nd}} \ne\varnothing\} d m = max { r ≤ m : G r nd = ∅ }
N ( Δ G ) ( m , n ) ≠ 0 ⟺ m − d m ≤ n ≤ m + 1. \mathcal{N}(\Delta G)(m,n) \ne0 \Longleftrightarrow m-d_m \le n \le m+1. N ( Δ G ) ( m , n ) = 0 ⟺ m − d m ≤ n ≤ m + 1. Proof. Note that r ≤ m r \le m r ≤ m G r n d ≠ ∅ G_r^{\mathrm{nd}} \ne\varnothing G r nd = ∅ 0 ≤ m − n + 1 ≤ r + 1 0 \le m-n+1 \le r+1 0 ≤ m − n + 1 ≤ r + 1 □ \square □
Corollary 5.12. The rank formula and EZ-decomposition imply
N ( Δ G ) ( m , 0 ) ≅ k [ G m n d ] \mathcal{N}(\Delta G)(m,0) \cong k[G_m^{\mathrm{nd}}] N ( Δ G ) ( m , 0 ) ≅ k [ G m nd ]
N ( Δ G ) ( m , m + 1 ) ≅ k [ G m ] \mathcal{N}(\Delta G)(m,m+1) \cong k[G_m] N ( Δ G ) ( m , m + 1 ) ≅ k [ G m ]
If G r n d ≠ ∅ G_r^{\mathrm{nd}} \ne\varnothing G r nd = ∅ N ( Δ G ) ( n + r , n ) ≠ 0 \mathcal{N}(\Delta G)(n+r,n) \ne0 N ( Δ G ) ( n + r , n ) = 0 n ∈ N n \in\mathbb{N} n ∈ N
In particular, all P m P_m P m N ( Δ G ) ( m , m + 1 ) ≅ Mod k [ Δ G ] ( P m , P m + 1 ) \mathcal{N}(\Delta G)(m,m+1) \cong\operatorname{Mod}_{k[\Delta G]}(P_m,P_{m+1}) N ( Δ G ) ( m , m + 1 ) ≅ Mod k [ Δ G ] ( P m , P m + 1 )
An interesting consequence of this observation is that modules of a crossed simplicial group cannot be equivalent to modules of a classical ring.
Corollary 5.13. k [ Δ G ] k[\Delta G] k [ Δ G ]
Proof. Suppose k [ Δ G ] k[\Delta G] k [ Δ G ] Mod k [ Δ G ] \operatorname{Mod}_{k[\Delta G]} Mod k [ Δ G ] M 1 , … , M n M_1,\ldots,M_n M 1 , … , M n M i M_i M i P j P_j P j S = P j 1 ⊕ ⋯ ⊕ P j r S = P_{j_1} \oplus\cdots\oplus P_{j_r} S = P j 1 ⊕ ⋯ ⊕ P j r M i M_i M i n > 1 + max { j 1 , … , j r } n > 1 + \max\{j_1,\ldots,j_r\} n > 1 + max { j 1 , … , j r } Mod k [ Δ G ] ( S , P n ) = 0 \operatorname{Mod}_{k[\Delta G]}(S,P_n) = 0 Mod k [ Δ G ] ( S , P n ) = 0 Mod k [ Δ G ] ( M i , P n ) = 0 \operatorname{Mod}_{k[\Delta G]}(M_i,P_n) = 0 Mod k [ Δ G ] ( M i , P n ) = 0 i i i P n P_n P n ( M i ) (M_i) ( M i ) □ \square □
Remark 5.14. We can visualize N ( Δ G ) \mathcal{N}(\Delta G) N ( Δ G ) k k k N ( Δ G ) ( m , n ) \mathcal{N}(\Delta G)(m,n) N ( Δ G ) ( m , n ) d = dim G d=\dim G d = dim G
Figure 5.14.
The entries above the diagonal are controlled by the Δ \Delta Δ Δ G \Delta G Δ G δ n \delta_n δ n G n − 1 G_{n-1} G n − 1 G G G Δ G \Delta G Δ G G r n ≠ ∅ G^{{n}}_r \ne\varnothing G r n = ∅ r r r
Classification of self-dual crossed simplicial groups
A particularly nice situation arises when N ( Δ G ) \mathcal{N}(\Delta G) N ( Δ G )
Lemma 5.15. Every equivalence ( Δ G ) o p ≃ Δ G (\Delta G)^{\mathrm{op}}\simeq\Delta G ( Δ G ) op ≃ Δ G
Proof. We construct the following invariants. By a chain of morphisms we mean a finite sequence ( f 1 , … , f r ) (f_1,\ldots,f_r) ( f 1 , … , f r ) f 1 f_1 f 1 f r f_r f r r r r
Let h ( n ) h(n) h ( n ) n ∈ Δ G n\in\Delta G n ∈ Δ G
Let d ( n ) d(n) d ( n ) n ∈ Δ G n\in\Delta G n ∈ Δ G
Clearly, we have h ( n ) = n = d ( n ) h(n)=n=d(n) h ( n ) = n = d ( n ) F : ( Δ G ) o p → Δ G F\colon(\Delta G)^{\mathrm{op}}\to\Delta G F : ( Δ G ) op → Δ G F F F h ( n ) h(n) h ( n ) d ( F ( n ) ) d(F(n)) d ( F ( n )) F ( n ) = d ( F ( n ) ) = h ( n ) = n F(n)=d(F(n))=h(n)=n F ( n ) = d ( F ( n )) = h ( n ) = n
Corollary 5.16. If ( Δ G ) o p ≃ Δ G (\Delta G)^{\mathrm{op}}\simeq\Delta G ( Δ G ) op ≃ Δ G dim G = 1 \dim G=1 dim G = 1
Proof. Let F : ( Δ G ) o p → Δ G F\colon(\Delta G)^{\mathrm{op}}\to\Delta G F : ( Δ G ) op → Δ G f ∈ Δ G ( n , 0 ) f\in\Delta G(n,0) f ∈ Δ G ( n , 0 ) n ≥ 2 n\ge2 n ≥ 2 F ( σ j ) = δ i ( j ) ∘ g j F(\sigma_j)=\delta_{i(j)}\circ g_j F ( σ j ) = δ i ( j ) ∘ g j F ( f ) = φ ∘ g F(f)=\varphi\circ g F ( f ) = φ ∘ g φ ∈ Δ ( 0 , n ) \varphi\in\Delta(0,n) φ ∈ Δ ( 0 , n ) j ↦ i ( j ) j\mapsto i(j) j ↦ i ( j ) i ( j ) = i ( ℓ ) i(j)=i(\ell) i ( j ) = i ( ℓ )
F ( σ j ) = F ( σ ℓ ) ∘ g ℓ − 1 ∘ g j . F(\sigma_j)=F(\sigma_\ell)\circ g_\ell^{-1}\circ g_j. F ( σ j ) = F ( σ ℓ ) ∘ g ℓ − 1 ∘ g j . Applying F − 1 F^{-1} F − 1 σ j = h ∘ σ ℓ \sigma_j=h\circ\sigma_\ell σ j = h ∘ σ ℓ h h h Proposition 5.7 implies j = ℓ j=\ell j = ℓ j j j i ( j ) ≠ φ ( 0 ) i(j)\ne\varphi(0) i ( j ) = φ ( 0 ) φ = δ i ( j ) ∘ φ ′ \varphi=\delta_{i(j)}\circ\varphi' φ = δ i ( j ) ∘ φ ′
f = F − 1 ( δ i ( j ) ∘ g j ∘ g j − 1 ∘ φ ′ ∘ g ) = F − 1 ( g j − 1 ∘ φ ′ ∘ g ) ∘ σ j f=F^{-1}\left(\delta_{i(j)}\circ g_j\circ g_j^{-1}\circ\varphi'\circ g\right)=F^{-1}\left(g_j^{-1}\circ\varphi'\circ g\right)\circ\sigma_j f = F − 1 ( δ i ( j ) ∘ g j ∘ g j − 1 ∘ φ ′ ∘ g ) = F − 1 ( g j − 1 ∘ φ ′ ∘ g ) ∘ σ j and thus f f f n = 1 n=1 n = 1 F ( σ 0 ) = δ i ( 0 ) ∘ g 0 F(\sigma_0)=\delta_{i(0)}\circ g_0 F ( σ 0 ) = δ i ( 0 ) ∘ g 0 φ ∘ g ∈ Δ G ( 0 , n ) \varphi\circ g\in\Delta G(0,n) φ ∘ g ∈ Δ G ( 0 , n ) φ ( 0 ) = i ( 0 ) \varphi(0)=i(0) φ ( 0 ) = i ( 0 ) F − 1 ( φ ∘ g ) F^{-1}(\varphi\circ g) F − 1 ( φ ∘ g ) g ∈ G 0 g\in G_0 g ∈ G 0 G 1 1 ≅ G 0 ≠ ∅ G^1_1\cong G_0\ne\varnothing G 1 1 ≅ G 0 = ∅ dim G = 1 \dim G=1 dim G = 1
Lemma 5.17. N ( Δ G ) \mathcal{N}(\Delta G) N ( Δ G ) skeletal .
Proof. If m ≅ n m\cong n m ≅ n N ( Δ G ) ( m , x ) ≠ 0 ⟺ N ( Δ G ) ( n , x ) ≠ 0 \mathcal{N}(\Delta G)(m,x)\ne0\Longleftrightarrow\mathcal{N}(\Delta G)(n,x)\ne0 N ( Δ G ) ( m , x ) = 0 ⟺ N ( Δ G ) ( n , x ) = 0 x x x x x x N ( Δ G ) ( m , x ) ≠ 0 \mathcal{N}(\Delta G)(m,x)\ne0 N ( Δ G ) ( m , x ) = 0 x = m + 1 x=m+1 x = m + 1 m + 1 = n + 1 m+1=n+1 m + 1 = n + 1 m = n m=n m = n
Theorem 5.18. Every equivalence N ( Δ G ) o p ≃ N ( Δ G ) \mathcal{N}(\Delta G)^{\mathrm{op}}\simeq\mathcal{N}(\Delta G) N ( Δ G ) op ≃ N ( Δ G )
Proof. Let F : N ( Δ G ) o p → N ( Δ G ) F:\mathcal{N}(\Delta G)^{\mathrm{op}}\to\mathcal{N}(\Delta G) F : N ( Δ G ) op → N ( Δ G ) N ( Δ G ) \mathcal{N}(\Delta G) N ( Δ G ) F F F r ∈ N r\in\mathbb{N} r ∈ N F ( r ) = F ( 0 ) + 1 F(r)=F(0)+1 F ( r ) = F ( 0 ) + 1 r > 0 r>0 r > 0
N ( Δ G ) ( r , 0 ) ≅ N ( Δ G ) ( F ( 0 ) , F ( 0 ) + 1 ) ≠ 0. \mathcal{N}(\Delta G)(r,0)\cong\mathcal{N}(\Delta G)(F(0),F(0)+1)\ne0. N ( Δ G ) ( r , 0 ) ≅ N ( Δ G ) ( F ( 0 ) , F ( 0 ) + 1 ) = 0. It follows that G r n d ≠ ∅ G_r^{\mathrm{nd}}\ne\varnothing G r nd = ∅ N ( Δ G ) ( F ( m ) , F ( m + r ) ) ≠ 0 \mathcal{N}(\Delta G)(F(m),F(m+r))\ne0 N ( Δ G ) ( F ( m ) , F ( m + r )) = 0 m m m F F F N \mathbb{N} N F ( x ) = x F(x)=x F ( x ) = x x < m x<m x < m F F F N ∖ [ m − 1 ] \mathbb{N}\setminus[m-1] N ∖ [ m − 1 ]
{ x ≥ m : N ( Δ G ) ( m , x ) ≠ 0 } = { m , m + 1 } \{x\ge m:\mathcal{N}(\Delta G)(m,x)\ne0\}=\{m,m+1\} { x ≥ m : N ( Δ G ) ( m , x ) = 0 } = { m , m + 1 } ⟺ { y ≥ m : N ( Δ G ) ( y , F ( m ) ) ≠ 0 } = { F ( m ) , F ( m + 1 ) } . \Longleftrightarrow\{y\ge m:\mathcal{N}(\Delta G)(y,F(m))\ne0\}=\{F(m),F(m+1)\}. ⟺ { y ≥ m : N ( Δ G ) ( y , F ( m )) = 0 } = { F ( m ) , F ( m + 1 )} . Hence, there are two objects in N ∖ [ m − 1 ] \mathbb{N}\setminus[m-1] N ∖ [ m − 1 ] F ( m ) F(m) F ( m ) F ( m ) > m F(m)>m F ( m ) > m F ( m ) − 1 F(m)-1 F ( m ) − 1 F ( m ) F(m) F ( m ) F ( m ) + r F(m)+r F ( m ) + r N ∖ [ m − 1 ] \mathbb{N}\setminus[m-1] N ∖ [ m − 1 ] F ( m ) F(m) F ( m ) F ( m ) ≤ m F(m)\le m F ( m ) ≤ m F ( m ) = m F(m)=m F ( m ) = m
Corollary 5.19. If N ( Δ G ) o p ≃ N ( Δ G ) \mathcal{N}(\Delta G)^{\mathrm{op}}\simeq\mathcal{N}(\Delta G) N ( Δ G ) op ≃ N ( Δ G ) dim G = 1 \dim G=1 dim G = 1
Proof. Theorem 5.18 implies N ( Δ G ) ( m , 0 ) = 0 ⟺ N ( Δ G ) ( 0 , m ) = 0 \mathcal{N}(\Delta G)(m,0)=0\Longleftrightarrow\mathcal{N}(\Delta G)(0,m)=0 N ( Δ G ) ( m , 0 ) = 0 ⟺ N ( Δ G ) ( 0 , m ) = 0 m ≥ 2 m\ge2 m ≥ 2 N ( Δ G ) ( m , 0 ) ≅ k [ G m n d ] \mathcal{N}(\Delta G)(m,0)\cong k[G_m^{\mathrm{nd}}] N ( Δ G ) ( m , 0 ) ≅ k [ G m nd ] N ( Δ G ) ( 0 , 1 ) ≠ 0 \mathcal{N}(\Delta G)(0,1)\ne0 N ( Δ G ) ( 0 , 1 ) = 0
We can now classify the crossed simplicial groups with self-dual normalization.
Theorem 5.20. Let Δ G \Delta G Δ G
Δ K ⟶ Δ G → p Δ H \Delta K\longrightarrow\Delta G\xrightarrow{p}\Delta H Δ K ⟶ Δ G p Δ H as in Theorem 5.6 where Δ K \Delta K Δ K p p p
There exists an equivalence N ( Δ G ) o p ≃ N ( Δ G ) \mathcal{N}(\Delta G)^{\mathrm{op}}\simeq\mathcal{N}(\Delta G) N ( Δ G ) op ≃ N ( Δ G )
The underlying simplicial set G G G
Δ H \Delta H Δ H K K K
There exists an equivalence ( Δ G ) o p ≃ Δ G (\Delta G)^{\mathrm{op}}\simeq\Delta G ( Δ G ) op ≃ Δ G
Proof. We begin with (1) ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒
(1) ⇒ \Rightarrow ⇒
(2) ⇒ \Rightarrow ⇒ dim G = 1 \dim G=1 dim G = 1 K K K G G G G → H G\to H G → H dim K ≤ 1 \dim K\le1 dim K ≤ 1 dim H ≤ 1 \dim H\le1 dim H ≤ 1 dim K = 0 \dim K=0 dim K = 0 K K K dim H = 1 \dim H=1 dim H = 1 dim H = 0 \dim H=0 dim H = 0 p : G → H p:G\to H p : G → H g ∈ G 1 g\in G_1 g ∈ G 1 p ( g ) = s 0 ( q ) p(g)=s_0(q) p ( g ) = s 0 ( q ) q ∈ H 0 q\in H_0 q ∈ H 0 p p p
we may choose a ∈ G 0 a \in G_0 a ∈ G 0 p ( a ) = q p(a)=q p ( a ) = q K 1 = s 0 ( K 0 ) K_1=s_0(K_0) K 1 = s 0 ( K 0 ) g = s 0 ( h ) s 0 ( a ) = s 0 ( h a ) g=s_0(h)s_0(a)=s_0(ha) g = s 0 ( h ) s 0 ( a ) = s 0 ( ha ) h ∈ K 0 h\in K_0 h ∈ K 0 dim G = 1 \dim G=1 dim G = 1 dim H = 1 \dim H=1 dim H = 1 H H H ∣ H n ∣ = ∣ H 0 ∣ + n ⋅ ∣ H 1 n d ∣ |H_n|=|H_0|+n\cdot|H^{\mathrm{nd}}_1| ∣ H n ∣ = ∣ H 0 ∣ + n ⋅ ∣ H 1 nd ∣ Δ H \Delta H Δ H [8 Table 2] , we deduce that Δ H \Delta H Δ H
(3) ⇒ \Rightarrow ⇒ § 7.3 . By Theorem 7.26 and Corollary 7.27 we recover Δ G \Delta G Δ G Δ P \Delta P Δ P Δ G \Delta G Δ G p n ↦ q n p_n\mapsto q_n p n ↦ q n N ( Δ G ) o p ≅ N ( Δ G ) \mathcal{N}(\Delta G)^{\mathrm{op}}\cong\mathcal{N}(\Delta G) N ( Δ G ) op ≅ N ( Δ G )
(4) ⇒ \Rightarrow ⇒
(3) ⇒ \Rightarrow ⇒
Normalization of the cyclic family
In this section, we apply Morita normalization to the cyclic family of planar crossed simplicial groups. Our starting point will be the Dwyer–Kan correspondence, which provides the normalization of Λ + \Lambda_+ Λ + [7 ] to our framework. Note that a more direct Dold–Kan correspondence for the paracyclic category is given by Getzler in [13 ] . We adopt Getzler’s notation of the Dwyer–Kan operator and the Karoubi operator and compare the approaches in Remark 6.11 .
Convention 6.1. In this section we will use Δ min \Delta_{\min} Δ m i n Δ max \Delta_{\max} Δ m a x q n q_n q n p n p_n p n
Definition 6.2. We denote by ( − ) ∘ : Λ ∞ o p → Λ ∞ (-)^\circ:\Lambda_\infty^{\mathrm{op}}\to\Lambda_\infty ( − ) ∘ : Λ ∞ op → Λ ∞ Proposition 7.9 which restricts to a self-duality Λ + o p → Λ + \Lambda_+^{\mathrm{op}}\to\Lambda_+ Λ + op → Λ + extra codegeneracies
σ n n : = ( δ 0 n ) ∘ = σ 0 n ∘ τ n ∈ Λ + ( n , n − 1 ) ( n ≥ 1 ) . \sigma_n^n := (\delta_0^n)^\circ=\sigma_0^n\circ\tau_n\in\Lambda_+(n,n-1)\qquad(n\ge1). σ n n := ( δ 0 n ) ∘ = σ 0 n ∘ τ n ∈ Λ + ( n , n − 1 ) ( n ≥ 1 ) . Furthermore, we define Δ ∘ \Delta^\circ Δ ∘ Δ \Delta Δ ( − ) ∘ (-)^\circ ( − ) ∘
Remark 6.3. Using the extra codegeneracies, the self-duality ( − ) ∘ (-)^\circ ( − ) ∘
( δ i n ) ∘ = σ n − i n , ( σ j n ) ∘ = δ n − j n , ( τ n ) ∘ = τ n (\delta_i^n)^\circ=\sigma_{n-i}^n,\qquad(\sigma_j^n)^\circ=\delta_{n-j}^n,\qquad(\tau_n)^\circ=\tau_n ( δ i n ) ∘ = σ n − i n , ( σ j n ) ∘ = δ n − j n , ( τ n ) ∘ = τ n for 0 ≤ i ≤ n 0\le i\le n 0 ≤ i ≤ n 0 ≤ j < n 0\le j<n 0 ≤ j < n
The following lemma is the equivalent of [7 Remark 3.6] .
Lemma 6.4. The diagram of inclusions
\begin{CD}
\Delta_{\min} @>>> \Delta\\
@VVV @VVV \\
\Delta^\circ@>>> \Lambda_+
\end{CD}
is a pushout in Δ min ↓ C a t \Delta_{\min}\downarrow\mathsf{Cat} Δ m i n ↓ Cat
Proof. Let P = Δ ∘ ∐ Δ min Δ \mathcal{P}=\Delta^\circ\coprod_{\Delta_{\min}}\Delta P = Δ ∘ ∐ Δ m i n Δ C a t \mathsf{Cat} Cat F : P → Λ + F:\mathcal{P}\to\Lambda_+ F : P → Λ + F F F Λ ∞ \Lambda_\infty Λ ∞ Corollary 7.12 ). In particular, we have a pushout square in Δ min ↓ C a t \Delta_{\min}\downarrow\mathsf{Cat} Δ m i n ↓ Cat
Definition 6.5. Let D u \mathbf{Du} Du ( u n : n → n + 1 ) n ∈ N (u_n:n\to n+1)_{n\in\mathbb{N}} ( u n : n → n + 1 ) n ∈ N ( d n : n + 1 → n ) n ∈ N (d_n:n+1\to n)_{n\in\mathbb{N}} ( d n : n + 1 → n ) n ∈ N u n + 1 ∘ u n = 0 u_{n+1}\circ u_n=0 u n + 1 ∘ u n = 0 d n ∘ d n + 1 = 0 d_n\circ d_{n+1}=0 d n ∘ d n + 1 = 0
Theorem 6.6 (Dwyer–Kan correspondence). The functor F : D u → N ( Λ + ) F:\mathbf{Du}\to\mathcal{N}(\Lambda_+) F : Du → N ( Λ + ) induced by
n ↦ n , u n ↦ [ δ 0 n + 1 ] , d n ↦ [ ( − 1 ) n + 1 σ n + 1 n + 1 ] n\mapsto n,\qquad u_n\mapsto[\delta^{n+1}_0],\qquad d_n\mapsto[(-1)^{n+1}\sigma^{n+1}_{n+1}] n ↦ n , u n ↦ [ δ 0 n + 1 ] , d n ↦ [( − 1 ) n + 1 σ n + 1 n + 1 ] is an isomorphism. In particular, the abstract Dold–Kan correspondence gives an equivalence
Fun ( Λ + o p , k Mod ) ≃ Mod k [ Λ + ] ≃ Mod N ( Λ + ) ≃ Mod D u . \operatorname{Fun}(\Lambda_+^{\mathrm{op}},k\operatorname{Mod})\simeq\operatorname{Mod}_{k[\Lambda_+]}\simeq\operatorname{Mod}_{\mathcal{N}(\Lambda_+)}\simeq\operatorname{Mod}_{\mathbf{Du}}. Fun ( Λ + op , k Mod ) ≃ Mod k [ Λ + ] ≃ Mod N ( Λ + ) ≃ Mod Du . Proof. Since N ( − ) \mathcal{N}(-) N ( − )
N ( Δ min ) → N ( Δ ) ↓ ↓ N ( Δ ∘ ) → N ( Λ + ) \begin{CD}
\mathcal{N}(\Delta_{\min}) @>>> \mathcal{N}(\Delta)\\
@VVV @VVV\\
\mathcal{N}(\Delta^\circ) @>>> \mathcal{N}(\Lambda_+)
\end{CD} N ( Δ m i n ) ↓ ⏐ N ( Δ ∘ ) N ( Δ ) ↓ ⏐ N ( Λ + ) is a pushout in R i n g N ( Δ min ) \mathbf{Ring}_{\mathcal{N}(\Delta_{\min})} Ring N ( Δ m i n ) k C a t k\mathbf{Cat} k Cat N ( Δ min ) \mathcal{N}(\Delta_{\min}) N ( Δ m i n ) N \mathbb{N} N Δ min \Delta_{\min} Δ m i n N ( Δ ) ≃ C h ≥ 0 \mathcal{N}(\Delta)\simeq\mathbf{Ch}_{\geq0} N ( Δ ) ≃ Ch ≥ 0 [ δ 0 n ] [\delta^n_0] [ δ 0 n ] ( − ) ∘ (-)^\circ ( − ) ∘ Λ + \Lambda_+ Λ + N ( Δ ∘ ) ≃ C h ≥ 0 \mathcal{N}(\Delta^\circ)\simeq\mathbf{Ch}_{\geq0} N ( Δ ∘ ) ≃ Ch ≥ 0 [ ( − 1 ) n σ n n ] [(-1)^n\sigma^n_n] [( − 1 ) n σ n n ] F F F
D u ≃ C h ≥ 0 ∐ k [ N ] C h ≥ 0 ⟶ N ( Λ + ) . \mathbf{Du}\simeq\mathbf{Ch}_{\geq0}\underset{k[\mathbb{N}]}{\coprod}\mathbf{Ch}_{\geq0}\longrightarrow\mathcal{N}(\Lambda_+). Du ≃ Ch ≥ 0 k [ N ] ∐ Ch ≥ 0 ⟶ N ( Λ + ) . ▢
The following two insights let us compute the components of π \pi π D u \mathbf{Du} Du [7 Theorem 6.3] .
Proposition 6.9. We have N ( C / ⟨ id r : r < n ⟩ ) ≅ N ( C ) / ⟨ id r : r < n ⟩ \mathcal{N}(\mathcal{C}/\langle\operatorname{id}_r:r<n\rangle)\cong\mathcal{N}(\mathcal{C})/\langle\operatorname{id}_r:r<n\rangle N ( C / ⟨ id r : r < n ⟩) ≅ N ( C ) / ⟨ id r : r < n ⟩ C ∈ Ring k [ Δ min ] \mathcal{C}\in\operatorname{Ring}_{k[\Delta_{\min}]} C ∈ Ring k [ Δ m i n ] k k k
C ( n , n ) / I ( n , n ) ≅ N ( C ) / J ( n , n ) \mathcal{C}(n,n)/I(n,n)\cong\mathcal{N}(\mathcal{C})/J(n,n) C ( n , n ) / I ( n , n ) ≅ N ( C ) / J ( n , n ) where I ⊆ C I\subseteq\mathcal{C} I ⊆ C J ⊆ N ( C ) J\subseteq\mathcal{N}(\mathcal{C}) J ⊆ N ( C ) ⟨ id r : r < n ⟩ \langle\operatorname{id}_r:r<n\rangle ⟨ id r : r < n ⟩
Proof. Let C \mathcal{C} C k [ Δ min ] k[\Delta_{\min}] k [ Δ m i n ] Remark 4.5 ) that q n − id n = ∑ i = 1 n δ i ∘ α i ′ q_n-\operatorname{id}_n=\sum_{i=1}^n\delta_i\circ\alpha'_i q n − id n = ∑ i = 1 n δ i ∘ α i ′ r < n r<n r < n ⟨ id r : r < n ⟩ = ⟨ q r : r < n ⟩ \langle\operatorname{id}_r:r<n\rangle=\langle q_r:r<n\rangle ⟨ id r : r < n ⟩ = ⟨ q r : r < n ⟩ ⟨ id r : r < n ⟩ ⊆ N ( C ) \langle\operatorname{id}_r:r<n\rangle\subseteq\mathcal{N}(\mathcal{C}) ⟨ id r : r < n ⟩ ⊆ N ( C ) [ q n ] [q_n] [ q n ] C / I \mathcal{C}/I C / I n n n □ \square □
We recover [7 Proposition 6.5] directly from the index categories.
Corollary 6.10. π n = ( id − x n ) n + 1 + ( id − y n ) n − id = ( id − x n ) n + 1 ∘ ( id − y n ) n \pi_n=(\operatorname{id}-x_n)^{n+1}+(\operatorname{id}-y_n)^n-\operatorname{id}=(\operatorname{id}-x_n)^{n+1}\circ(\operatorname{id}-y_n)^n π n = ( id − x n ) n + 1 + ( id − y n ) n − id = ( id − x n ) n + 1 ∘ ( id − y n ) n
Proof. The isomorphisms N ( k [ Λ + ] / I ) ≅ N ( Λ + ) / I ≅ D u / I \mathcal{N}(k[\Lambda_+]/I)\cong\mathcal{N}(\Lambda_+)/I\cong\mathbf{Du}/I N ( k [ Λ + ] / I ) ≅ N ( Λ + ) / I ≅ Du / I I = ⟨ id r : r < n ⟩ I=\langle\operatorname{id}_r:r<n\rangle I = ⟨ id r : r < n ⟩ k k k
φ : k [ Λ + ] ( n , n ) ⟶ k [ Λ + ] ( n , n ) / I ( n , n ) ≅ D u ( n , n ) / I ( n , n ) ≅ k [ x n ] \varphi:k[\Lambda_+](n,n)\longrightarrow k[\Lambda_+](n,n)/I(n,n)\cong\mathbf{Du}(n,n)/I(n,n)\cong k[x_n] φ : k [ Λ + ] ( n , n ) ⟶ k [ Λ + ] ( n , n ) / I ( n , n ) ≅ Du ( n , n ) / I ( n , n ) ≅ k [ x n ] where φ ( f ) = F − 1 ( [ f ] ) m o d y n \varphi(f)=F^{-1}([f])\bmod y_n φ ( f ) = F − 1 ([ f ]) mod y n φ ( τ n n + 1 ) = φ ( τ n ) n + 1 \varphi(\tau_n^{n+1})=\varphi(\tau_n)^{n+1} φ ( τ n n + 1 ) = φ ( τ n ) n + 1 F − 1 ( [ τ n ] ) F^{-1}([\tau_n]) F − 1 ([ τ n ]) τ n = σ n + 1 ∘ δ 0 \tau_n=\sigma_{n+1}\circ\delta_0 τ n = σ n + 1 ∘ δ 0 [ σ n + 1 ] ∘ [ δ 0 ] [\sigma_{n+1}]\circ[\delta_0] [ σ n + 1 ] ∘ [ δ 0 ] N ( Λ + ) \mathcal{N}(\Lambda_+) N ( Λ + ) q n + 1 ∘ δ 0 = Rev ( p n + 1 ∘ δ n + 1 ) q_{n+1}\circ\delta_0=\operatorname{Rev}(p_{n+1}\circ\delta_{n+1}) q n + 1 ∘ δ 0 = Rev ( p n + 1 ∘ δ n + 1 ) q n + 1 ∘ δ 0 = ( ∑ i = 0 n + 1 ( − 1 ) i δ i ) ∘ q n q_{n+1}\circ\delta_0=(\sum_{i=0}^{n+1}(-1)^i\delta_i)\circ q_n q n + 1 ∘ δ 0 = ( ∑ i = 0 n + 1 ( − 1 ) i δ i ) ∘ q n
σ n + 1 ∘ q n + 1 ∘ δ 0 = ( σ n + 1 ∘ δ 0 + ( − 1 ) n + 1 σ n ∘ δ n + 1 ) ∘ q n + ∑ i = 1 n ( − 1 ) i δ i ∘ c i \sigma_{n+1}\circ q_{n+1}\circ\delta_0=(\sigma_{n+1}\circ\delta_0+(-1)^{n+1}\sigma_n\circ\delta_{n+1})\circ q_n+\sum_{i=1}^{n}(-1)^i\delta_i\circ c_i σ n + 1 ∘ q n + 1 ∘ δ 0 = ( σ n + 1 ∘ δ 0 + ( − 1 ) n + 1 σ n ∘ δ n + 1 ) ∘ q n + i = 1 ∑ n ( − 1 ) i δ i ∘ c i = ( τ n + ( − 1 ) n + 1 id n ) ∘ q n + ψ =(\tau_n+(-1)^{n+1}\operatorname{id}_n)\circ q_n+\psi = ( τ n + ( − 1 ) n + 1 id n ) ∘ q n + ψ for certain c i c_i c i [ ψ ] = 0 [\psi]=0 [ ψ ] = 0 [ σ n + 1 ] ∘ [ δ 0 ] = [ τ n ] + ( − 1 ) n + 1 id [\sigma_{n+1}]\circ[\delta_0]=[\tau_n]+(-1)^{n+1}\operatorname{id} [ σ n + 1 ] ∘ [ δ 0 ] = [ τ n ] + ( − 1 ) n + 1 id F ( u n ) = [ δ 0 ] F(u_n)=[\delta_0] F ( u n ) = [ δ 0 ] F ( d n ) = ( − 1 ) n + 1 [ σ n + 1 ] F(d_n)=(-1)^{n+1}[\sigma_{n+1}] F ( d n ) = ( − 1 ) n + 1 [ σ n + 1 ] F − 1 ( [ τ n ] ) = ( − 1 ) n ( id − x n ) F^{-1}([\tau_n])=(-1)^n(\operatorname{id}-x_n) F − 1 ([ τ n ]) = ( − 1 ) n ( id − x n ) π n m o d y n = ( ( − 1 ) n ( id − x n ) ) n + 1 = ( id − x n ) n + 1 \pi_n\bmod y_n=((-1)^n(\operatorname{id}-x_n))^{n+1}=(\operatorname{id}-x_n)^{n+1} π n mod y n = (( − 1 ) n ( id − x n ) ) n + 1 = ( id − x n ) n + 1 π \pi π Remark 6.8 via
c n = 1 , f n ( t ) = ( 1 − t ) n + 1 − 1 , g n ( t ) = ( 1 − t ) n − 1. c_n=1,\qquad f_n(t)=(1-t)^{n+1}-1,\qquad g_n(t)=(1-t)^n-1. c n = 1 , f n ( t ) = ( 1 − t ) n + 1 − 1 , g n ( t ) = ( 1 − t ) n − 1. Corollary 6.12. We summarize the normalization of the cyclic family.
N ( Λ + ) ≅ D u , \mathcal{N}(\Lambda_+)\cong\mathbf{Du}, N ( Λ + ) ≅ Du , N ( Λ ∞ ) ≅ D u [ π − 1 ] ≅ D u [ κ − 1 ] , \mathcal{N}(\Lambda_\infty)\cong\mathbf{Du}[\pi^{-1}]\cong\mathbf{Du}[\kappa^{-1}], N ( Λ ∞ ) ≅ Du [ π − 1 ] ≅ Du [ κ − 1 ] , N ( Λ ) ≅ D u / ⟨ π − id ⟩ , \mathcal{N}(\Lambda)\cong\mathbf{Du}/\langle\pi-\operatorname{id}\rangle, N ( Λ ) ≅ Du / ⟨ π − id ⟩ , N ( Λ N ) ≅ D u / ⟨ π N − id ⟩ . \mathcal{N}(\Lambda_N)\cong\mathbf{Du}/\langle\pi^N-\operatorname{id}\rangle. N ( Λ N ) ≅ Du / ⟨ π N − id ⟩ . Self-duality of crossed simplicial groups
Following the work of Elmendorf [10 ] , we are going to construct the paracyclic category by suitably extending Δ \Delta Δ Remark 2.16 ) that we have a chain of adjunctions δ i + 1 n ⊣ σ i n ⊣ δ i n \delta_{i+1}^{n} \dashv\sigma_{i}^{n} \dashv\delta_{i}^{n} δ i + 1 n ⊣ σ i n ⊣ δ i n 0 ≤ i < n 0 \le i < n 0 ≤ i < n δ n n \delta_{n}^{n} δ n n δ 0 n \delta_{0}^{n} δ 0 n φ : [ m ] → [ n ] \varphi\colon[m] \to[n] φ : [ m ] → [ n ] φ ~ : Z → Z \widetilde{\varphi}\colon\mathbb{Z} \to\mathbb{Z} φ : Z → Z φ ~ ( x + m + 1 ) = φ ~ ( x ) + n + 1 \widetilde{\varphi}(x+m+1)=\widetilde{\varphi}(x)+n+1 φ ( x + m + 1 ) = φ ( x ) + n + 1
Proposition 7.1. Let f : Z → Z f\colon\mathbb{Z} \to\mathbb{Z} f : Z → Z weakly increasing . Then f f f f f f L f ⊣ f Lf \dashv f L f ⊣ f L f ( x ) = min { y ∈ Z : x ≤ f ( y ) } Lf(x)=\min\{y \in\mathbb{Z}: x \le f(y)\} L f ( x ) = min { y ∈ Z : x ≤ f ( y )}
Proof. The adjunction condition is L f ( x ) ≤ y ⟺ x ≤ f ( y ) Lf(x) \le y \Longleftrightarrow x \le f(y) L f ( x ) ≤ y ⟺ x ≤ f ( y ) x , y ∈ Z x,y \in\mathbb{Z} x , y ∈ Z S ( x ) ≔ { y ∈ Z : x ≤ f ( y ) } S(x) \coloneqq\{y \in\mathbb{Z}: x \le f(y)\} S ( x ) : = { y ∈ Z : x ≤ f ( y )} L f Lf L f L f ( x ) = min S ( x ) Lf(x)=\min S(x) L f ( x ) = min S ( x ) S ( x ) S(x) S ( x ) x x x f f f S ( x ) S(x) S ( x ) f f f □ \square □
Corollary 7.2. Every weakly increasing map f : Z → Z f\colon\mathbb{Z} \to\mathbb{Z} f : Z → Z L f ⊣ f ⊣ R f Lf \dashv f \dashv Rf L f ⊣ f ⊣ R f R f ( y ) = max { x ∈ Z : f ( x ) ≤ y } Rf(y)=\max\{x \in\mathbb{Z}: f(x) \le y\} R f ( y ) = max { x ∈ Z : f ( x ) ≤ y } f ↦ L f f \mapsto Lf f ↦ L f f ↦ R f f \mapsto Rf f ↦ R f R f ( y ) = L f ( y + 1 ) − 1 Rf(y)=Lf(y+1)-1 R f ( y ) = L f ( y + 1 ) − 1
Proof. Since the poset category Z \mathbb{Z} Z f f f f f f L f Lf L f R f Rf R f R L f = f = L R f RLf=f=LRf R L f = f = L R f R f Rf R f R f ( y ) = L f ( y + 1 ) − 1 Rf(y)=Lf(y+1)-1 R f ( y ) = L f ( y + 1 ) − 1 x , y ∈ Z x,y \in\mathbb{Z} x , y ∈ Z
x ≤ R f ( y ) ⟺ f ( x ) ≤ y ⟺ ¬ ( y + 1 ≤ f ( x ) ) ⟺ ¬ ( L f ( y + 1 ) ≤ x ) ⟺ x + 1 ≤ L f ( y + 1 ) ⟺ x ≤ L f ( y + 1 ) − 1. \begin{aligned}
x \le Rf(y) &\Longleftrightarrow f(x) \le y\\
&\Longleftrightarrow\neg(y+1 \le f(x))\\
&\Longleftrightarrow\neg(Lf(y+1) \le x)\\
&\Longleftrightarrow x+1 \le Lf(y+1)\\
&\Longleftrightarrow x \le Lf(y+1)-1.
\end{aligned} x ≤ R f ( y ) ⟺ f ( x ) ≤ y ⟺ ¬ ( y + 1 ≤ f ( x )) ⟺ ¬ ( L f ( y + 1 ) ≤ x ) ⟺ x + 1 ≤ L f ( y + 1 ) ⟺ x ≤ L f ( y + 1 ) − 1. hence the result follows by uniqueness of the adjoint. □ \square □
Proposition 7.3. Let f : Z → Z f\colon\mathbb{Z} \to\mathbb{Z} f : Z → Z weakly increasing . If f ( x + m + 1 ) = f ( x ) + n + 1 f(x+m+1)=f(x)+n+1 f ( x + m + 1 ) = f ( x ) + n + 1 x ∈ Z x \in\mathbb{Z} x ∈ Z L f ( x + n + 1 ) = L f ( x ) + m + 1 Lf(x+n+1)=Lf(x)+m+1 L f ( x + n + 1 ) = L f ( x ) + m + 1 x ∈ Z x \in\mathbb{Z} x ∈ Z R f Rf R f
Proof. We apply the adjunction:
L f ( x + n + 1 ) ≤ y ⟺ x ≤ f ( y ) − n − 1 = f ( y − m − 1 ) ⟺ L f ( x ) + m + 1 ≤ y . \begin{aligned}
Lf(x+n+1) \le y &\Longleftrightarrow x \le f(y)-n-1=f(y-m-1)\\
&\Longleftrightarrow Lf(x)+m+1 \le y.
\end{aligned} L f ( x + n + 1 ) ≤ y ⟺ x ≤ f ( y ) − n − 1 = f ( y − m − 1 ) ⟺ L f ( x ) + m + 1 ≤ y . Since this holds for all x , y ∈ Z x,y \in\mathbb{Z} x , y ∈ Z L f ( x + n + 1 ) = L f ( x ) + m + 1 Lf(x+n+1)=Lf(x)+m+1 L f ( x + n + 1 ) = L f ( x ) + m + 1 R f Rf R f □ \square □
The cyclic family.
Definition 7.4. The paracyclic category Λ ∞ \Lambda_{\infty} Λ ∞ f ∈ Λ ∞ ( m , n ) f \in\Lambda_{\infty}(m,n) f ∈ Λ ∞ ( m , n ) f : Z → Z f\colon\mathbb{Z} \to\mathbb{Z} f : Z → Z f ( x + m + 1 ) = f ( x ) + n + 1 f(x+m+1)=f(x)+n+1 f ( x + m + 1 ) = f ( x ) + n + 1 x ∈ Z x \in\mathbb{Z} x ∈ Z Δ \Delta Δ Λ ∞ \Lambda_{\infty} Λ ∞
extending the morphisms of Δ \Delta Δ τ n ∈ Λ ∞ ( n , n ) \tau_n \in\Lambda_\infty(n,n) τ n ∈ Λ ∞ ( n , n ) τ : Z → Z \tau:\mathbb{Z}\to\mathbb{Z} τ : Z → Z τ ( x ) = x + 1 \tau(x)=x+1 τ ( x ) = x + 1
Theorem 7.5. The assignment f ↦ L f f\mapsto Lf f ↦ L f L : Λ ∞ o p → Λ ∞ L:\Lambda_\infty^{\mathrm{op}}\to\Lambda_\infty L : Λ ∞ op → Λ ∞ R R R
Proof. This follows immediately from Corollary 7.2 and Proposition 7.3. □
Definition 7.7. For every sequence of integers a = ( a n ) n ∈ N ⊆ Z a=(a_n)_{n\in\mathbb{N}}\subseteq\mathbb{Z} a = ( a n ) n ∈ N ⊆ Z T a : Λ ∞ → Λ ∞ T_a:\Lambda_\infty\to\Lambda_\infty T a : Λ ∞ → Λ ∞ f ∈ Λ ∞ ( m , n ) f\in\Lambda_\infty(m,n) f ∈ Λ ∞ ( m , n ) T a f T_af T a f T a f ( x ) ≔ a n − f ( a m − x ) T_af(x)\coloneqq a_n-f(a_m-x) T a f ( x ) : = a n − f ( a m − x ) x ∈ Z x\in\mathbb{Z} x ∈ Z
Definition 7.8. We denote by Λ + \Lambda_+ Λ + Λ ∞ \Lambda_\infty Λ ∞
Λ + ( m , n ) = { f ∈ Λ ∞ ( m , n ) : f ( 0 ) ≥ 0 } . \Lambda_+(m,n)=\{f\in\Lambda_\infty(m,n):f(0)\geq0\}. Λ + ( m , n ) = { f ∈ Λ ∞ ( m , n ) : f ( 0 ) ≥ 0 } . Proposition 7.9. Let a = ( a n ) a=(a_n) a = ( a n ) b = ( b n ) b=(b_n) b = ( b n ) a n = n a_n=n a n = n b n = 0 b_n=0 b n = 0
T a ∘ L T_a\circ L T a ∘ L ( p n ) (p_n) ( p n )
T b ∘ L = ( − ) ∘ T_b\circ L=(-)^\circ T b ∘ L = ( − ) ∘ ( q n ) (q_n) ( q n ) Λ + \Lambda_+ Λ +
Furthermore, both self-dualities preserve τ n \tau_n τ n
Proof. T a T_a T a Δ ⊆ Λ ∞ \Delta\subseteq\Lambda_\infty Δ ⊆ Λ ∞ Rev \operatorname{Rev} Rev T b ∘ L T_b\circ L T b ∘ L ( q n ) (q_n) ( q n ) T b L f ( 0 ) ≥ 0 ⟺ f ( 0 ) ≥ 0 T_bLf(0)\geq0\Longleftrightarrow f(0)\geq0 T b L f ( 0 ) ≥ 0 ⟺ f ( 0 ) ≥ 0 τ n \tau_n τ n T c ( τ n ) = τ n − 1 T_c(\tau_n)=\tau_n^{-1} T c ( τ n ) = τ n − 1 c = ( c n ) c=(c_n) c = ( c n )
Remark 7.10. The functor ( − ) ∘ : Λ + o p → Λ + (-)^\circ:\Lambda_+^{\mathrm{op}}\to\Lambda_+ ( − ) ∘ : Λ + op → Λ +
Proposition 7.11. The automorphisms of the object n ∈ Λ ∞ n\in\Lambda_\infty n ∈ Λ ∞ { τ n a : a ∈ Z } \{\tau_n^a:a\in\mathbb{Z}\} { τ n a : a ∈ Z }
Proof. Let f ∈ Λ ∞ ( n , n ) f\in\Lambda_\infty(n,n) f ∈ Λ ∞ ( n , n ) f : Z → Z f:\mathbb{Z}\to\mathbb{Z} f : Z → Z f ( x + 1 ) = f ( x ) + 1 f(x+1)=f(x)+1 f ( x + 1 ) = f ( x ) + 1 x ∈ Z x\in\mathbb{Z} x ∈ Z f ( x ) = f ( 0 ) + x f(x)=f(0)+x f ( x ) = f ( 0 ) + x f = τ n a f=\tau_n^a f = τ n a a = f ( 0 ) a=f(0) a = f ( 0 )
Corollary 7.12. Λ ∞ \Lambda_\infty Λ ∞ Aut Λ ∞ ( n ) ≅ Z \operatorname{Aut}_{\Lambda_\infty}(n)\cong\mathbb{Z} Aut Λ ∞ ( n ) ≅ Z
Proof. Let f ∈ Λ ∞ ( m , n ) f\in\Lambda_\infty(m,n) f ∈ Λ ∞ ( m , n ) L f ∈ Λ ∞ ( n , m ) Lf\in\Lambda_\infty(n,m) L f ∈ Λ ∞ ( n , m ) L f = τ m a ∘ ψ Lf=\tau_m^a\circ\psi L f = τ m a ∘ ψ a = L f ( 0 ) a=Lf(0) a = L f ( 0 ) ψ ( 0 ) = 0 \psi(0)=0 ψ ( 0 ) = 0 ψ = L φ \psi=L\varphi ψ = L φ φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) f f f f = φ ∘ R ( τ m ) a f=\varphi\circ R(\tau_m)^a f = φ ∘ R ( τ m ) a R ( τ n ) = τ n − 1 R(\tau_n)=\tau_n^{-1} R ( τ n ) = τ n − 1 Λ ∞ \Lambda_\infty Λ ∞
Definition 7.13. For N ∈ N ≥ 1 N\in\mathbb{N}_{\geq1} N ∈ N ≥ 1 N N N
Λ N ≔ Λ ∞ / ⟨ τ n N ( n + 1 ) = id n : n ≥ 0 ⟩ . \Lambda_N\coloneqq\Lambda_\infty/\langle\tau_n^{N(n+1)}=\operatorname{id}_n:n\geq0\rangle. Λ N : = Λ ∞ / ⟨ τ n N ( n + 1 ) = id n : n ≥ 0 ⟩ . The cyclic category is defined as Λ ≔ Λ 1 \Lambda\coloneqq\Lambda_1 Λ : = Λ 1
Proposition 7.14. Λ N \Lambda_N Λ N Aut Λ N ( n ) ≅ Z / N ( n + 1 ) \operatorname{Aut}_{\Lambda_N}(n) \cong\mathbb{Z}/N(n+1) Aut Λ N ( n ) ≅ Z / N ( n + 1 )
Proof. It is readily verified that ⟨ τ n N ( n + 1 ) = id n : n ≥ 0 ⟩ \langle\tau_n^{N(n+1)} = \operatorname{id}_n : n \ge0\rangle ⟨ τ n N ( n + 1 ) = id n : n ≥ 0 ⟩ f , f ′ ∈ Λ ∞ ( m , n ) f, f' \in\Lambda_\infty(m,n) f , f ′ ∈ Λ ∞ ( m , n )
f = f ′ ∘ τ m ℓ N ( m + 1 ) f = f' \circ\tau_m^{\ell N(m+1)} f = f ′ ∘ τ m ℓ N ( m + 1 ) for some ℓ ∈ Z \ell\in\mathbb{Z} ℓ ∈ Z Λ ∞ \Lambda_\infty Λ ∞
The dihedral family.
Definition 7.15. We denote by ρ : Z → Z \rho:\mathbb{Z} \to\mathbb{Z} ρ : Z → Z ρ ( x ) = − x \rho(x)=-x ρ ( x ) = − x paradihedral category Ξ ∞ \Xi_\infty Ξ ∞ n ∈ N n \in\mathbb{N} n ∈ N
Ξ ∞ ( m , n ) = Λ ∞ ( m , n ) ∪ ( ρ ∘ Λ ∞ ( m , n ) ) \Xi_\infty(m,n)=\Lambda_\infty(m,n)\cup\left(\rho\circ\Lambda_\infty(m,n)\right) Ξ ∞ ( m , n ) = Λ ∞ ( m , n ) ∪ ( ρ ∘ Λ ∞ ( m , n ) ) with composition as maps of sets. We denote by ρ n ∈ Ξ ∞ ( n , n ) \rho_n \in\Xi_\infty(n,n) ρ n ∈ Ξ ∞ ( n , n ) ρ \rho ρ
Proposition 7.16. Ξ ∞ \Xi_\infty Ξ ∞ Aut Ξ ∞ ( n ) ≅ D ∞ \operatorname{Aut}_{\Xi_\infty}(n)\cong D_\infty Aut Ξ ∞ ( n ) ≅ D ∞
Proof. By definition, every morphism in Ξ ∞ ( m , n ) \Xi_\infty(m,n) Ξ ∞ ( m , n ) ρ n ϵ ∘ f \rho_n^\epsilon\circ f ρ n ϵ ∘ f ϵ ∈ { 0 , 1 } \epsilon\in\{0,1\} ϵ ∈ { 0 , 1 } f ∈ Λ ∞ ( m , n ) f\in\Lambda_\infty(m,n) f ∈ Λ ∞ ( m , n ) T b : Λ ∞ → Λ ∞ T_b:\Lambda_\infty\to\Lambda_\infty T b : Λ ∞ → Λ ∞ Proposition 7.9 maps
T b ( f ) = ρ n ∘ f ∘ ρ m . T_b(f)=\rho_n\circ f\circ\rho_m. T b ( f ) = ρ n ∘ f ∘ ρ m . Using the unique normal form T b ϵ ( f ) = φ ∘ τ m a T_b^\epsilon(f)=\varphi\circ\tau_m^a T b ϵ ( f ) = φ ∘ τ m a
ρ n ϵ ∘ f = T b ϵ ( f ) ∘ ρ m ϵ = φ ∘ τ m a ∘ ρ m ϵ \rho_n^\epsilon\circ f=T_b^\epsilon(f)\circ\rho_m^\epsilon=\varphi\circ\tau_m^a\circ\rho_m^\epsilon ρ n ϵ ∘ f = T b ϵ ( f ) ∘ ρ m ϵ = φ ∘ τ m a ∘ ρ m ϵ for unique φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) a ∈ Z a\in\mathbb{Z} a ∈ Z ϵ ∈ { 0 , 1 } \epsilon\in\{0,1\} ϵ ∈ { 0 , 1 } Aut Ξ ∞ ( n ) ≅ Z / 2 ⋉ Z ≅ D ∞ \operatorname{Aut}_{\Xi_\infty}(n)\cong\mathbb{Z}/2\ltimes\mathbb{Z}\cong D_\infty Aut Ξ ∞ ( n ) ≅ Z /2 ⋉ Z ≅ D ∞
Definition 7.17. For N ∈ N ≥ 1 N\in\mathbb{N}_{\ge1} N ∈ N ≥ 1 N N N
Ξ N ≔ Ξ ∞ / ⟨ τ n N ( n + 1 ) = id n : n ≥ 0 ⟩ . \Xi_N\coloneqq\Xi_\infty/\langle\tau_n^{N(n+1)}=\operatorname{id}_n:n\ge0\rangle. Ξ N : = Ξ ∞ / ⟨ τ n N ( n + 1 ) = id n : n ≥ 0 ⟩ . The dihedral category is defined as Ξ ≔ Ξ 1 \Xi\coloneqq\Xi_1 Ξ : = Ξ 1
Proposition 7.18. Ξ N \Xi_N Ξ N Aut Ξ N ( n ) ≅ D N ( n + 1 ) \operatorname{Aut}_{\Xi_N}(n)\cong D_N(n+1) Aut Ξ N ( n ) ≅ D N ( n + 1 )
Proof. It is readily verified that ⟨ τ n N ( n + 1 ) = id n : n ≥ 0 ⟩ \langle\tau_n^{N(n+1)}=\operatorname{id}_n:n\ge0\rangle ⟨ τ n N ( n + 1 ) = id n : n ≥ 0 ⟩ f , f ′ ∈ Ξ ∞ ( m , n ) f,f'\in\Xi_\infty(m,n) f , f ′ ∈ Ξ ∞ ( m , n )
f = f ′ ∘ τ m ℓ N ( m + 1 ) f=f'\circ\tau_m^{\ell N(m+1)} f = f ′ ∘ τ m ℓ N ( m + 1 ) for some ℓ ∈ Z \ell\in\mathbb{Z} ℓ ∈ Z Ξ ∞ \Xi_\infty Ξ ∞ ρ n ∘ τ n = τ n − 1 ∘ ρ n \rho_n\circ\tau_n=\tau_n^{-1}\circ\rho_n ρ n ∘ τ n = τ n − 1 ∘ ρ n
Remark 7.19. The way we have constructed Ξ ∞ \Xi_\infty Ξ ∞ Λ ∞ \Lambda_\infty Λ ∞ L ^ : Ξ ∞ o p → Ξ ∞ \widehat{L}:\Xi_\infty^{\mathrm{op}}\to\Xi_\infty L : Ξ ∞ op → Ξ ∞ L L L f ∈ Λ ∞ ( m , n ) f\in\Lambda_\infty(m,n) f ∈ Λ ∞ ( m , n )
L f ∘ L ^ ( ρ n ) = L ^ ( ρ n ∘ f ) = L ^ ( T b ( f ) ∘ ρ m ) = L ^ ( ρ m ) ∘ L ( T b ( f ) ) = L ^ ( ρ m ) ∘ T ^ b ( R f ) = L ^ ( ρ m ) ∘ ρ m ∘ τ m − 1 ∘ L f ∘ τ n ∘ ρ n . \begin{aligned}
Lf\circ\widehat{L}(\rho_n)=\widehat{L}(\rho_n\circ f)=\widehat{L}(T_b(f)\circ\rho_m)=\widehat{L}(\rho_m)\circ L(T_b(f))=\widehat{L}(\rho_m)\circ\widehat{T}_b(Rf) \\
&=\widehat{L}(\rho_m)\circ\rho_m\circ\tau_m^{-1}\circ Lf\circ\tau_n\circ\rho_n.
\end{aligned} L f ∘ L ( ρ n ) = L ( ρ n ∘ f ) = L ( T b ( f ) ∘ ρ m ) = L ( ρ m ) ∘ L ( T b ( f )) = L ( ρ m ) ∘ T b ( R f ) = L ( ρ m ) ∘ ρ m ∘ τ m − 1 ∘ L f ∘ τ n ∘ ρ n . A natural choice for the extension is therefore L ^ ( ρ n ) = τ n ∘ ρ n \widehat{L}(\rho_n)=\tau_n\circ\rho_n L ( ρ n ) = τ n ∘ ρ n
Proposition 7.20. The self-duality L : Λ ∞ o p → Λ ∞ L:\Lambda_\infty^{\mathrm{op}}\to\Lambda_\infty L : Λ ∞ op → Λ ∞ L ^ : Ξ ∞ o p → Ξ ∞ \widehat{L}:\Xi_\infty^{\mathrm{op}}\to\Xi_\infty L : Ξ ∞ op → Ξ ∞
L ^ ( δ i n ) = { σ i n if i < n , σ 0 n ∘ τ n if i = n , L ^ ( σ j n ) = δ j + 1 n , L ^ ( τ n ) = τ n − 1 , L ^ ( ρ n ) = τ n ∘ ρ n . \widehat{L}(\delta_i^n)=
\begin{cases}
\sigma_i^n & \text{if }i<n,\\
\sigma_0^n\circ\tau_n & \text{if }i=n,
\end{cases}
\qquad
\widehat{L}(\sigma_j^n)=\delta_{j+1}^n,
\qquad
\widehat{L}(\tau_n)=\tau_n^{-1},
\qquad
\widehat{L}(\rho_n)=\tau_n\circ\rho_n. L ( δ i n ) = { σ i n σ 0 n ∘ τ n if i < n , if i = n , L ( σ j n ) = δ j + 1 n , L ( τ n ) = τ n − 1 , L ( ρ n ) = τ n ∘ ρ n . Its inverse is the extension R ^ : Ξ ∞ o p → Ξ ∞ \widehat{R}:\Xi_\infty^{\mathrm{op}}\to\Xi_\infty R : Ξ ∞ op → Ξ ∞ R : Λ ∞ o p → Λ ∞ R:\Lambda_\infty^{\mathrm{op}}\to\Lambda_\infty R : Λ ∞ op → Λ ∞ R ^ ( ρ n ) = ρ n ∘ τ n \widehat{R}(\rho_n)=\rho_n\circ\tau_n R ( ρ n ) = ρ n ∘ τ n
Proof. Since every morphism in Ξ ∞ ( m , n ) \Xi_{\infty}(m,n) Ξ ∞ ( m , n ) f ∘ ρ m ϵ f \circ\rho_m^\epsilon f ∘ ρ m ϵ f ∈ Λ ∞ ( m , n ) f \in\Lambda_{\infty}(m,n) f ∈ Λ ∞ ( m , n ) ϵ ∈ { 0 , 1 } \epsilon\in\{0,1\} ϵ ∈ { 0 , 1 } L ^ \widehat{L} L
L ^ ( f ∘ ρ ) ∘ L ^ ( g ∘ ρ ) = τ ∘ ρ ∘ L f ∘ τ ∘ ρ ∘ L g = τ ∘ T b ( L f ) ∘ τ − 1 ∘ L g = τ ∘ R ( T b ( f ) ) ∘ τ − 1 ∘ L g = L ( T b ( f ) ) ∘ L g = L ^ ( g ∘ T b ( f ) ) = L ^ ( g ∘ ρ ∘ f ∘ ρ ) . \begin{aligned}
\widehat{L}(f \circ\rho) \circ\widehat{L}(g \circ\rho) &= \tau\circ\rho\circ Lf \circ\tau\circ\rho\circ Lg \\
&= \tau\circ T_b(Lf) \circ\tau^{-1} \circ Lg \\
&= \tau\circ R(T_b(f)) \circ\tau^{-1} \circ Lg \\
&= L(T_b(f)) \circ Lg \\
&= \widehat{L}(g \circ T_b(f)) \\
&= \widehat{L}(g \circ\rho\circ f \circ\rho).
\end{aligned} L ( f ∘ ρ ) ∘ L ( g ∘ ρ ) = τ ∘ ρ ∘ L f ∘ τ ∘ ρ ∘ Lg = τ ∘ T b ( L f ) ∘ τ − 1 ∘ Lg = τ ∘ R ( T b ( f )) ∘ τ − 1 ∘ Lg = L ( T b ( f )) ∘ Lg = L ( g ∘ T b ( f )) = L ( g ∘ ρ ∘ f ∘ ρ ) . The argument for R ^ \widehat{R} R L ^ \widehat{L} L L L L □ \square □
Corollary 7.21. Extending the morphisms in Definition 7.7 to Ξ ∞ \Xi_{\infty} Ξ ∞
T a ∘ L ^ T_a \circ\widehat{L} T a ∘ L ( p n ) (p_n) ( p n ) T a ∘ L T_a \circ L T a ∘ L Λ ∞ \Lambda_{\infty} Λ ∞
T b ∘ L ^ = ( − ) ∘ T_b \circ\widehat{L} = (-)^\circ T b ∘ L = ( − ) ∘ ( q n ) (q_n) ( q n ) T b ∘ L T_b \circ L T b ∘ L Λ ∞ \Lambda_{\infty} Λ ∞
Lifting self-duality to extensions
Throughout this section we fix an extension p : Δ G → Δ H p\colon\Delta G \to\Delta H p : Δ G → Δ H Δ K \Delta K Δ K
Δ H ∈ { Λ , Ξ } \Delta H \in\{\Lambda,\Xi\} Δ H ∈ { Λ , Ξ }
K K K
Since Aut Λ ( 0 ) \operatorname{Aut}_{\Lambda}(0) Aut Λ ( 0 ) Aut Ξ ( 0 ) ≅ C 2 = { 1 , − 1 } \operatorname{Aut}_{\Xi}(0) \cong C_2 = \{1,-1\} Aut Ξ ( 0 ) ≅ C 2 = { 1 , − 1 } ϵ : G 0 → C 2 \epsilon\colon G_0 \to C_2 ϵ : G 0 → C 2 G 0 → p H 0 ↪ C 2 G_0 \xrightarrow{p} H_0 \hookrightarrow C_2 G 0 p H 0 ↪ C 2 Δ P \Delta P Δ P C a t \mathbf{Cat} Cat
Δ P → B G 0 ↓ ↓ ϵ Ξ ∞ → sgn B C 2 \begin{CD}
\Delta P @>>> BG_0 \\
@VVV @VV\epsilon V \\
\Xi_{\infty} @>{\operatorname{sgn}}>> BC_2
\end{CD} Δ P ↓ ⏐ Ξ ∞ sgn B G 0 ↓ ⏐ ϵ B C 2 Here, B G BG B G sgn \operatorname{sgn} sgn Ξ ∞ \Xi_{\infty} Ξ ∞ Δ P \Delta P Δ P
Objects of Δ P \Delta P Δ P n ∈ N n \in\mathbb{N} n ∈ N
Morphisms in Δ P ( m , n ) \Delta P(m,n) Δ P ( m , n ) ( f , a ) (f,a) ( f , a ) f ∈ Ξ ∞ ( m , n ) f \in\Xi_{\infty}(m,n) f ∈ Ξ ∞ ( m , n ) a ∈ G 0 a \in G_0 a ∈ G 0 sgn ( f ) = ϵ ( a ) \operatorname{sgn}(f) = \epsilon(a) sgn ( f ) = ϵ ( a )
Composition is given by ( g , b ) ∘ ( f , a ) = ( g ∘ f , b ∘ a ) (g,b) \circ(f,a) = (g \circ f,b \circ a) ( g , b ) ∘ ( f , a ) = ( g ∘ f , b ∘ a )
Note that Δ P \Delta P Δ P Ξ ∞ \Xi_{\infty} Ξ ∞ Δ P \Delta P Δ P
( Δ P ) o p → Δ P , n ↦ n , ( f , a ) ↦ ( L ^ f , a − 1 ) . (\Delta P)^{\mathrm{op}} \to\Delta P,\qquad n \mapsto n,\qquad(f,a) \mapsto(\widehat{L}f,a^{-1}). ( Δ P ) op → Δ P , n ↦ n , ( f , a ) ↦ ( L f , a − 1 ) . Furthermore, Δ P \Delta P Δ P
π : Δ P → Δ H , n ↦ n , ( f , a ) ↦ [ f ] . \pi\colon\Delta P \to\Delta H,\qquad n \mapsto n,\qquad(f,a) \mapsto[f]. π : Δ P → Δ H , n ↦ n , ( f , a ) ↦ [ f ] . We will explicitly construct a lift:
Δ P → π Δ H p ↑ Δ G \begin{CD}
\Delta P @>{\pi}>> \Delta H \\
@. @A{p}AA \\
@. \Delta G
\end{CD} Δ P π Δ H p ⏐ ↑ Δ G Definition 7.22. We denote by v i n ∈ Δ ( 0 , n ) v_i^n \in\Delta(0,n) v i n ∈ Δ ( 0 , n ) i i i i ∈ [ n ] i \in[n] i ∈ [ n ]
Convention 7.23. Since K K K K 0 K_0 K 0 K n K_n K n
K 0 → ∼ K n , k ↦ k n : = k ◃ ω n . K_0 \xrightarrow{\sim} K_n,\qquad k \mapsto k_n := k \triangleleft\omega_n. K 0 ∼ K n , k ↦ k n := k ◃ ω n . Note that k ↦ k n k \mapsto k_n k ↦ k n
Lemma 7.24. Let f ∈ Δ H ( m , n ) f \in\Delta H(m,n) f ∈ Δ H ( m , n ) v i m v_i^m v i m
p − 1 ( f ) ⟶ p − 1 ( f ∘ v i m ) , f ˉ ↦ f ˉ ∘ v i m . p^{-1}(f) \longrightarrow p^{-1}(f \circ v_i^m),\qquad\bar f \mapsto\bar f \circ v_i^m. p − 1 ( f ) ⟶ p − 1 ( f ∘ v i m ) , f ˉ ↦ f ˉ ∘ v i m . Proof. Write f = φ ∘ g f=\varphi\circ g f = φ ∘ g g ˉ ∈ G m \bar g\in G_m g ˉ ∈ G m f ˉ : = φ ∘ g ˉ \bar f:=\varphi\circ\bar g f ˉ := φ ∘ g ˉ f f f f f f
p − 1 ( f ) = { φ ∘ g ˉ ∘ k m : k ∈ K 0 } = { f ˉ ∘ k m : k ∈ K 0 } . p^{-1}(f)=\{\varphi\circ\bar g\circ k_m:k\in K_0\}=\{\bar f\circ k_m:k\in K_0\}. p − 1 ( f ) = { φ ∘ g ˉ ∘ k m : k ∈ K 0 } = { f ˉ ∘ k m : k ∈ K 0 } . Now write k m ∘ v i m = v j m ∘ a k_m\circ v_i^m=v_j^m\circ a k m ∘ v i m = v j m ∘ a a ∈ G 0 a\in G_0 a ∈ G 0 p p p v i m = v j m ∘ p ( a ) v_i^m=v_j^m\circ p(a) v i m = v j m ∘ p ( a ) i = j i=j i = j a ∈ K 0 a\in K_0 a ∈ K 0 a = k m ◃ v i m a=k_m\triangleleft v_i^m a = k m ◃ v i m K K K a = k a=k a = k
k m ∘ v i m = v i m ∘ k k_m\circ v_i^m=v_i^m\circ k k m ∘ v i m = v i m ∘ k for all v i m v_i^m v i m k ∈ K 0 k\in K_0 k ∈ K 0 f ˉ ∘ v i m \bar f\circ v_i^m f ˉ ∘ v i m f ∘ v i m f\circ v_i^m f ∘ v i m
{ f ˉ ∘ v i m ∘ k : k ∈ K 0 } = { f ˉ ∘ k m ∘ v i m : k ∈ K 0 } . \{\bar f\circ v_i^m\circ k:k\in K_0\}=\{\bar f\circ k_m\circ v_i^m:k\in K_0\}. { f ˉ ∘ v i m ∘ k : k ∈ K 0 } = { f ˉ ∘ k m ∘ v i m : k ∈ K 0 } . Using the lemma, we see that lifts of morphisms are uniquely determined if we specify their value on a vertex.
Construction 7.25. We construct a functor Φ : Δ P → Δ G \Phi:\Delta P\to\Delta G Φ : Δ P → Δ G p ∘ Φ = π p\circ\Phi=\pi p ∘ Φ = π n ≥ 1 n\ge1 n ≥ 1 t n ∈ G n t_n\in G_n t n ∈ G n τ n ∈ H n \tau_n\in H_n τ n ∈ H n t n ∘ v 0 n = v 1 n t_n\circ v_0^n=v_1^n t n ∘ v 0 n = v 1 n t n ∘ δ n = δ 0 t_n\circ\delta_n=\delta_0 t n ∘ δ n = δ 0 p p p
t n ∘ v i n = v i + 1 n ( 0 ≤ i < n ) . t_n\circ v_i^n=v_{i+1}^n\qquad(0\le i<n). t n ∘ v i n = v i + 1 n ( 0 ≤ i < n ) . We define z ∈ K 0 z\in K_0 z ∈ K 0 t 1 ∘ v 1 1 = v 0 1 ∘ z t_1\circ v_1^1=v_0^1\circ z t 1 ∘ v 1 1 = v 0 1 ∘ z
t n ∘ v n n = v 0 n ∘ z . t_n\circ v_n^n=v_0^n\circ z. t n ∘ v n n = v 0 n ∘ z . Indeed, let φ , ψ ∈ Δ ( 1 , n ) \varphi,\psi\in\Delta(1,n) φ , ψ ∈ Δ ( 1 , n ) ( n − 1 , n ) (n-1,n) ( n − 1 , n ) ( 0 , n ) (0,n) ( 0 , n ) τ n ∘ φ = ψ ∘ τ 1 \tau_n\circ\varphi=\psi\circ\tau_1 τ n ∘ φ = ψ ∘ τ 1 t n ∘ φ t_n\circ\varphi t n ∘ φ ψ ∘ t 1 \psi\circ t_1 ψ ∘ t 1 t 0 : = z t_0:=z t 0 := z
For a ∈ G 0 a\in G_0 a ∈ G 0 a n : = a ◃ ω n ∈ G n a_n:=a\triangleleft\omega_n\in G_n a n := a ◃ ω n ∈ G n r a , n r_{a,n} r a , n ϵ ( a ) = 1 \epsilon(a)=1 ϵ ( a ) = 1 j ↦ n − j j\mapsto n-j j ↦ n − j Δ H \Delta H Δ H
a n ∘ v i n = v r a , n ( i ) n ∘ a ( 0 ≤ i ≤ n ) . a_n\circ v_i^n=v_{r_{a,n}(i)}^n\circ a\qquad(0\le i\le n). a n ∘ v i n = v r a , n ( i ) n ∘ a ( 0 ≤ i ≤ n ) . Indeed, composing with ω n \omega_n ω n a a a p p p a ∘ z = z ϵ ( a ) ∘ a a \circ z = z^{\epsilon(a)} \circ a a ∘ z = z ϵ ( a ) ∘ a v j n v_j^n v j n j ∈ Z j \in\mathbb{Z} j ∈ Z j = q ( n + 1 ) + i j=q(n+1)+i j = q ( n + 1 ) + i 0 ≤ i ≤ n 0 \le i \le n 0 ≤ i ≤ n
v j n : = v i n ∘ z q ∈ Δ G ( 0 , n ) . v_j^n := v_i^n \circ z^q \in\Delta G(0,n). v j n := v i n ∘ z q ∈ Δ G ( 0 , n ) . The previous equations involving vertex operators hold in the same way for j ∈ Z j \in\mathbb{Z} j ∈ Z φ ∈ Δ ( m , n ) \varphi\in\Delta(m,n) φ ∈ Δ ( m , n ) Z \mathbb{Z} Z
φ ∘ v j m = v φ ( j ) n ( j ∈ Z ) . \varphi\circ v_j^m = v_{\varphi(j)}^n \qquad(j \in\mathbb{Z}). φ ∘ v j m = v φ ( j ) n ( j ∈ Z ) . For ( f , a ) ∈ Δ P ( m , n ) (f,a) \in\Delta P(m,n) ( f , a ) ∈ Δ P ( m , n ) Φ ( f , a ) ∈ Δ G ( m , n ) \Phi(f,a) \in\Delta G(m,n) Φ ( f , a ) ∈ Δ G ( m , n )
p ( Φ ( f , a ) ) = π ( f , a ) , Φ ( f , a ) ∘ v 0 m = v f ( 0 ) n ∘ a . p(\Phi(f,a))=\pi(f,a), \qquad\Phi(f,a)\circ v_0^m=v_{f(0)}^n\circ a. p ( Φ ( f , a )) = π ( f , a ) , Φ ( f , a ) ∘ v 0 m = v f ( 0 ) n ∘ a . In fact, the assignment satisfies the stronger property
Φ ( f , a ) ∘ v j m = v f ( j ) n ∘ a ( j ∈ Z ) . \Phi(f,a)\circ v_j^m=v_{f(j)}^n\circ a \qquad(j\in\mathbb{Z}). Φ ( f , a ) ∘ v j m = v f ( j ) n ∘ a ( j ∈ Z ) . Indeed, let f = φ ∘ τ m ℓ ∘ ρ a , m ∈ Ξ ∞ ( m , n ) f=\varphi\circ\tau_m^\ell\circ\rho_{a,m}\in\Xi_\infty(m,n) f = φ ∘ τ m ℓ ∘ ρ a , m ∈ Ξ ∞ ( m , n ) π ( f , a ) \pi(f,a) π ( f , a ) φ ∘ t m ℓ ∘ a m \varphi\circ t_m^\ell\circ a_m φ ∘ t m ℓ ∘ a m Δ G \Delta G Δ G
φ ∘ t m ℓ ∘ a m ∘ v j m = φ ∘ t m ℓ ∘ v ρ a , m ( j ) m ∘ a = v φ ( ρ a , m ( j ) + ℓ ) n ∘ a = v f ( j ) n ∘ a . \begin{aligned}
\varphi\circ t_m^\ell\circ a_m\circ v_j^m
&=\varphi\circ t_m^\ell\circ v_{\rho_{a,m}(j)}^m\circ a\\
&=v_{\varphi(\rho_{a,m}(j)+\ell)}^n\circ a\\
&=v_{f(j)}^n\circ a.
\end{aligned} φ ∘ t m ℓ ∘ a m ∘ v j m = φ ∘ t m ℓ ∘ v ρ a , m ( j ) m ∘ a = v φ ( ρ a , m ( j ) + ℓ ) n ∘ a = v f ( j ) n ∘ a . Functoriality of Φ \Phi Φ
Theorem 7.26. There exists an element z ∈ K 0 z \in K_0 z ∈ K 0 a z a − 1 = z ϵ ( a ) aza^{-1}=z^{\epsilon(a)} a z a − 1 = z ϵ ( a ) a ∈ G 0 a\in G_0 a ∈ G 0 Δ P / ⟨ ( τ n n + 1 , 1 ) = ( id n , z ) : n ≥ 0 ⟩ ≅ Δ G \Delta P/\langle(\tau_n^{n+1},1)=(\operatorname{id}_n,z):n\ge0\rangle\cong\Delta G Δ P / ⟨( τ n n + 1 , 1 ) = ( id n , z ) : n ≥ 0 ⟩ ≅ Δ G
Proof. We will show that Φ : Δ P → Δ G \Phi:\Delta P\to\Delta G Φ : Δ P → Δ G Φ \Phi Φ F ∈ Δ G ( m , n ) F\in\Delta G(m,n) F ∈ Δ G ( m , n ) f ∈ Ξ ∞ ( m , n ) f\in\Xi_\infty(m,n) f ∈ Ξ ∞ ( m , n ) p ( F ) ∈ Δ H ( m , n ) p(F)\in\Delta H(m,n) p ( F ) ∈ Δ H ( m , n ) 0 ≤ f ( 0 ) ≤ n 0\le f(0)\le n 0 ≤ f ( 0 ) ≤ n
F ∘ v 0 m = v f ( 0 ) n ∘ a F\circ v_0^m=v_{f(0)}^n\circ a F ∘ v 0 m = v f ( 0 ) n ∘ a for some a ∈ G 0 a\in G_0 a ∈ G 0 ϵ ( a ) = sgn ( f ) \epsilon(a)=\operatorname{sgn}(f) ϵ ( a ) = sgn ( f ) F = Φ ( f , a ) F=\Phi(f,a) F = Φ ( f , a ) Φ \Phi Φ
It is left to characterize the kernel equivalence relation of Φ \Phi Φ
Φ ( f , a ) = Φ ( g , b ) ⟺ g = τ n ( n + 1 ) ℓ ∘ f and b = z − ℓ ∘ a for some ℓ ∈ Z . \Phi(f,a)=\Phi(g,b)\Longleftrightarrow g=\tau_n^{(n+1)\ell}\circ f\text{ and }b=z^{-\ell}\circ a\text{ for some }\ell\in\mathbb{Z}. Φ ( f , a ) = Φ ( g , b ) ⟺ g = τ n ( n + 1 ) ℓ ∘ f and b = z − ℓ ∘ a for some ℓ ∈ Z . Indeed, equality after applying p p p f f f g g g τ n n + 1 \tau_n^{n+1} τ n n + 1
v f ( 0 ) n ∘ a = v g ( 0 ) n ∘ b = v f ( 0 ) n ∘ z ℓ ∘ b v_{f(0)}^n\circ a=v_{g(0)}^n\circ b=v_{f(0)}^n\circ z^\ell\circ b v f ( 0 ) n ∘ a = v g ( 0 ) n ∘ b = v f ( 0 ) n ∘ z ℓ ∘ b which by uniqueness of this form, is equivalent to b = z − ℓ ∘ a b=z^{-\ell}\circ a b = z − ℓ ∘ a ( τ n n + 1 , 1 ) ∼ ( id n , z ) (\tau_n^{n+1},1)\sim(\operatorname{id}_n,z) ( τ n n + 1 , 1 ) ∼ ( id n , z ) n ≥ 0 n\ge0 n ≥ 0
Corollary 7.27. The lifted order adjoint self-duality of Δ P \Delta P Δ P
( Δ G ) o p ⟶ Δ G , n ↦ n , Φ ( f , a ) ↦ Φ ( L ˉ ( f ) , a − 1 ) . (\Delta G)^{\mathrm{op}}\longrightarrow\Delta G,\qquad n\mapsto n,\qquad\Phi(f,a)\mapsto\Phi(\bar{L}(f),a^{-1}). ( Δ G ) op ⟶ Δ G , n ↦ n , Φ ( f , a ) ↦ Φ ( L ˉ ( f ) , a − 1 ) . Proof. The self-duality on Δ P \Delta P Δ P
( τ n n + 1 , 1 ) ↦ ( τ n n + 1 , 1 ) − 1 and ( id n , z ) ↦ ( id n , z ) − 1 (\tau_n^{n+1},1)\mapsto(\tau_n^{n+1},1)^{-1}\quad\text{and}\quad(\operatorname{id}_n,z)\mapsto(\operatorname{id}_n,z)^{-1} ( τ n n + 1 , 1 ) ↦ ( τ n n + 1 , 1 ) − 1 and ( id n , z ) ↦ ( id n , z ) − 1 so it preserves the generating relations. Its inverse does likewise, so it descends to a self-duality on Δ G \Delta G Δ G
References
TU Dresden, Institut für Geometrie, Zellescher Weg 12–14, 01062 Dresden, Germany
Email address: florian.warg@tu-dresden.de
