Introduction
Thomason’s model structure equips the category of small categories with the homotopy theory of spaces. Its higher-dimensional analogue asks for the same result on the entire category of strict globular n n n
Fix a universe of small sets and enlarge the universe when forming categories of small objects. For a positive integer n n n n C a t n\mathrm{Cat} n Cat n n n ω C a t \omega\mathrm{Cat} ω Cat ω \omega ω n n n n = ∞ n = \infty n = ∞ n C a t = ω C a t n\mathrm{Cat} = \omega\mathrm{Cat} n Cat = ω Cat
c n : s S e t ⇄ n C a t : N n , ( N n X ) m = Hom ω C a t ( O m , X ) , c_n : \mathrm{sSet} \rightleftarrows n\mathrm{Cat} : N_n,\qquad(N_nX)_m = \operatorname{Hom}_{\omega\mathrm{Cat}}(\mathcal{O}_m,X), c n : sSet ⇄ n Cat : N n , ( N n X ) m = Hom ω Cat ( O m , X ) , where O m \mathcal{O}_m O m m m m S d ⊣ E x \mathrm{Sd} \dashv\mathrm{Ex} Sd ⊣ Ex
L n = c n S d 2 , R n = E x 2 N n . L_n = c_n\mathrm{Sd}^{2},\qquad R_n = \mathrm{Ex}^{2}N_n. L n = c n Sd 2 , R n = Ex 2 N n . Write W K Q W_{\mathrm{KQ}} W KQ F i b K Q \mathrm{Fib}_{\mathrm{KQ}} Fib KQ
W n = R n − 1 ( W K Q ) = N n − 1 ( W K Q ) , F n = R n − 1 ( F i b K Q ) , C n = ⊥ ( F n ∩ W n ) . (1.1) W_n = R_n^{-1}(W_{\mathrm{KQ}}) = N_n^{-1}(W_{\mathrm{KQ}}),\qquad F_n = R_n^{-1}(\mathrm{Fib}_{\mathrm{KQ}}),\qquad C_n = {}^\perp(F_n \cap W_n). \tag*{(1.1)} W n = R n − 1 ( W KQ ) = N n − 1 ( W KQ ) , F n = R n − 1 ( Fib KQ ) , C n = ⊥ ( F n ∩ W n ) . (1.1) Here ⊥ {}^\perp ⊥ W n W_n W n E x \mathrm{Ex} Ex
Theorem 1.1. For every n ∈ { 1 , 2 , 3 , … } ∪ { ∞ } n \in\{1,2,3,\ldots\} \cup\{\infty\} n ∈ { 1 , 2 , 3 , … } ∪ { ∞ } ( C n , W n , F n ) (C_n,W_n,F_n) ( C n , W n , F n ) (1.1) form a proper combinatorial model structure on n C a t n\mathrm{Cat} n Cat ( L n I , L n J ) (L_nI,L_nJ) ( L n I , L n J )
I = { ∂ Δ [ m ] ↪ Δ [ m ] : m ≥ 0 } , J = { Λ k [ m ] ↪ Δ [ m ] : m ≥ 1 , 0 ≤ k ≤ m } . I = \{\partial\Delta[m] \hookrightarrow\Delta[m] : m \ge0\},\qquad J = \{\Lambda^k[m] \hookrightarrow\Delta[m] : m \ge1,\ 0 \le k \le m\}. I = { ∂ Δ [ m ] ↪ Δ [ m ] : m ≥ 0 } , J = { Λ k [ m ] ↪ Δ [ m ] : m ≥ 1 , 0 ≤ k ≤ m } . Moreover,
L n : s S e t K Q ⇄ ( n C a t , C n , W n , F n ) : R n L_n : \mathrm{sSet}_{\mathrm{KQ}} \rightleftarrows(n\mathrm{Cat},C_n,W_n,F_n) : R_n L n : sSet KQ ⇄ ( n Cat , C n , W n , F n ) : R n is a Quillen equivalence.
History and significance
Thomason established the case n = 1 n = 1 n = 1 c 1 S d 2 ⊣ E x 2 N 1 c_1\mathrm{Sd}^{2} \dashv\mathrm{Ex}^{2}N_1 c 1 Sd 2 ⊣ Ex 2 N 1 [17 ] . His proof separates two issues that persist in higher dimensions: the homotopy behavior of the adjunction unit and the comparison between a pushout of nerves and the nerve of a categorical pushout. For the latter he introduced Dwyer maps and proved the comparison for their arbitrary cobase changes [17 Definition 4.1 and Proposition 4.3] . Cisinski corrected the retract argument used for properness, while preserving the model structure and its properness [7 ] . Street’s orientals extend the nerve to strict higher categories [16 ] , providing the weak equivalences in the question considered here.
Ara and Maltsiniotis developed an abstract transfer theorem, drawing on the homotopy theory of categories developed by Grothendieck and Cisinski, and established the proper combinatorial model structure in dimension two. Using Chiche’s comparison theorem for the homotopy categories of categories and 2-categories, they also obtained the Quillen equivalence [4 Theorems 4.11, 6.27, 6.30, and Corollary 6.32] . Worytkiewicz, Hess, Parent, and Tonks had proposed the two-dimensional structure earlier [18 ] ; their later corrigendum addresses the gaps identified by Ara and Maltsiniotis [19 ] . For arbitrary 1 ≤ n ≤ ∞ 1 \le n \le\infty 1 ≤ n ≤ ∞ [4 Scholie 5.14] .
Ara and Maltsiniotis proved the poset-unit condition in all these dimensions [5 Corollary 10.11 and Scholie 10.12] . Gagna subsequently gave a direct simplicial proof and showed that the Street nerve induces an equivalence with homotopy types after localizing at the nerve weak equivalences [9 Theorems 5.6 and 6.9] . Thus strict higher categories already described the homotopy category of spaces in every dimension. The prescribed model structure requires the additional lifting and factorization statements, whose construction depends on controlling actual categorical pushouts. Ara’s 2023 account records this remaining sufficient condition and the general conjecture [3 §§9.2.5–9.2.6] ; Guetta and Maltsiniotis also state the ω \omega ω [11 Introduction, p. 6] .
The shapes of higher cells matter in this question. Fiore and Paoli constructed a Thomason model structure on small n n n n n n [8 Theorem 9.28] . Their objects are obtained by iterating the internal-category construction; the present theorem concerns strict globular categories and the Street nerve.
Theorem 1.1 supplies both weak factorization systems on all strict n n n ω \omega ω n n n
Stationary cylinders and the pushout argument
For a poset E E E U E = c n N E U_E = c_nNE U E = c n N E N N N A ⊆ E A \subseteq E A ⊆ E U A → U E U_A \to U_E U A → U E [4 Scholie 5.14, condition (d′)] . The cylinders used for contraction fix U A U_A U A U A → X U_A \to X U A → X X X X
The construction starts with a restricted but explicit directed prism. Suppose f , g : E → E f,g\colon E \to E f , g : E → E f ≤ g f \leq g f ≤ g A A A g g g E ∖ A E \setminus A E ∖ A u u u w w w
[ w 0 , … , w p ] ⊗ u ⟼ [ f w 0 , … , f w p , g w p ] [w_0,\ldots,w_p]\otimes u \longmapsto[fw_0,\ldots,fw_p,gw_p] [ w 0 , … , w p ] ⊗ u ⟼ [ f w 0 , … , f w p , g w p ] together with the endpoint maps induced by f f f g g g U A U_A U A f ∣ A = g ∣ A f|_A = g|_A f ∣ A = g ∣ A
When E ∖ A E \setminus A E ∖ A id A \operatorname{id}_A id A A A A
The remaining comparison follows the geometry of Thomason’s Dwyer maps [17 Propositions 4.2–4.3] . Two subdivisions put each generating boundary or horn inclusion into the form N A ↪ N B NA \hookrightarrow NB N A ↪ N B A A A B B B U A → X U_A \to X U A → X
At the chain level we use Steiner’s description of orientals [14 , 15 ] and the biclosed Gray tensor product, with the completed extension theorem of Ara and Maltsiniotis [6 Theorem A.15] . Positive-chain constructions already give simplicial nerve homotopies in Gagna’s proof [9 Sections 6.3–6.8] . Here the restricted prism hypotheses give strict relative cylinders; the choices fixing A A A [4 Lemma 2.8 and Propositions 5.3 and 5.5] ; they allow us to glue the weak-equivalence conclusions without choosing compatible contractions on overlapping pieces.
Organization
Section 2 records the foundations and the precise simplicial gluing lemma. Sections 3 and 4 construct the stationary cylinders and identify labelled pieces. Section 5 proves couniversality and the unit comparison for posets. Section 6 proves the nerve comparison for subdivided boundary and horn attachments, with explicit control of the monomorphic gluing legs. Section 7 supplies the model axioms, properness, and the unit comparison for arbitrary simplicial sets. Appendix A proves the required presentability and smallness assertions by guarded term presentations. Appendix B discusses the relationship with the elementary-expansion method for Grothendieck groupoids; its claims are not inputs to the proof.
Strict categories, directed chains, and simplicial gluing
The pushout argument needs three foundations: directed chain maps must give strict right cylinders, these cylinders must descend to finite dimensions, and their nerve comparisons must survive gluing and filtered colimits. We fix the conventions and prove the needed local statements before constructing the relative prism.
We work with small objects in a fixed universe. An ω \omega ω ω \omega ω n n n n n n n n n n C a t ↪ ω C a t n\mathrm{Cat} \hookrightarrow\omega\mathrm{Cat} n Cat ↪ ω Cat τ n \tau_n τ n τ n X \tau_nX τ n X X X X n n n n n n n n n X X X
s n ( z ) = t n ( z ) ( z an ( n + 1 ) -cell of X ) . s_n(z)=t_n(z) \qquad(z\text{ an }(n+1)\text{-cell of }X). s n ( z ) = t n ( z ) ( z an ( n + 1 ) -cell of X ) . Here the congruence is closed under every defined composition. Its generating pairs are parallel n n n n n n X X X n n n τ ∞ = id \tau_\infty=\operatorname{id} τ ∞ = id n C a t = ω C a t n\mathrm{Cat}=\omega\mathrm{Cat} n Cat = ω Cat n = ∞ n=\infty n = ∞
Write D i D_i D i ω \omega ω i i i Hom ω C a t ( D i , X ) \operatorname{Hom}_{\omega\mathrm{Cat}}(D_i,X) Hom ω Cat ( D i , X ) i i i X X X D 0 D_0 D 0 D 1 D_1 D 1 0 → 1 0\to1 0 → 1 O m \mathcal{O}_m O m m m m
( N n X ) m = Hom ω C a t ( O m , X ) = Hom n C a t ( τ n O m , X ) . (N_nX)_m=\operatorname{Hom}_{\omega\mathrm{Cat}}(\mathcal{O}_m,X)=\operatorname{Hom}_{n\mathrm{Cat}}(\tau_nO_m,X). ( N n X ) m = Hom ω Cat ( O m , X ) = Hom n Cat ( τ n O m , X ) . For ω \omega ω N ∞ N_{\infty} N ∞
c ∞ K = colim ( Δ [ m ] → K ) ∈ Δ / K O m , c n K = τ n c ∞ K = colim ( Δ [ m ] → K ) ∈ Δ / K τ n O m . (2.1) c_{\infty}K=\underset{(\Delta[m]\to K)\in\Delta/K}{\operatorname{colim}}\mathcal{O}_{m},\qquad c_{n}K=\tau_{n}c_{\infty}K=\underset{(\Delta[m]\to K)\in\Delta/K}{\operatorname{colim}}\tau_{n}\mathcal{O}_{m}. \tag*{(2.1)} c ∞ K = ( Δ [ m ] → K ) ∈ Δ/ K colim O m , c n K = τ n c ∞ K = ( Δ [ m ] → K ) ∈ Δ/ K colim τ n O m . (2.1) These formulas follow from the universal property of a colimit and the degreewise definition of the nerve. In particular, for a poset E E E
U E ∞ = c ∞ N E , U E = c n N E = τ n U E ∞ . U^{\infty}_{E}=c_{\infty}NE,\qquad U_{E}=c_{n}NE=\tau_{n}U^{\infty}_{E}. U E ∞ = c ∞ N E , U E = c n N E = τ n U E ∞ . The fixed dimension is suppressed in U U U f f f U ( f ) = c n N ( f ) U(f)=c_{n}N(f) U ( f ) = c n N ( f ) U ∞ ( f ) = c ∞ N ( f ) U^{\infty}(f)=c_{\infty}N(f) U ∞ ( f ) = c ∞ N ( f ) N N N N ∞ D 1 = Δ [ 1 ] N_{\infty}D_{1}=\Delta[1] N ∞ D 1 = Δ [ 1 ]
The directed tensor convention
An augmented directed complex is a nonnegative chain complex K K K ϵ : K 0 → Z \epsilon:K_{0}\to\mathbb{Z} ϵ : K 0 → Z K p + ⊂ K p K^{+}_{p}\subset K_{p} K p + ⊂ K p d c = d + c − d − c dc=d^{+}c-d^{-}c d c = d + c − d − c unital when ϵ ( ( d − ) p b ) = ϵ ( ( d + ) p b ) = 1 \epsilon((d^{-})^{p}b)=\epsilon((d^{+})^{p}b)=1 ϵ (( d − ) p b ) = ϵ (( d + ) p b ) = 1 b b b p p p strongly loop-free when the relations
a < b if a occurs in d − b , b < a if a occurs in d + b a<b\quad\text{if }a\text{ occurs in }d^{-}b,\qquad b<a\quad\text{if }a\text{ occurs in }d^{+}b a < b if a occurs in d − b , b < a if a occurs in d + b are contained in a strict partial order on the whole basis. A complex with such a basis is called a strong Steiner complex .
Steiner’s functor ν \nu ν ω \omega ω p p p ν K \nu K ν K
( x 0 − , x 0 + ∣ x 1 − , x 1 + ∣ ⋯ ∣ x p − , x p + ) , x r ± ∈ K r + , (2.2) (x^{-}_{0},x^{+}_{0}\mid x^{-}_{1},x^{+}_{1}\mid\cdots\mid x^{-}_{p},x^{+}_{p}),\qquad x^{\pm}_{r}\in K^{+}_{r}, \tag*{(2.2)} ( x 0 − , x 0 + ∣ x 1 − , x 1 + ∣ ⋯ ∣ x p − , x p + ) , x r ± ∈ K r + , (2.2) such that
ϵ x 0 − = ϵ x 0 + = 1 , d x r − = d x r + = x r − 1 + − x r − 1 − ( 1 ≤ r ≤ p ) , x p − = x p + . \epsilon x^{-}_{0}=\epsilon x^{+}_{0}=1,\qquad dx^{-}_{r}=dx^{+}_{r}=x^{+}_{r-1}-x^{-}_{r-1}\quad(1\leq r\leq p),\qquad x^{-}_{p}=x^{+}_{p}. ϵ x 0 − = ϵ x 0 + = 1 , d x r − = d x r + = x r − 1 + − x r − 1 − ( 1 ≤ r ≤ p ) , x p − = x p + . An identity appends a zero pair to the table. A positive, augmentation-preserving chain map acts on every entry, giving the corresponding strict functor [14 Definitions 2.6 and 2.8, Proposition 2.7] . For a basis element b b b p p p atom has top pair ( b , b ) (b,b) ( b , b ) ⟨ b ⟩ r ± = ( d ± ) p − r b \langle b\rangle^{\pm}_{r}=(d^{\pm})^{p-r}b ⟨ b ⟩ r ± = ( d ± ) p − r b 0 ≤ r ≤ p 0\leq r\leq p 0 ≤ r ≤ p ν K \nu K ν K [14 §3 before Definition 3.2, Theorem 6.1] . The concrete complexes needed here are as follows:
The normalized simplex complex M [ m ] M[m] M [ m ] p p p [ i 0 , … , i p ] [i_{0},\ldots,i_{p}] [ i 0 , … , i p ] 0 ≤ i 0 < ⋯ < i p ≤ m 0\leq i_{0}<\cdots<i_{p}\leq m 0 ≤ i 0 < ⋯ < i p ≤ m
d [ i 0 , … , i p ] = ∑ j = 0 p ( − 1 ) j [ i 0 , … , i j ^ , … , i p ] ( p > 0 ) , ϵ [ i ] = 1. d[i_{0},\ldots,i_{p}]=\sum_{j=0}^{p}(-1)^{j}[i_{0},\ldots,\widehat{i_{j}},\ldots,i_{p}]\quad(p>0),\qquad\epsilon[i]=1. d [ i 0 , … , i p ] = j = 0 ∑ p ( − 1 ) j [ i 0 , … , i j , … , i p ] ( p > 0 ) , ϵ [ i ] = 1. It is strong Steiner and ν M [ m ] = O m \nu M[m]=\mathcal{O}_{m} ν M [ m ] = O m G [ i ] G[i] G [ i ] i i i i i i ν G [ i ] = D i \nu G[i] = D_i ν G [ i ] = D i i = 0 i = 0 i = 0
The strong Steiner properties and tensor closure follow from [14 Examples 3.8–3.10] . The disk identification follows from [14 Example 4.7 and Theorem 5.11] . For the oriental identification and its cosimplicial action, including repeated entries, see [15 Theorem 3.2 and §4 before Theorem 4.1] .
The tensor of augmented directed complexes uses the ordinary nonnegative chain-complex tensor, with separate augmentation:
d ( x ⊗ y ) = d x ⊗ y + ( − 1 ) ∣ x ∣ x ⊗ d y , ϵ ( x ⊗ y ) = ϵ ( x ) ϵ ( y ) ( ∣ x ∣ = ∣ y ∣ = 0 ) . (2.3) d(x \otimes y) = dx \otimes y + (-1)^{|x|}x \otimes dy,\qquad\epsilon(x \otimes y) = \epsilon(x)\epsilon(y) \qquad(|x| = |y| = 0). \tag*{(2.3)} d ( x ⊗ y ) = d x ⊗ y + ( − 1 ) ∣ x ∣ x ⊗ d y , ϵ ( x ⊗ y ) = ϵ ( x ) ϵ ( y ) ( ∣ x ∣ = ∣ y ∣ = 0 ) . (2.3) Its positive submonoid is generated by the tensors of positive elements. Tensor products of the indicated bases are the relevant positive bases.
We use the biclosed Gray tensor product on all strict ω \omega ω [6 Theorem A.15 and §A.16] . It has unit D 0 D_0 D 0
ν K ⊗ ν L ≅ ν ( K ⊗ L ) ( K , L strong Steiner ) . (2.4) \nu K \otimes\nu L \cong\nu(K \otimes L) \qquad(K, L\text{ strong Steiner}). \tag*{(2.4)} ν K ⊗ ν L ≅ ν ( K ⊗ L ) ( K , L strong Steiner ) . (2.4) Naturality here is for every morphism between strong Steiner complexes, including maps sending basis elements to sums or zero. This is the completed extension theorem; the general extension is not being inferred from the incomplete proof of [14 Theorem 7.3] . In particular, preservation of colimits in this paper is a property of the Gray tensor, not a claimed preservation property of ν \nu ν
It follows that D i ⊗ D 1 D_i \otimes D_1 D i ⊗ D 1 O m ⊗ D 1 \mathcal{O}_m \otimes D_1 O m ⊗ D 1
Throughout, the interval is the right tensor factor. Write e 0 , e 1 e_0,e_1 e 0 , e 1 M [ 1 ] M[1] M [ 1 ] u u u d u = e 1 − e 0 du = e_1 - e_0 d u = e 1 − e 0 e a : D 0 → D 1 e_a : D_0 \to D_1 e a : D 0 → D 1 ϵ : D 1 → D 0 \epsilon: D_1 \to D_0 ϵ : D 1 → D 0
π X = id X ⊗ ϵ : X ⊗ D 1 ⟶ X ⊗ D 0 ≅ X . \pi_X = \operatorname{id}_X \otimes\epsilon: X \otimes D_1 \longrightarrow X \otimes D_0 \cong X. π X = id X ⊗ ϵ : X ⊗ D 1 ⟶ X ⊗ D 0 ≅ X . No interchange of the two tensor factors is used.
Right cylinders and truncation
Biclosedness supplies an ω \omega ω H ( Y ) H(Y) H ( Y )
Hom ω C a t ( X , H ( Y ) ) ≅ Hom ω C a t ( X ⊗ D 1 , Y ) . (2.5) \operatorname{Hom}_{\omega\mathrm{Cat}}(X,H(Y)) \cong\operatorname{Hom}_{\omega\mathrm{Cat}}(X \otimes D_1,Y). \tag*{(2.5)} Hom ω Cat ( X , H ( Y )) ≅ Hom ω Cat ( X ⊗ D 1 , Y ) . (2.5) In the notation of [6 §A.18] , H ( Y ) = Hom o p l a x ( D 1 , Y ) H(Y) = \operatorname{Hom}_{\mathrm{oplax}}(D_1,Y) H ( Y ) = Hom oplax ( D 1 , Y ) (2.5) fixes the convention. Evaluation at e 0 , e 1 e_0,e_1 e 0 , e 1 π X \pi_X π X n C a t n\mathrm{Cat} n Cat H ( Y ) H(Y) H ( Y ) n C a t n\mathrm{Cat} n Cat Y Y Y
Lemma 2.1 (Internal hom and reflection). Let n < ∞ n < \infty n < ∞ and let Y Y Y be a strict n n n category. Then H ( Y ) H(Y) H ( Y ) is a strict n n n category. Consequently the reflection unit X → τ n X X \to\tau_n X X → τ n X induces a natural bijection
Hom ω C a t ( ( τ n X ) ⊗ D 1 , Y ) → ≅ Hom ω C a t ( X ⊗ D 1 , Y ) . (2.6) \operatorname{Hom}_{\omega\mathrm{Cat}}((\tau_n X) \otimes D_1,Y) \xrightarrow{\cong} \operatorname{Hom}_{\omega\mathrm{Cat}}(X \otimes D_1,Y). \tag*{(2.6)} Hom ω Cat (( τ n X ) ⊗ D 1 , Y ) ≅ Hom ω Cat ( X ⊗ D 1 , Y ) . (2.6) This bijection respects endpoints and restriction along functors in the first variable. For n = ∞ n = \infty n = ∞ the same assertion holds with τ n = id \tau_n = \operatorname{id} τ n = id
Proof. We prove the special case needed here directly; the general internal-hom truncation assertion is also [6 Proposition A.29] . An i i i H ( Y ) H(Y) H ( Y ) P : D i ⊗ D 1 → Y P : D_i \otimes D_1 \to Y P : D i ⊗ D 1 → Y s : D i − 1 → D i s : D_{i-1} \to D_i s : D i − 1 → D i p : D i → D i − 1 p : D_i \to D_{i-1} p : D i → D i − 1 i i i ( i − 1 ) (i-1) ( i − 1 ) p s = id ps = \operatorname{id} p s = id q = s p q = sp q = s p D i D_i D i P P P P ( q ⊗ id ) P(q \otimes\operatorname{id}) P ( q ⊗ id ) i > n i > n i > n
The atoms of D i ⊗ D 1 D_i \otimes D_1 D i ⊗ D 1 D i − 1 ⊗ D 1 D_{i-1} \otimes D_1 D i − 1 ⊗ D 1 i i i ( i − 1 ) (i-1) ( i − 1 ) D i D_i D i e 0 e_0 e 0 e 1 e_1 e 1 u u u i , i , i + 1 i,i,i+1 i , i , i + 1 i − 1 , i − 1 , i i-1,i-1,i i − 1 , i − 1 , i i > n + 1 i > n+1 i > n + 1 n n n i = n + 1 i = n+1 i = n + 1 n n n e 0 e_0 e 0 e 1 e_1 e 1 n n n ( n + 1 ) (n+1) ( n + 1 ) P P P n n n P P P P ( q ⊗ id ) P(q \otimes\operatorname{id}) P ( q ⊗ id ) n n n q ⊗ id q \otimes\operatorname{id} q ⊗ id
The two functors therefore agree on all atoms of dimension at most n n n n n n n n n n n n P = P ( q ⊗ id ) P = P(q \otimes\operatorname{id}) P = P ( q ⊗ id ) H ( Y ) H(Y) H ( Y ) n n n
Now transpose a cylinder to a functor X → H ( Y ) X \to H(Y) X → H ( Y ) H ( Y ) H(Y) H ( Y ) n n n τ n X \tau_n X τ n X Y Y Y
Remark 2.2. Even when X X X Y Y Y n n n X ⊗ D 1 X \otimes D_1 X ⊗ D 1 ω C a t \omega\mathrm{Cat} ω Cat n n n Y Y Y X → H ( Y ) X \to H(Y) X → H ( Y ) n C a t n\mathrm{Cat} n Cat n C a t n\mathrm{Cat} n Cat
The Alexander–Whitney map
For the Gray-cylinder-to-Street-nerve construction, see Ara [3 Sections 5.2.2–5.2.6, especially Proposition 5.2.3 and the right-cylinder formula in 5.2.6] . The verification needed here is included below; the classical Alexander–Whitney diagonal is credited separately in the proof.
Lemma 2.3 (A right Gray cylinder gives a nerve homotopy). There is a natural simplicial map
J X : N ∞ X × Δ [ 1 ] ⟶ N ∞ ( X ⊗ D 1 ) J_X : N_\infty X \times\Delta[1] \longrightarrow N_\infty(X \otimes D_1) J X : N ∞ X × Δ [ 1 ] ⟶ N ∞ ( X ⊗ D 1 ) whose restriction at the endpoint a a a is N ∞ ( id X ⊗ e a ) N_\infty(\operatorname{id}_X \otimes e_a) N ∞ ( id X ⊗ e a ) and for which N ∞ ( π X ) J X N_\infty(\pi_X)J_X N ∞ ( π X ) J X is the projection to N ∞ X N_\infty X N ∞ X Consequently a strict functor P : X ⊗ D 1 → Y P : X \otimes D_1 \to Y P : X ⊗ D 1 → Y induces a simplicial homotopy between the nerves of its endpoint functors. If a functor a : A → X a : A \to X a : A → X satisfies P ( a ⊗ id ) = h π A P(a \otimes\operatorname{id}) = h\pi_A P ( a ⊗ id ) = h π A for some h : A → Y h : A \to Y h : A → Y the homotopy restricts along N ∞ a N_\infty a N ∞ a to the constant homotopy of N ∞ h N_\infty h N ∞ h The same assertions apply to N n N_n N n when X , Y X,Y X , Y are n n n categories.
Proof. For each m m m [10 IV.2, Remark 2.5]
AW m [ i 0 , … , i p ] = ∑ a = 0 p [ i 0 , … , i a ] ⊗ [ i a , … , i p ] . (2.7) \operatorname{AW}_m[i_0,\ldots,i_p]=\sum_{a=0}^{p}[i_0,\ldots,i_a]\otimes[i_a,\ldots,i_p]. \tag*{(2.7)} AW m [ i 0 , … , i p ] = a = 0 ∑ p [ i 0 , … , i a ] ⊗ [ i a , … , i p ] . (2.7) It is a positive chain map M [ m ] → M [ m ] ⊗ M [ m ] M[m]\to M[m]\otimes M[m] M [ m ] → M [ m ] ⊗ M [ m ] AW m d \operatorname{AW}_m d AW m d ( − 1 ) j (-1)^j ( − 1 ) j i j i_j i j a a a
( − 1 ) a [ i 0 , … , i a − 1 ] ⊗ [ i a , … , i p ] , (-1)^a[i_0,\ldots,i_{a-1}]\otimes[i_a,\ldots,i_p], ( − 1 ) a [ i 0 , … , i a − 1 ] ⊗ [ i a , … , i p ] , whereas deleting the initial vertex of the second factor at cut a − 1 a-1 a − 1 ( − 1 ) a − 1 (-1)^{a-1} ( − 1 ) a − 1 (2.3) . The degree-zero assertion is immediate. Every summand in (2.7) is positive. On vertices the augmentation is one; applying the augmentation to the first or second tensor factor selects, respectively, only the cut a = 0 a=0 a = 0 a = p a=p a = p
The diagonal is natural for every monotone ordinal map. For a tuple whose image is strictly increasing, this follows term by term. If the image has equal consecutive entries in positions j , j + 1 j,j+1 j , j + 1 a ≤ j a\leq j a ≤ j a ≥ j + 1 a\geq j+1 a ≥ j + 1
Apply ν \nu ν (2.4) to obtain natural functors δ m : O m → O m ⊗ O m \delta_m:\mathcal{O}_m\to\mathcal{O}_m\otimes\mathcal{O}_m δ m : O m → O m ⊗ O m m m m F : O m → X F:\mathcal{O}_m\to X F : O m → X θ : [ m ] → [ 1 ] \theta:[m]\to[1] θ : [ m ] → [ 1 ]
J X ( F , θ ) = ( F ⊗ O θ ) δ m , J_X(F,\theta)=(F\otimes\mathcal{O}_\theta)\delta_m, J X ( F , θ ) = ( F ⊗ O θ ) δ m , where O θ : O m → O 1 = D 1 \mathcal{O}_\theta:\mathcal{O}_m\to\mathcal{O}_1=D_1 O θ : O m → O 1 = D 1 J X J_X J X θ \theta θ a a a e a e_a e a π X \pi_X π X N ∞ ( π X ) J X N_\infty(\pi_X)J_X N ∞ ( π X ) J X N ∞ P N_\infty P N ∞ P X X X
Finiteness and gluing in simplicial sets
We use the Kan–Quillen model structure on sSet. Its cofibrations are the monomorphisms, every object is cofibrant, and the model structure is proper. We also use the classical Kan replacement K → Ex ∞ K K\to\operatorname{Ex}^{\infty}K K → Ex ∞ K [10 II.8.6 and III.4.8] .
Lemma 2.4 (Presentability and filtered colimits). For every 1 ≤ n ≤ ∞ 1\leq n\leq\infty 1 ≤ n ≤ ∞ n C a t n\mathrm{Cat} n Cat τ n O m \tau_n\mathcal{O}_m τ n O m n C a t n\mathrm{Cat} n Cat N n N_n N n Ex r N n \operatorname{Ex}^rN_n Ex r N n r ≥ 0 r\geq0 r ≥ 0 f f f n C a t n\mathrm{Cat} n Cat N n f N_nf N n f
Proof. Strict n n n ω \omega ω [1 , 2 ] ; for its finitary partial-operation formulation, see also [12 pp. 52–53] .
For the cellwise colimit assertion one can argue directly. Form the filtered colimit of the sets of cells in each dimension. Any finite collection of representatives is represented at a common stage. If their boundaries agree in the colimit, those finitely many equalities hold at some later common stage, where the required composition is defined. Define its colimit value by that composition. A further common stage proves independence of choices, and each strict category law follows at a stage containing its finitely many entries. This constructs the colimit with its universal property. If all inputs are n n n n n n
The oriental O m \mathcal{O}_m O m [ m ] [m] [ m ] O m \mathcal{O}_m O m O m \mathcal{O}_m O m ω C a t \omega\mathrm{Cat} ω Cat n C a t ↪ ω C a t \mathord{n\mathrm{Cat}}\hookrightarrow\omega\mathrm{Cat} n Cat ↪ ω Cat τ n O m \tau_n\mathcal{O}_m τ n O m N n N_n N n [4 Proposition 5.13(a)] . Since ( Ex K ) m = Hom s S e t ( Sd Δ [ m ] , K ) (\operatorname{Ex}K)_m=\operatorname{Hom}_{\mathrm{sSet}}(\operatorname{Sd}\Delta[m],K) ( Ex K ) m = Hom sSet ( Sd Δ [ m ] , K ) Sd Δ [ m ] \operatorname{Sd}\Delta[m] Sd Δ [ m ] Ex \operatorname{Ex} Ex
Here are details of the weak-equivalence assertion. The functor Q = Ex ∞ Q=\operatorname{Ex}^{\infty} Q = Ex ∞ Ex \operatorname{Ex} Ex K i K_i K i x x x colim i K i \operatorname{colim}_i K_i colim i K i x i ∈ K i x_i\in K_i x i ∈ K i i → j i\to j i → j x j x_j x j k ≥ 1 k\ge1 k ≥ 1
colim ( i → j ) π k ( K j , x j ) ≅ π k ( colim j K j , x ) . \operatorname*{colim}_{(i\to j)}\pi_k(K_j,x_j)\cong\pi_k\left(\operatorname*{colim}_j K_j,x\right). ( i → j ) colim π k ( K j , x j ) ≅ π k ( j colim K j , x ) . Indeed, use the finite pointed sphere Δ [ k ] / ∂ Δ [ k ] \Delta[k]/\partial\Delta[k] Δ [ k ] / ∂ Δ [ k ] Q Q Q K → Q K K\to QK K → Q K N n N_n N n
Lemma 2.5 (Gluing with one monomorphic leg). Consider a commutative map of spans of simplicial sets
A ← C ↪ i B a ↓ c ↓ ↓ b A ′ ← C ′ ↪ i ′ B ′ \begin{array}{ccccc}
A & \xleftarrow{\ \ } & C & \xhookrightarrow{i} & B \\
{\scriptstyle a}\,\downarrow & & {\scriptstyle c}\,\downarrow & & \downarrow\,{\scriptstyle b} \\
A' & \xleftarrow{\ \ } & C' & \xhookrightarrow{i'} & B'
\end{array} A a ↓ A ′ C c ↓ C ′ i i ′ B ↓ b B ′ where i i i i ′ i' i ′ a , b , c a,b,c a , b , c
A ⨿ C B ⟶ A ′ ⨿ C ′ B ′ A\amalg_C B\longrightarrow A'\amalg_{C'} B' A ⨿ C B ⟶ A ′ ⨿ C ′ B ′ is a weak equivalence. There is no condition on the maps C → A C \to A C → A C ′ → A ′ C' \to A' C ′ → A ′
Proof. This is the simplicial gluing lemma [10 II.8.8 and II.8.12] ; the following proof keeps track of the specified cofibrations. Put
E = B ⨿ C C ′ . E = B \amalg_C C'. E = B ⨿ C C ′ . By simplicial left properness, B → E B \to E B → E c c c i i i E → B ′ E \to B' E → B ′ C ′ → E C' \to E C ′ → E C ′ → B ′ C' \to B' C ′ → B ′ C ′ / s S e t C'/\mathrm{sSet} C ′ / sSet C ′ → A ′ C' \to A' C ′ → A ′
C ′ / s S e t ⟶ A ′ / s S e t : ( C ′ → T ) ⟼ ( A ′ → A ′ ⨿ C ′ T ) . C'/\mathrm{sSet} \longrightarrow A'/\mathrm{sSet} : (C' \to T) \longmapsto(A' \to A' \amalg_{C'} T). C ′ / sSet ⟶ A ′ / sSet : ( C ′ → T ) ⟼ ( A ′ → A ′ ⨿ C ′ T ) . Indeed, its right adjoint is restriction of the structure map, which preserves fibrations and trivial fibrations. It preserves weak equivalences between cofibrant objects, so
A ′ ⨿ C B ≅ A ′ ⨿ C ′ E ⟶ A ′ ⨿ C ′ B ′ A' \amalg_C B \cong A' \amalg_{C'} E \longrightarrow A' \amalg_{C'} B' A ′ ⨿ C B ≅ A ′ ⨿ C ′ E ⟶ A ′ ⨿ C ′ B ′ is a weak equivalence. Also A → A ⨿ C B A \to A \amalg_C B A → A ⨿ C B i i i a a a A ⨿ C B → A ′ ⨿ C B A \amalg_C B \to A' \amalg_C B A ⨿ C B → A ′ ⨿ C B
To see the homotopy-pushout assertion, factor the possibly nonmonomorphic map C → A C \to A C → A C ↪ A ~ C \hookrightarrow\widetilde{A} C ↪ A A ~ → A \widetilde{A} \to A A → A A ~ ← C ↪ B \widetilde{A} \leftarrow C \hookrightarrow B A ← C ↪ B A ⨿ C B A \amalg_C B A ⨿ C B X ← X ↪ Z X \leftarrow X \hookrightarrow Z X ← X ↪ Z Y ← X ↪ Z Y \leftarrow X \hookrightarrow Z Y ← X ↪ Z X → Y X \to Y X → Y id X \operatorname{id}_X id X id Z \operatorname{id}_Z id Z
Positive relative prisms
Throughout this section the interval is the right tensor factor. At the chain level this is the antihomotopy convention of [6 Appendix B.4.1–B.4.2] . Gagna constructs simplicial homotopies by positive concatenation of chains under an order condition on the endpoint maps [9 Sections 6.3–6.8] . We use a related positive formula under the sieve and constancy hypotheses below to construct a strict right Gray cylinder stationary on the entire old subcategory. When its endpoint maps fix that subcategory, it glues to the constant cylinder on an arbitrary attaching target. Write ϵ : D 1 → D 0 \epsilon: D_1 \to D_0 ϵ : D 1 → D 0
π B = id B ⊗ ϵ : B ⊗ D 1 ⟶ B . \pi_B = \operatorname{id}_B \otimes\epsilon: B \otimes D_1 \longrightarrow B. π B = id B ⊗ ϵ : B ⊗ D 1 ⟶ B . For i : A → B i : A \to B i : A → B v : A → C v : A \to C v : A → C P : B ⊗ D 1 → C P : B \otimes D_1 \to C P : B ⊗ D 1 → C stationary along i i i v v v if P ( i ⊗ id ) = v π A P(i \otimes\operatorname{id}) = v\pi_A P ( i ⊗ id ) = v π A A A A A A A
Lemma 3.1 (Relative prism). Let A ⊂ E A \subset E A ⊂ E f , g : E → E f,g : E \to E f , g : E → E
f ( e ) ≤ g ( e ) ( e ∈ E ) , f ∣ A = g ∣ A , g ∣ E ∖ A is constant . f(e) \leq g(e) \quad(e \in E), \qquad f|_A = g|_A, \qquad g|_{E \setminus A}\text{ is constant}. f ( e ) ≤ g ( e ) ( e ∈ E ) , f ∣ A = g ∣ A , g ∣ E ∖ A is constant . The constancy condition is vacuous when E ∖ A E \setminus A E ∖ A 1 ≤ n ≤ ∞ 1 \le n \le\infty 1 ≤ n ≤ ∞ ω \omega ω
P f , g : U E ⊗ D 1 ⟶ U E P_{f,g}: U_E \otimes D_1 \longrightarrow U_E P f , g : U E ⊗ D 1 ⟶ U E whose restrictions to the two interval endpoints are U ( f ) U(f) U ( f ) U ( g ) U(g) U ( g ) i : A ↪ E i : A \hookrightarrow E i : A ↪ E v = f i = g i : A → E v = fi = gi : A \to E v = f i = g i : A → E
P f , g ( U ( i ) ⊗ id D 1 ) = U ( v ) π U A . (3.1) P_{f,g}(U(i) \otimes\operatorname{id}_{D_1}) = U(v)\pi_{U_A}. \tag*{(3.1)} P f , g ( U ( i ) ⊗ id D 1 ) = U ( v ) π U A . (3.1) Consequently its endpoints induce simplicially homotopic maps on Street nerves, with the homotopy stationary relative to N n U ( v ) N_nU(v) N n U ( v )
Proof. Simplexwise chain map. We first construct the cylinder in ω C a t \omega\mathrm{Cat} ω Cat τ n \tau_n τ n T T T M [ T ] M[T] M [ T ] p p p [ t 0 , … , t p ] [t_0,\ldots,t_p] [ t 0 , … , t p ] p > 0 p>0 p > 0 1 1 1
Fix a simplex σ = ( v 0 ≤ ⋯ ≤ v m ) \sigma=(v_0\le\cdots\le v_m) σ = ( v 0 ≤ ⋯ ≤ v m ) N E NE N E
T σ = { f ( v i ) , g ( v i ) : 0 ≤ i ≤ m } ⊂ E . T_\sigma=\{f(v_i),g(v_i):0\le i\le m\}\subset E. T σ = { f ( v i ) , g ( v i ) : 0 ≤ i ≤ m } ⊂ E . This subset is totally ordered. Indeed, the indices whose vertices belong to A A A g g g b b b f ( v i ) f(v_i) f ( v i ) f ( v m ) ≤ b f(v_m)\le b f ( v m ) ≤ b T σ T_\sigma T σ f f f T σ T_\sigma T σ f f f T σ ↪ E T_\sigma\hookrightarrow E T σ ↪ E
ν M [ T σ ] ⟶ U E ∞ . \nu M[T_\sigma]\longrightarrow U_E^\infty. ν M [ T σ ] ⟶ U E ∞ . Here ν M [ T σ ] \nu M[T_\sigma] ν M [ T σ ] T σ T_\sigma T σ
Let e 0 , e 1 e_0,e_1 e 0 , e 1 u u u M [ 1 ] M[1] M [ 1 ] d u = e 1 − e 0 du=e_1-e_0 d u = e 1 − e 0
C σ : M [ m ] ⊗ M [ 1 ] ⟶ M [ T σ ] (3.2) C_\sigma:M[m]\otimes M[1]\longrightarrow M[T_\sigma] \tag*{(3.2)} C σ : M [ m ] ⊗ M [ 1 ] ⟶ M [ T σ ] (3.2) on the tensor basis. For x = [ i 0 , … , i p ] x=[i_0,\ldots,i_p] x = [ i 0 , … , i p ] i 0 < ⋯ < i p i_0<\cdots<i_p i 0 < ⋯ < i p w j = v i j w_j=v_{i_j} w j = v i j
C σ ( x ⊗ e 0 ) = F ( w ) : = [ f ( w 0 ) , … , f ( w p ) ] , C σ ( x ⊗ e 1 ) = G ( w ) : = [ g ( w 0 ) , … , g ( w p ) ] , C σ ( x ⊗ u ) = h ( w ) : = [ f ( w 0 ) , … , f ( w p ) , g ( w p ) ] . (3.3) \begin{aligned}
C_\sigma(x\otimes e_0)&=F(w):=[f(w_0),\ldots,f(w_p)],\\
C_\sigma(x\otimes e_1)&=G(w):=[g(w_0),\ldots,g(w_p)],\\
C_\sigma(x\otimes u)&=h(w):=[f(w_0),\ldots,f(w_p),g(w_p)]. \tag*{(3.3)}
\end{aligned} C σ ( x ⊗ e 0 ) C σ ( x ⊗ e 1 ) C σ ( x ⊗ u ) = F ( w ) := [ f ( w 0 ) , … , f ( w p )] , = G ( w ) := [ g ( w 0 ) , … , g ( w p )] , = h ( w ) := [ f ( w 0 ) , … , f ( w p ) , g ( w p )] . (3.3) Each image is a positive basis element or zero: all displayed tuples are weakly increasing in T σ T_\sigma T σ C σ C_\sigma C σ
It remains to check the differential on x ⊗ u x\otimes u x ⊗ u p = 0 p=0 p = 0
d [ f ( w 0 ) , g ( w 0 ) ] = [ g ( w 0 ) ] − [ f ( w 0 ) ] . d[f(w_0),g(w_0)]=[g(w_0)]-[f(w_0)]. d [ f ( w 0 ) , g ( w 0 )] = [ g ( w 0 )] − [ f ( w 0 )] . For p > 0 p>0 p > 0
B = [ f ( w 0 ) , … , f ( w p − 1 ) , g ( w p ) ] , B ′ = [ f ( w 0 ) , … , f ( w p − 1 ) , g ( w p − 1 ) ] . \begin{aligned}
B&=[f(w_0),\ldots,f(w_{p-1}),g(w_p)],\\
B'&=[f(w_0),\ldots,f(w_{p-1}),g(w_{p-1})].
\end{aligned} B B ′ = [ f ( w 0 ) , … , f ( w p − 1 ) , g ( w p )] , = [ f ( w 0 ) , … , f ( w p − 1 ) , g ( w p − 1 )] . We claim that B − B ′ = G ( w ) B-B'=G(w) B − B ′ = G ( w ) w p − 1 ∈ A w_{p-1}\in A w p − 1 ∈ A w 0 , … , w p − 1 w_0,\ldots,w_{p-1} w 0 , … , w p − 1 A A A B = G ( w ) B=G(w) B = G ( w ) B ′ = 0 B'=0 B ′ = 0 w p − 1 ∉ A w_{p-1}\notin A w p − 1 ∈ / A w p ∉ A w_p\notin A w p ∈ / A g g g B = B ′ B=B' B = B ′ G ( w ) G(w) G ( w ) B − B ′ = G ( w ) B-B'=G(w) B − B ′ = G ( w ) d h ( w ) dh(w) d h ( w ) f ( w 0 ) , … , f ( w p − 1 ) f(w_0),\ldots,f(w_{p-1}) f ( w 0 ) , … , f ( w p − 1 )
d h ( w ) = ∑ j = 0 p ( − 1 ) j h ( w 0 , … , w j ^ , … , w p ) + ( − 1 ) p ( B − B ′ − F ( w ) ) = ∑ j = 0 p ( − 1 ) j h ( w 0 , … , w j ^ , … , w p ) + ( − 1 ) p ( G ( w ) − F ( w ) ) . (3.4) \begin{aligned}
dh(w)&=\sum_{j=0}^{p}(-1)^j h(w_0,\ldots,\widehat{w_j},\ldots,w_p)+(-1)^p\left(B-B'-F(w)\right)\\
&=\sum_{j=0}^{p}(-1)^j h(w_0,\ldots,\widehat{w_j},\ldots,w_p)+(-1)^p\left(G(w)-F(w)\right). \tag*{(3.4)}
\end{aligned} d h ( w ) = j = 0 ∑ p ( − 1 ) j h ( w 0 , … , w j , … , w p ) + ( − 1 ) p ( B − B ′ − F ( w ) ) = j = 0 ∑ p ( − 1 ) j h ( w 0 , … , w j , … , w p ) + ( − 1 ) p ( G ( w ) − F ( w ) ) . (3.4) This is exactly the image under C σ C_\sigma C σ
d ( x ⊗ u ) = d x ⊗ u + ( − 1 ) p x ⊗ ( e 1 − e 0 ) . d(x\otimes u)=dx\otimes u+(-1)^p x\otimes(e_1-e_0). d ( x ⊗ u ) = d x ⊗ u + ( − 1 ) p x ⊗ ( e 1 − e 0 ) . Thus (3.2) is a positive, augmentation-preserving chain map in every degree. The strong Steiner tensor comparison gives the strict functor
O m ⊗ D 1 ≅ ν ( M [ m ] ⊗ M [ 1 ] ) → ν C σ ν M [ T σ ] ⟶ U E ∞ . (3.5) \mathcal{O}_m\otimes D_1\cong\nu(M[m]\otimes M[1])\xrightarrow{\nu C_\sigma}\nu M[T_\sigma]\longrightarrow U^\infty_E. \tag*{(3.5)} O m ⊗ D 1 ≅ ν ( M [ m ] ⊗ M [ 1 ]) ν C σ ν M [ T σ ] ⟶ U E ∞ . (3.5) Compatibility with simplex operators. We next verify compatibility with all simplex operators. Let α : [ q ] → [ m ] \alpha:[q]\to[m] α : [ q ] → [ m ] ρ = σ α \rho=\sigma\alpha ρ = σ α j : T ρ ↪ T σ j:T_\rho\hookrightarrow T_\sigma j : T ρ ↪ T σ
M ( j ) C ρ = C σ ( M ( α ) ⊗ id M [ 1 ] ) . (3.6) M(j)C_\rho=C_\sigma\left(M(\alpha)\otimes\operatorname{id}_{M[1]}\right). \tag*{(3.6)} M ( j ) C ρ = C σ ( M ( α ) ⊗ id M [ 1 ] ) . (3.6) To check it, take a basis tuple [ a 0 , … , a p ] [a_0,\ldots,a_p] [ a 0 , … , a p ] M [ q ] M[q] M [ q ] α \alpha α (3.6) , on tensoring with e 0 e_0 e 0 e 1 e_1 e 1 u u u (3.3) , selected by the indices α ( a 0 ) , … , α ( a p ) \alpha(a_0),\ldots,\alpha(a_p) α ( a 0 ) , … , α ( a p ) α \alpha α f f f e 0 e_0 e 0 u u u g g g e 1 e_1 e 1 σ \sigma σ ν \nu ν (3.6) into an equality of the strict functors (3.5) after mapping into U E ∞ U^\infty_E U E ∞
The zero chains in this argument retain their lower boundary data after applying ν \nu ν (2.2) , the entries satisfy d x r ± = x r − 1 + − x r − 1 − dx_r^\pm=x_{r-1}^+-x_{r-1}^- d x r ± = x r − 1 + − x r − 1 − q q q q > 0 q>0 q > 0 ( q − 1 ) (q-1) ( q − 1 ) ( q − 1 ) (q-1) ( q − 1 ) E = { a < b } E=\{a<b\} E = { a < b } A = { a } A=\{a\} A = { a } f f f a a a g = id g=\operatorname{id} g = id h ( a , b ) = [ a , a , b ] = 0 h(a,b)=[a,a,b]=0 h ( a , b ) = [ a , a , b ] = 0 [ a , b ] [a,b] [ a , b ] (3.6) compares every degree of the chain maps, so it establishes agreement on these retained boundaries as well.
Assembly and stationarity. The oriental presentation and cocontinuity of the Gray tensor in its first variable now give
U E ∞ ⊗ D 1 ≅ colim σ : Δ [ m ] → N E ( O m ⊗ D 1 ) . U^\infty_E\otimes D_1\cong\underset{\sigma:\Delta[m]\to NE}{\operatorname{colim}}\left(\mathcal{O}_m\otimes D_1\right). U E ∞ ⊗ D 1 ≅ σ : Δ [ m ] → N E colim ( O m ⊗ D 1 ) . The compatible maps (3.5) consequently induce P f , g ∞ : U E ∞ ⊗ D 1 → U E ∞ P^\infty_{f,g}:U^\infty_E\otimes D_1\to U^\infty_E P f , g ∞ : U E ∞ ⊗ D 1 → U E ∞ ω \omega ω ν \nu ν U ∞ ( f ) U^\infty(f) U ∞ ( f ) U ∞ ( g ) U^\infty(g) U ∞ ( g )
On a simplex entirely in A A A F F F G G G h ( w ) = 0 h(w)=0 h ( w ) = 0 f ( w p ) = g ( w p ) f(w_p)=g(w_p) f ( w p ) = g ( w p )
C σ = F ∘ ( id M [ m ] ⊗ ϵ ) , ϵ ( e 0 ) = ϵ ( e 1 ) = 1 , ϵ ( u ) = 0 , C_\sigma=F\circ(\operatorname{id}_{M[m]}\otimes\epsilon),\qquad\epsilon(e_0)=\epsilon(e_1)=1,\qquad\epsilon(u)=0, C σ = F ∘ ( id M [ m ] ⊗ ϵ ) , ϵ ( e 0 ) = ϵ ( e 1 ) = 1 , ϵ ( u ) = 0 , where the chain augmentation of the interval is denoted by the same letter as its categorical realization. Applying ν \nu ν U A ∞ U_A^\infty U A ∞ U ∞ ( v ) U^\infty(v) U ∞ ( v )
Finite-dimensional reflection. For finite n n n q E : U E ∞ → U E q_E:U_E^\infty\to U_E q E : U E ∞ → U E q E P f , g ∞ q_E P_{f,g}^\infty q E P f , g ∞
U E ∞ ⊗ D 1 → q E ⊗ id U E ⊗ D 1 → P f , g U E . U_E^\infty\otimes D_1\xrightarrow{q_E\otimes\operatorname{id}}U_E\otimes D_1\xrightarrow{P_{f,g}}U_E. U E ∞ ⊗ D 1 q E ⊗ id U E ⊗ D 1 P f , g U E . Its endpoint identities follow by precomposing with q E q_E q E (3.1) , precompose both sides with q A ⊗ id q_A\otimes\operatorname{id} q A ⊗ id U E U_E U E n = ∞ n=\infty n = ∞ E E E A A A
Lemma 3.2 (Gluing a stationary cylinder). Let i : B 0 → B i:B_0\to B i : B 0 → B a : B 0 → X a:B_0\to X a : B 0 → X n n n
B 0 → i B a ↓ ↓ b X → j Y \begin{CD}
B_0 @>i>> B\\
@VaVV @VVbV\\
X @>j>> Y
\end{CD} B 0 a ↓ ⏐ X i j B ↓ ⏐ b Y Suppose F 0 , F 1 : B → B F_0,F_1:B\to B F 0 , F 1 : B → B F 0 i = F 1 i = i F_0i=F_1i=i F 0 i = F 1 i = i P : B ⊗ D 1 → B P:B\otimes D_1\to B P : B ⊗ D 1 → B F 0 , F 1 F_0,F_1 F 0 , F 1 P ( i ⊗ id ) = i π B 0 P(i\otimes\operatorname{id})=i\pi_{B_0} P ( i ⊗ id ) = i π B 0 Q : Y ⊗ D 1 → Y Q:Y\otimes D_1\to Y Q : Y ⊗ D 1 → Y X X X F ‾ 0 , F ‾ 1 : Y → Y \overline{F}_0,\overline{F}_1:Y\to Y F 0 , F 1 : Y → Y
F ‾ δ j = j , F ‾ δ b = b F δ ( δ = 0 , 1 ) . \overline{F}_\delta j=j,\qquad\overline{F}_\delta b=bF_\delta\qquad(\delta=0,1). F δ j = j , F δ b = b F δ ( δ = 0 , 1 ) . Its restriction along b ⊗ id b\otimes\operatorname{id} b ⊗ id b P bP b P a : B 0 → X a:B_0\to X a : B 0 → X N n F ‾ 0 N_n\overline{F}_0 N n F 0 N n F ‾ 1 N_n\overline{F}_1 N n F 1 N n j N_nj N n j f ∣ A = g ∣ A = id A f|_A=g|_A=\operatorname{id}_A f ∣ A = g ∣ A = id A U A → X U_A\to X U A → X
Proof. Let H ( Y ) H(Y) H ( Y ) ( − ) ⊗ D 1 → Y ( - )\otimes D_1\to Y ( − ) ⊗ D 1 → Y n n n n = ∞ n=\infty n = ∞
b P : B ⊗ D 1 → Y , j π X : X ⊗ D 1 → Y bP:B\otimes D_1\to Y,\qquad j\pi_X:X\otimes D_1\to Y b P : B ⊗ D 1 → Y , j π X : X ⊗ D 1 → Y transpose to maps B → H ( Y ) B\to H(Y) B → H ( Y ) X → H ( Y ) X\to H(Y) X → H ( Y ) n C a t n\mathrm{Cat} n Cat B 0 B_0 B 0
b i π B 0 = j a π B 0 . bi\pi_{B_0}=ja\pi_{B_0}. bi π B 0 = j a π B 0 . The pushout universal property in n C a t n\mathrm{Cat} n Cat Y → H ( Y ) Y\to H(Y) Y → H ( Y ) Q Q Q Q ( j ⊗ id ) = j π X Q(j\otimes\operatorname{id})=j\pi_X Q ( j ⊗ id ) = j π X B B B X → X ′ X\to X' X → X ′ Y → Y ′ Y\to Y' Y → Y ′ Y → H ( Y ′ ) Y\to H(Y') Y → H ( Y ′ ) X X X B B B i i i a a a X X X
Labelled pieces of algebraic pushouts
We fix 1 ≤ n ≤ ∞ 1 \le n \le\infty 1 ≤ n ≤ ∞
The sieve and nerve constructions have established antecedents in [4 Propositions 5.3 and 5.5] . We give the calculations explicitly to identify the labels, fibers and intersections after an arbitrary attachment.
A full sub-n n n C C C S S S 0 0 0 0 0 0 S S S sieve if a 1-cell x → y x \to y x → y y ∈ S y \in S y ∈ S x ∈ S x \in S x ∈ S cosieve if a 1-cell x → y x \to y x → y x ∈ S x \in S x ∈ S y ∈ S y \in S y ∈ S C → D 1 C \to D_1 C → D 1
The following fiber adjoints are those of [4 Lemma 2.8] .
Lemma 4.1 (Fibers and objects preserve colimits). For ϵ ∈ { 0 , 1 } \epsilon\in\{0,1\} ϵ ∈ { 0 , 1 }
F ϵ : n C a t / D 1 ⟶ n C a t , ( C → p D 1 ) ⟼ C ϵ = p − 1 ( ϵ ) , F_\epsilon: n\mathrm{Cat}/D_1 \longrightarrow n\mathrm{Cat}, \qquad(C \xrightarrow{p} D_1) \longmapsto C_\epsilon= p^{-1}(\epsilon), F ϵ : n Cat / D 1 ⟶ n Cat , ( C p D 1 ) ⟼ C ϵ = p − 1 ( ϵ ) , preserves all small colimits. The object functor Ob : n C a t → S e t \operatorname{Ob} : n\mathrm{Cat} \to\mathrm{Set} Ob : n Cat → Set n = ∞ n = \infty n = ∞
Proof. We construct a right adjoint Q 0 Q_0 Q 0 F 0 F_0 F 0 B B B ∗ \ast ∗ B ( b , b ′ ) B(b,b') B ( b , b ′ )
Q 0 ( B ) ( b , ∗ ) = 1 , Q 0 ( B ) ( ∗ , b ) = ∅ , Q 0 ( B ) ( ∗ , ∗ ) = 1. Q_0(B)(b,\ast) = 1, \qquad Q_0(B)(\ast,b) = \varnothing, \qquad Q_0(B)(\ast,\ast) = 1. Q 0 ( B ) ( b , ∗ ) = 1 , Q 0 ( B ) ( ∗ , b ) = ∅ , Q 0 ( B ) ( ∗ , ∗ ) = 1. Here the homs are strict ( n − 1 ) (n-1) ( n − 1 ) n n n n = 1 n=1 n = 1 ω \omega ω n = ∞ n=\infty n = ∞ ∅ \varnothing ∅ B B B B B B Q 0 ( B ) Q_0(B) Q 0 ( B ) D 1 D_1 D 1
For p : C → D 1 p : C \to D_1 p : C → D 1 C 0 → B C_0 \to B C 0 → B D 1 D_1 D 1 C → Q 0 ( B ) C \to Q_0(B) C → Q 0 ( B ) C 1 C_1 C 1 ∗ \ast ∗ C 0 C_0 C 0 C 1 C_1 C 1 C 1 C_1 C 1 C 0 C_0 C 0 1 → 0 1 \to0 1 → 0
Hom n C a t / D 1 ( C , Q 0 ( B ) ) ≅ Hom n C a t ( C 0 , B ) . \operatorname{Hom}_{n\mathrm{Cat}/D_1}(C,Q_0(B)) \cong\operatorname{Hom}_{n\mathrm{Cat}}(C_0,B). Hom n Cat / D 1 ( C , Q 0 ( B )) ≅ Hom n Cat ( C 0 , B ) . Thus F 0 F_0 F 0 F 1 F_1 F 1 B B B B B B Q 1 Q_1 Q 1 B = ∅ B = \varnothing B = ∅
For a set S S S K ( S ) K(S) K ( S ) S S S n n n Ob C → S \operatorname{Ob} C \to S Ob C → S C → K ( S ) C \to K(S) C → K ( S ) Ob \operatorname{Ob} Ob K K K
The next elementary property of orientals is needed before taking a colimit of simplex pieces. It is not enough merely to know their sets of vertices.
Lemma 4.2 (Full oriental segments). Let I I I be an initial or a final segment of [ m ] = { 0 , … , m } [m]=\{0,\ldots,m\} [ m ] = { 0 , … , m } If I I I is nonempty and has q + 1 q+1 q + 1 vertices, the face inclusion identifies τ n O q \tau_n \mathcal{O}_q τ n O q with the full sub-n -category of τ n O m \tau_n \mathcal{O}_m τ n O m on I I I If I I I is empty, that full subcategory is empty. Here and below τ n = i d \tau_n=\mathrm{id} τ n = id when n = ∞ n=\infty n = ∞
Proof. Recall the oriental presentation from Section 2: its atoms are indexed by strictly increasing tuples, with 0-source the first vertex and 0-target the last vertex. All cells are finite composites and identities of these atoms. In particular every 1-cell has nondecreasing endpoints.
Suppose a cell has 0-source a a a b b b [ a , b ] [a,b] [ a , b ] ( a , c ) (a,c) ( a , c ) ( c , b ) (c,b) ( c , b ) a ≤ c ≤ b a\leq c\leq b a ≤ c ≤ b a , b ∈ I a,b\in I a , b ∈ I [ a , b ] [a,b] [ a , b ] I I I
For finite n ≥ 1 n\geq1 n ≥ 1 n n n I ↪ [ m ] I\hookrightarrow[m] I ↪ [ m ] I I I I I I
Lemma 4.3 (Poset pieces). There is a canonical identification Ob U P = P \operatorname{Ob}U_P=P Ob U P = P for every poset P P P If S ⊂ P S\subset P S ⊂ P is a sieve, the map U S → U P U_S\to U_P U S → U P is the inclusion of the full 0-fiber of the membership label U P → D 1 U_P\to D_1 U P → D 1 If S S S
Proof. Use the canonical simplex presentation
U P = colim σ : Δ [ m ] → N P τ n O m . (4.1) U_P=\mathop{\operatorname{colim}}_{\sigma:\Delta[m]\to NP}\tau_n \mathcal{O}_m. \tag*{(4.1)} U P = colim σ : Δ [ m ] → N P τ n O m . (4.1) By Lemma 4.1, its object set is the colimit of the vertex sets [ m ] [m] [ m ] j j j σ \sigma σ σ ( j ) \sigma(j) σ ( j ) P P P
For a sieve S S S P → [ 1 ] P\to[1] P → [ 1 ] S S S U P U_P U P U [ 1 ] = D 1 U_{[1]}=D_1 U [ 1 ] = D 1 n Cat / D 1 n\operatorname{Cat}/D_1 n Cat / D 1 n Cat n\operatorname{Cat} n Cat σ \sigma σ S S S I σ I_\sigma I σ
We verify that this colimit of segment orientals is canonically U S U_S U S S S S U S U_S U S S S S S S S σ \sigma σ I σ I_\sigma I σ N P NP N P U S → U P U_S \to U_P U S → U P S S S F 1 F_1 F 1
Here is the form of the fiber calculation that we will use in pushouts. It also records precisely which hypotheses are needed of a label on an arbitrary attaching target.
Proposition 4.4 (Labelled attachments). Let A ⊂ P A \subset P A ⊂ P a : U A → X a : U_A \to X a : U A → X
Y = X ⨿ U A U P . Y = X \amalg_{U_A} U_P. Y = X ⨿ U A U P . Suppose a monotone map λ : P → [ 1 ] \lambda: P \to[1] λ : P → [ 1 ] ξ : X → D 1 \xi: X \to D_1 ξ : X → D 1
ξ a = U ( λ ∣ A ) : U A ⟶ D 1 , U [ 1 ] = D 1 . \xi a = U(\lambda|_A) : U_A \longrightarrow D_1,\qquad U_{[1]} = D_1. ξ a = U ( λ ∣ A ) : U A ⟶ D 1 , U [ 1 ] = D 1 . Write P ϵ = λ − 1 ( ϵ ) P_\epsilon= \lambda^{-1}(\epsilon) P ϵ = λ − 1 ( ϵ ) A ϵ = A ∩ P ϵ A_\epsilon= A \cap P_\epsilon A ϵ = A ∩ P ϵ X ϵ = ξ − 1 ( ϵ ) X_\epsilon= \xi^{-1}(\epsilon) X ϵ = ξ − 1 ( ϵ ) Y → D 1 Y \to D_1 Y → D 1
Y ϵ ≅ X ϵ ⨿ U A ϵ U P ϵ . (4.2) Y_\epsilon\cong X_\epsilon\amalg_{U_{A_\epsilon}} U_{P_\epsilon}. \tag*{(4.2)} Y ϵ ≅ X ϵ ⨿ U A ϵ U P ϵ . (4.2) Moreover, independently of these labels,
Ob Y ≅ Ob X ⨿ A P ≅ Ob X ⨿ ( P ∖ A ) . (4.3) \operatorname{Ob} Y \cong\operatorname{Ob} X \amalg_A P \cong\operatorname{Ob} X \amalg(P \setminus A). \tag*{(4.3)} Ob Y ≅ Ob X ⨿ A P ≅ Ob X ⨿ ( P ∖ A ) . (4.3) For two such pairs of compatible labels, write λ ‾ 1 , λ ‾ 2 : Y → D 1 \overline{\lambda}_1,\overline{\lambda}_2 : Y \to D_1 λ 1 , λ 2 : Y → D 1 max ∘ ( λ ‾ 1 , λ ‾ 2 ) \max\circ(\overline{\lambda}_1,\overline{\lambda}_2) max ∘ ( λ 1 , λ 2 ) min ∘ ( λ ‾ 1 , λ ‾ 2 ) \min\circ(\overline{\lambda}_1,\overline{\lambda}_2) min ∘ ( λ 1 , λ 2 ) (4.2) , using the corresponding maximum or minimum on both original legs, computes these intersections as well.
Proof. The compatibility assumption makes the defining pushout a pushout in n C a t / D 1 \mathsf{nCat}/D_1 nCat / D 1 F ϵ F_\epsilon F ϵ (4.2) . This identifies the canonical map from the displayed pushout to Y Y Y A ↪ P A \hookrightarrow P A ↪ P P P P X X X P ∖ A P \setminus A P ∖ A Ob a \operatorname{Ob} a Ob a
Two labels on Y Y Y Y → D 1 × D 1 Y \to D_1 \times D_1 Y → D 1 × D 1 max , min : [ 1 ] × [ 1 ] → [ 1 ] \max,\min: [1] \times[1] \to[1] max , min : [ 1 ] × [ 1 ] → [ 1 ] Y Y Y
For use in the next two sections, we spell out both cover consequences.
Corollary 4.5 (Sieve and cosieve covers after attachment). With the notation of Proposition 4.4, the following are canonical identifications of subcategories of Y Y Y
(i) Suppose P 1 , P 2 P_1, P_2 P 1 , P 2 P P P A A A P P P
Y i = X ⨿ U A U P i ( i = 1 , 2 ) Y_i = X \amalg_{U_A} U_{P_i}\qquad(i=1,2) Y i = X ⨿ U A U P i ( i = 1 , 2 ) are sieves covering Y Y Y
Y 1 ∩ Y 2 = X ⨿ U A U P 1 ∩ P 2 . Y_1 \cap Y_2 = X \amalg_{U_A} U_{P_1 \cap P_2}. Y 1 ∩ Y 2 = X ⨿ U A U P 1 ∩ P 2 . (ii) Suppose T 1 , T 2 T_1,T_2 T 1 , T 2 P P P P P P A ⊂ T 1 A \subset T_1 A ⊂ T 1 A ∩ T 2 = ∅ A \cap T_2 = \varnothing A ∩ T 2 = ∅
Y 1 = X ⨿ U A U T 1 , Y 2 = U T 2 Y_1 = X \amalg_{U_A} U_{T_1}, \qquad Y_2 = U_{T_2} Y 1 = X ⨿ U A U T 1 , Y 2 = U T 2 are cosieves covering Y Y Y Y 1 ∩ Y 2 = U T 1 ∩ T 2 Y_1 \cap Y_2 = U_{T_1\cap T_2} Y 1 ∩ Y 2 = U T 1 ∩ T 2
If A A A P P P X → Y X \to Y X → Y
Proof. For (i), label membership in each P i P_i P i X X X T i T_i T i X X X T 1 T_1 T 1 X X X T 2 T_2 T 2 (4.2) therefore gives U T 2 U_{T_2} U T 2 U T 1 ∩ T 2 U_{T_1\cap T_2} U T 1 ∩ T 2 (4.3) proves the object cover. Finally, when A A A A A A X X X P ∖ A P \setminus A P ∖ A X ⨿ U A U A = X X \amalg_{U_A} U_A = X X ⨿ U A U A = X
In particular, an attaching functor may identify two old objects, collapse an old arrow to an identity, or identify parallel old cells. These operations cause no ambiguity in the cover calculations: the labels agree on every old cell before the pushout is taken. The fiber adjunctions compute the resulting subcategories after all those identifications.
Lemma 4.6 (Street nerves of two-piece covers). Let C 1 , C 2 ⊂ C C_1,C_2 \subset C C 1 , C 2 ⊂ C C C C C 12 = C 1 ∩ C 2 C_{12}=C_1 \cap C_2 C 12 = C 1 ∩ C 2
N n C 12 ↪ N n C 2 ↓ ∪ ↓ ∪ N n C 1 ↪ N n C \begin{array}{ccc}
N_nC_{12} & \xhookrightarrow{\ \ } & N_nC_2 \\
\overset{\cup}{\downarrow} & & \overset{\cup}{\downarrow} \\
N_nC_1 & \xhookrightarrow{\ \ } & N_nC
\end{array} N n C 12 ↓ ∪ N n C 1 N n C 2 ↓ ∪ N n C is an ordinary pushout of simplicial sets. All four maps are monomorphisms, so it is also a homotopy pushout in the Kan–Quillen model structure.
Proof. A full inclusion induces a monomorphism of Street nerves, and the intersection of the two nerves is N n C 12 N_n C_{12} N n C 12 z : O m → C z\colon \mathcal{O}_m \to C z : O m → C z ( m ) z(m) z ( m ) j ≤ m j \le m j ≤ m j → m j \to m j → m z ( j ) z(j) z ( j ) z ( 0 ) z(0) z ( 0 ) 0 → j 0 \to j 0 → j m = 0 m=0 m = 0
Couniversality and the unit on posets
Fix n ∈ { 1 , 2 , … } ∪ { ∞ } n \in\{1,2,\ldots\} \cup\{\infty\} n ∈ { 1 , 2 , … } ∪ { ∞ } Thomason equivalence is a strict functor whose Street nerve is a weak homotopy equivalence. This terminology uses no model structure on n C a t n\mathrm{Cat} n Cat
For an inclusion i : A ↪ E i\colon A \hookrightarrow E i : A ↪ E r : E → A r\colon E \to A r : E → A
a ≤ e ⟺ a ≤ r ( e ) ( a ∈ A , e ∈ E ) , r i = id A . (5.1) a \le e \quad\Longleftrightarrow\quad a \le r(e) \qquad(a \in A,\ e \in E), \qquad ri = \operatorname{id}_A. \tag*{(5.1)} a ≤ e ⟺ a ≤ r ( e ) ( a ∈ A , e ∈ E ) , r i = id A . (5.1) We identify A A A E E E r ( e ) ≤ e r(e) \leq e r ( e ) ≤ e sieve when e ≤ a ∈ A e \leq a \in A e ≤ a ∈ A e ∈ A e \in A e ∈ A a : U A → X a: U_A \to X a : U A → X
Y = X ⨿ U A U E , j : X ⟶ Y , v : U E ⟶ Y . (5.2) Y = X \amalg_{U_A} U_E,\qquad j:X \longrightarrow Y,\qquad v:U_E \longrightarrow Y. \tag*{(5.2)} Y = X ⨿ U A U E , j : X ⟶ Y , v : U E ⟶ Y . (5.2) The symbol a a a A A A b b b
Lemma 5.1 (A greatest new element). Suppose that A ↪ E A \hookrightarrow E A ↪ E E ∖ A E \setminus A E ∖ A (5.2) , the map j j j p : Y → X p:Y \to X p : Y → X N n ( j p ) N_n(jp) N n ( j p ) id N n Y \operatorname{id}_{N_nY} id N n Y j j j A A A E E E
Proof. Let x x x E ∖ A E \setminus A E ∖ A t = r ( x ) ∈ A t=r(x) \in A t = r ( x ) ∈ A b ∈ A b \in A b ∈ A y ∈ E ∖ A y \in E \setminus A y ∈ E ∖ A b ≤ y b \leq y b ≤ y b ≤ y ≤ x b \leq y \leq x b ≤ y ≤ x (5.1) gives b ≤ t b \leq t b ≤ t k , g : E → E k,g:E \to E k , g : E → E
k ( e ) = { e , e ∈ A , t , e ∉ A , g ( e ) = { e , e ∈ A , x , e ∉ A . k(e)=
\begin{cases}
e, & e \in A,\\
t, & e \notin A,
\end{cases}
\qquad
g(e)=
\begin{cases}
e, & e \in A,\\
x, & e \notin A.
\end{cases} k ( e ) = { e , t , e ∈ A , e ∈ / A , g ( e ) = { e , x , e ∈ A , e ∈ / A . Both are monotone. Indeed, for a comparable pair e ≤ e ′ e \leq e' e ≤ e ′ e ≤ t ≤ x e \leq t \leq x e ≤ t ≤ x A A A
id E ≤ g , k ≤ g . (5.3) \operatorname{id}_E \leq g,\qquad k \leq g. \tag*{(5.3)} id E ≤ g , k ≤ g . (5.3) and all three maps restrict to id A \operatorname{id}_A id A g g g E ∖ A E \setminus A E ∖ A Figure 1 shows why this common upper map is useful: id E \operatorname{id}_E id E k k k g g g
Figure 1. The shaded square nodes form the old sieve A = { a , b } A = \{a,b\} A = { a , b } r ( y ) = a r(y) = a r ( y ) = a r ( x ) = t = b r(x) = t = b r ( x ) = t = b k ( y ) = b k(y) = b k ( y ) = b y y y x x x
Apply Lemma 3.2 to the two comparisons in (5.3) . Let G , K : Y → Y G,K:Y \to Y G , K : Y → Y X X X j j j U E U_E U E v c n N ( g ) v c_nN(g) v c n N ( g ) v c n N ( k ) v c_nN(k) v c n N ( k ) X X X
id Y ⟶ G ⟵ K . \operatorname{id}_Y \longrightarrow G \longleftarrow K. id Y ⟶ G ⟵ K . By Lemma 2.3, these give simplicial homotopies with the same endpoint pattern on Street nerves. The image of k k k A A A k = i k ˉ k=i\bar{k} k = i k ˉ k ˉ : E → A \bar{k}:E \to A k ˉ : E → A id X \operatorname{id}_X id X a c n N ( k ˉ ) : U E → X a c_nN(\bar{k}):U_E \to X a c n N ( k ˉ ) : U E → X U A U_A U A p : Y → X p:Y \to X p : Y → X p j = id X pj=\operatorname{id}_X p j = id X j p = K jp=K j p = K X X X j j j U E U_E U E
j p v = j a c n N ( k ˉ ) = v c n N ( i ) c n N ( k ˉ ) = v c n N ( k ) . jp\,v=j\,a c_nN(\bar{k})=v c_nN(i)c_nN(\bar{k})=v c_nN(k). j p v = j a c n N ( k ˉ ) = v c n N ( i ) c n N ( k ˉ ) = v c n N ( k ) . After realization, concatenate the homotopy from id Y \operatorname{id}_Y id Y G G G j p jp j p G G G ∣ N n p ∣ |N_np| ∣ N n p ∣ ∣ N n j ∣ |N_nj| ∣ N n j ∣
Theorem 5.2 (Couniversality). If i : A ↪ E i:A \hookrightarrow E i : A ↪ E c n N ( i ) c_nN(i) c n N ( i ) a : U A → X a:U_A \to X a : U A → X
X ⟶ X ⨿ U A U E X \longrightarrow X \amalg_{U_A} U_E X ⟶ X ⨿ U A U E is a Thomason equivalence. This holds for every finite n ≥ 1 n \geq1 n ≥ 1 n = ∞ n=\infty n = ∞
Proof. First suppose that E E E ∣ E ∖ A ∣ |E \setminus A| ∣ E ∖ A ∣ X X X a a a j j j
Otherwise the nonempty finite poset E ∖ A E \setminus A E ∖ A x 1 , x 2 x_1,x_2 x 1 , x 2 x i x_i x i E E E A A A
E 1 = E ∖ { x 1 } , E 2 = E ∖ { x 2 } , E 12 = E ∖ { x 1 , x 2 } . E_1 = E \setminus\{x_1\}, \qquad E_2 = E \setminus\{x_2\}, \qquad E_{12} = E \setminus\{x_1,x_2\}. E 1 = E ∖ { x 1 } , E 2 = E ∖ { x 2 } , E 12 = E ∖ { x 1 , x 2 } . These are sieves of E E E A A A E 1 ∪ E 2 = E E_1 \cup E_2 = E E 1 ∪ E 2 = E E 1 ∩ E 2 = E 12 E_1 \cap E_2 = E_{12} E 1 ∩ E 2 = E 12 r r r E T E_T E T T = 1 , 2 , 12 T = 1,2,12 T = 1 , 2 , 12 A A A E T ∖ A E_T \setminus A E T ∖ A
Y T = X ⨿ U A U E T , j T : X ⟶ Y T . Y_T = X \amalg_{U_A} U_{E_T}, \qquad j_T : X \longrightarrow Y_T. Y T = X ⨿ U A U E T , j T : X ⟶ Y T . By induction every N n j T N_nj_T N n j T
Corollary 4.5(i), applied to the sieves E 1 E_1 E 1 E 2 E_2 E 2 A A A E E E Y i → Y Y_i \to Y Y i → Y Y 12 Y_{12} Y 12 Y Y Y a a a
Lemma 4.6 now identifies the bottom pushout in the following diagram with N n Y N_nY N n Y
N n X = N n X = N n X N n j 1 ↓ ↓ N n j 12 ↓ N n j 2 N n Y 1 ↩ N n Y 12 ↪ N n Y 2 (5.4) \begin{array}{ccccc}
N_nX & = & N_nX & = & N_nX \\
{\scriptstyle N_nj_1}\,\downarrow & & \downarrow\,{\scriptstyle N_nj_{12}} & & \downarrow\,{\scriptstyle N_nj_2} \\
N_nY_1 & \xhookleftarrow{\ \ } & N_nY_{12} & \xhookrightarrow{\ \ } & N_nY_2 \tag*{(5.4)}
\end{array} N n X N n j 1 ↓ N n Y 1 = N n X ↓ N n j 12 N n Y 12 = N n X ↓ N n j 2 N n Y 2 (5.4) The vertical maps are the canonical maps induced by X X X N n X N_nX N n X N n j N_nj N n j
Now let E E E q = i r : E → E q = ir : E \to E q = i r : E → E q 2 = q q^2 = q q 2 = q F \mathcal{F} F F ⊂ E F \subset E F ⊂ E q ( F ) ⊂ F q(F) \subset F q ( F ) ⊂ F ∅ \varnothing ∅ T T T T ∪ q ( T ) T \cup q(T) T ∪ q ( T ) F F F A F = A ∩ F A_F = A \cap F A F = A ∩ F A F A_F A F F F F r ( F ) ⊂ A F r(F) \subset A_F r ( F ) ⊂ A F r F : F → A F r_F : F \to A_F r F : F → A F F F F E E E
Use the restriction of a a a
Y F = X ⨿ U A F U F , j F : X → Y F . Y_F = X \amalg_{U_{A_F}} U_F, \qquad j_F : X \to Y_F. Y F = X ⨿ U A F U F , j F : X → Y F . Every j F j_F j F F ⊂ F ′ F \subset F' F ⊂ F ′ Y F → Y F ′ Y_F \to Y_{F'} Y F → Y F ′ X X X
N E = colim F ∈ F N F , N A = colim F ∈ F N A F . NE = \operatorname{colim}_{F \in\mathcal{F}} NF,\qquad NA = \operatorname{colim}_{F \in\mathcal{F}} NA_F. N E = colim F ∈ F N F , N A = colim F ∈ F N A F . Applying the colimit-preserving left adjoint c n c_n c n
colim F ∈ F Y F ≅ X ⨿ U A U E = Y . (5.5) \operatorname{colim}_{F \in\mathcal{F}} Y_F \cong X \amalg_{U_A} U_E = Y. \tag*{(5.5)} colim F ∈ F Y F ≅ X ⨿ U A U E = Y . (5.5) For clarity, the fixed copy of X X X Y F → Z Y_F \to Z Y F → Z X → Z X \to Z X → Z F \mathcal{F} F U F → Z U_F \to Z U F → Z U A F U_{A_F} U A F j F j_F j F j j j
By Lemma 2.4, N n N_n N n N n j N_nj N n j Y F → Y F ′ Y_F \to Y_{F'} Y F → Y F ′ n = ∞ n = \infty n = ∞
The following unit comparison is known; see [5 Corollary 10.11] , [9 Theorem 6.9] and [3 Proposition 9.2.3] . We include an independent proof using the same prism and sieve-cover method.
Proposition 5.3 (The unit on every poset). For every small poset S S S
η S : N S ⟶ N n U S \eta_S : NS \longrightarrow N_nU_S η S : N S ⟶ N n U S is a weak homotopy equivalence. This assertion is natural in S S S n ≥ 1 n \ge1 n ≥ 1 n = ∞ n = \infty n = ∞
Proof. First take S S S S S S S S S x x x id S ≤ const x \operatorname{id}_S \le\operatorname{const}_x id S ≤ const x N S NS N S ∅ \varnothing ∅ id U S \operatorname{id}_{U_S} id U S c n N ( const x ) c_nN(\operatorname{const}_x) c n N ( const x ) U { x } = O 0 U_{\{x\}} = \mathcal{O}_0 U { x } = O 0 n n n N n U S N_nU_S N n U S η S \eta_S η S
In the remaining case choose two distinct maximal elements x 1 , x 2 x_1,x_2 x 1 , x 2 S S S S i = S ∖ { x i } S_i = S \setminus\{x_i\} S i = S ∖ { x i } S 12 = S ∖ { x 1 , x 2 } S_{12} = S \setminus\{x_1,x_2\} S 12 = S ∖ { x 1 , x 2 } S S S N S NS N S U ⋅ U_{\,\cdot\,} U ⋅ U S U_S U S U S 12 U_{S_{12}} U S 12
N S 1 ↩ N S 12 ↪ N S 2 η S 1 ↓ ↓ η S 12 ↓ η S 2 N n U S 1 ↩ N n U S 12 ↪ N n U S 2 \begin{array}{ccccc}
NS_1 & \xhookleftarrow{\ \ } & NS_{12} & \xhookrightarrow{\ \ } & NS_2 \\
{\scriptstyle \eta_{S_1}}\,\downarrow & & \downarrow\,{\scriptstyle \eta_{S_{12}}} & & \downarrow\,{\scriptstyle \eta_{S_2}} \\
N_nU_{S_1} & \xhookleftarrow{\ \ } & N_nU_{S_{12}} & \xhookrightarrow{\ \ } & N_nU_{S_2}
\end{array} N S 1 η S 1 ↓ N n U S 1 N S 12 ↓ η S 12 N n U S 12 N S 2 ↓ η S 2 N n U S 2 The vertical maps are weak equivalences by induction; all horizontal legs are monomorphisms. The induced map on pushouts is a weak equivalence by Lemma 2.5. Under the cover identifications it is exactly η S \eta_S η S N S i NS_i N S i c n c_n c n N n N_n N n η S \eta_S η S
Comparison for subdivided attachments
We now compare an ordinary pushout of Street nerves with the Street nerve of an actual attachment. All attaching functors in this section are arbitrary. In particular, they may identify old objects and cells. The comparison will supply both the acyclicity of horn attachments and the preservation of weak equivalences under boundary attachments.
We first treat a sieve A ⊂ B A \subset B A ⊂ B B B B [17 Proposition 4.3] ; Theorem 5.2 supplies its homotopy input in strict higher dimensions. After proving this general comparison, we identify the twice-subdivided boundaries and horns as poset pairs with these properties.
Proposition 6.1 (Attachment comparison). Let A ⊂ B A \subset B A ⊂ B Z Z Z A ↪ Z A \hookrightarrow Z A ↪ Z a : U A → X a: U_A \to X a : U A → X n C a t n\mathrm{Cat} n Cat
V B ( X ) = X ⨿ U A U B , P B ( X ) = N n X ⨿ N n U A N n U B . V_B(X) = X \amalg_{U_A} U_B,\qquad P_B(X) = N_nX \amalg_{N_nU_A} N_nU_B. V B ( X ) = X ⨿ U A U B , P B ( X ) = N n X ⨿ N n U A N n U B . The canonical map
γ B , X : P B ( X ) ⟶ N n V B ( X ) (6.1) \gamma_{B,X}: P_B(X) \longrightarrow N_nV_B(X) \tag*{(6.1)} γ B , X : P B ( X ) ⟶ N n V B ( X ) (6.1) is a simplicial weak equivalence. The map N n U A → N n U B N_nU_A \to N_nU_B N n U A → N n U B f : X → X ′ f: X \to X' f : X → X ′ U A U_A U A
Proof. First set
V Z ( X ) = X ⨿ U A U Z , P Z ( X ) = N n X ⨿ N n U A N n U Z . V_Z(X) = X \amalg_{U_A} U_Z,\qquad P_Z(X) = N_nX \amalg_{N_nU_A} N_nU_Z. V Z ( X ) = X ⨿ U A U Z , P Z ( X ) = N n X ⨿ N n U A N n U Z . Theorem 5.2, applied to A ↪ Z A \hookrightarrow Z A ↪ Z X → V Z ( X ) X \to V_Z(X) X → V Z ( X ) W n W_n W n U A U_A U A U A → U Z U_A \to U_Z U A → U Z W n W_n W n A A A Z Z Z U A → U Z U_A \to U_Z U A → U Z N n X → P Z ( X ) N_nX \to P_Z(X) N n X → P Z ( X )
N n X ⟶ P Z ( X ) ↘ ↓ γ Z , X N n V Z ( X ) \begin{array}{ccc}
N_nX & \longrightarrow & P_Z(X) \\
& \searrow & \downarrow\,{\scriptstyle \gamma_{Z,X}} \\
& & N_nV_Z(X)
\end{array} N n X ⟶ ↘ P Z ( X ) ↓ γ Z , X N n V Z ( X ) then shows by two-out-of-three that γ Z , X \gamma_{Z,X} γ Z , X
To pass from Z Z Z B B B
C = B ∖ A , T = Z ∖ A = Z ∩ C . C = B \setminus A,\qquad T = Z \setminus A = Z \cap C. C = B ∖ A , T = Z ∖ A = Z ∩ C . Both Z Z Z C C C B B B B B B C C C A ⊂ Z A \subset Z A ⊂ Z T T T U Z , U C ⊂ U B U_Z,U_C \subset U_B U Z , U C ⊂ U B U T U_T U T P = B P = B P = B T 1 = Z T_1 = Z T 1 = Z T 2 = C T_2 = C T 2 = C A ⊂ Z A \subset Z A ⊂ Z A ∩ C = ∅ A \cap C = \varnothing A ∩ C = ∅
V Z ( X ) , U C ⊂ V B ( X ) , V Z ( X ) ∩ U C = U T , (6.2) V_Z(X),\quad U_C \subset V_B(X),\quad V_Z(X) \cap U_C = U_T, \tag*{(6.2)} V Z ( X ) , U C ⊂ V B ( X ) , V Z ( X ) ∩ U C = U T , (6.2) as actual full cosieves and their full intersection. These two cosieves cover all objects of V B ( X ) V_B(X) V B ( X ) a a a
Apply Lemma 4.6 to the two cosieve covers just obtained. It gives ordinary simplicial pushouts
N n U B = N n U Z ⨿ N n U T N n U C , (6.3) N_n U_B = N_n U_Z \amalg_{N_n U_T} N_n U_C, \tag*{(6.3)} N n U B = N n U Z ⨿ N n U T N n U C , (6.3) N n V B ( X ) = N n V Z ( X ) ⨿ N n U T N n U C . (6.4) N_n V_B(X) = N_n V_Z(X) \amalg_{N_n U_T} N_n U_C. \tag*{(6.4)} N n V B ( X ) = N n V Z ( X ) ⨿ N n U T N n U C . (6.4) All maps from the intersections to the pieces in these covers are monomorphisms. Reassociating the pushout in the definition of P B ( X ) P_B(X) P B ( X ) (6.3) gives
P B ( X ) = P Z ( X ) ⨿ N n U T N n U C . P_B(X) = P_Z(X) \amalg_{N_n U_T} N_n U_C. P B ( X ) = P Z ( X ) ⨿ N n U T N n U C . Under these canonical identifications, (6.1) is exactly the map induced by γ Z , X \gamma_{Z,X} γ Z , X N n U C N_n U_C N n U C
P Z ( X ) ↪ P B ( X ) γ Z , X ↓ ↓ γ B , X N n V Z ( X ) ↪ N n V B ( X ) \begin{array}{ccc}
P_Z(X) & \xhookrightarrow{\ \ } & P_B(X) \\
{\scriptstyle \gamma_{Z,X}}\,\downarrow & & \downarrow\,{\scriptstyle \gamma_{B,X}} \\
N_nV_Z(X) & \xhookrightarrow{\ \ } & N_nV_B(X)
\end{array} P Z ( X ) γ Z , X ↓ N n V Z ( X ) P B ( X ) ↓ γ B , X N n V B ( X ) is a pushout: its lower right corner is also N n V Z ( X ) ⨿ P Z ( X ) P B ( X ) N_n V_Z(X) \amalg_{P_Z(X)} P_B(X) N n V Z ( X ) ⨿ P Z ( X ) P B ( X )
N n U T ↪ N n U C . (6.5) N_n U_T \hookrightarrow N_n U_C. \tag*{(6.5)} N n U T ↪ N n U C . (6.5) Left properness of simplicial sets, as recorded in Lemma 2.5, now makes γ B , X \gamma_{B,X} γ B , X N n U A → N n X N_n U_A \to N_n X N n U A → N n X N n U T → P Z ( X ) N_n U_T \to P_Z(X) N n U T → P Z ( X )
The remaining assertion that N n U A → N n U B N_n U_A \to N_n U_B N n U A → N n U B γ B , X \gamma_{B,X} γ B , X N n X N_n X N n X N n U B N_n U_B N n U B f : X → X ′ f : X \to X' f : X → X ′ U A U_A U A N n f N_n f N n f V B ( X ) → V B ( X ′ ) V_B(X) \to V_B(X') V B ( X ) → V B ( X ′ )
P B ( X ) → γ B , X N n V B ( X ) ↓ ↓ P B ( X ′ ) → γ B , X ′ N n V B ( X ′ ) . (6.6) \begin{CD}
P_B(X) @>{\gamma_{B,X}}>> N_n V_B(X) \\
@VVV @VVV \\
P_B(X') @>{\gamma_{B,X'}}>> N_n V_B(X'). \tag*{(6.6)}
\end{CD} P B ( X ) ↓ ⏐ P B ( X ′ ) γ B , X γ B , X ′ N n V B ( X ) ↓ ⏐ N n V B ( X ′ ) . (6.6) The comparison is thus canonical and natural, while the local contractions used in Theorem 5.2 establish that it is a weak equivalence.
To apply the comparison to the generating maps, we now describe their second subdivisions. We use the classical face-chain description and intersection retraction of [17 Proposition 4.2] ; see also [4 Lemmas 4.8 and 4.10] . The proof below records the precise poset pair for each boundary and horn, including the empty case.
Lemma 6.2 (The subdivision posets). Let d : K ↪ Δ [ m ] d: K \hookrightarrow\Delta[m] d : K ↪ Δ [ m ] m ≥ 0 m \ge0 m ≥ 0 Λ k [ m ] ↪ Δ [ m ] \Lambda^k[m] \hookrightarrow\Delta[m] Λ k [ m ] ↪ Δ [ m ] m ≥ 1 m \ge1 m ≥ 1 0 ≤ k ≤ m 0 \le k \le m 0 ≤ k ≤ m A ⊂ B A \subset B A ⊂ B
Sd 2 d = ( N A ↪ N B ) . \operatorname{Sd}^{2}d = (NA \hookrightarrow NB). Sd 2 d = ( N A ↪ N B ) . Here A A A Z Z Z B B B A ↪ Z A \hookrightarrow Z A ↪ Z
Proof. Put M = { 0 , … , m } M = \{0,\ldots,m\} M = { 0 , … , m } P P P P K P_K P K K K K
P K = { { F : ∅ ≠ F ⊊ M } , K = ∂ Δ [ m ] , { F : ∅ ≠ F ⊆ M , M ∖ { k } ⊈ F } , K = Λ k [ m ] . P_K =
\begin{cases}
\{F : \varnothing\ne F \subsetneq M\}, & K = \partial\Delta[m],\\
\{F : \varnothing\ne F \subseteq M,\ M \setminus\{k\} \not\subseteq F\}, & K = \Lambda^k[m].
\end{cases} P K = { { F : ∅ = F ⊊ M } , { F : ∅ = F ⊆ M , M ∖ { k } ⊆ F } , K = ∂ Δ [ m ] , K = Λ k [ m ] . Indeed, a nonempty face belongs to the horn precisely when it is contained in M ∖ { j } M \setminus\{j\} M ∖ { j } j ≠ k j \ne k j = k P K P_K P K P P P M M M M ∖ { k } M \setminus\{k\} M ∖ { k } P K P_K P K P P P
The barycentric subdivision of an ordered simplicial complex has a vertex for every nonempty face and a nondegenerate simplex for every strict chain of such faces. Consequently
Sd Δ [ m ] = N P , Sd K = N P K . \operatorname{Sd}\Delta[m] = NP,\qquad\operatorname{Sd}K = NP_K. Sd Δ [ m ] = N P , Sd K = N P K . One can see these identifications directly on a simplex: the vertices of its subdivision are its nonempty subsets, and its simplices are chains of those subsets. The identifications agree on every face, so they glue over the faces of K K K N P NP N P P P P
B = { σ : σ is a nonempty chain in P } , A = { σ ∈ B : σ ⊆ P K } , σ ≤ τ ⟺ σ ⊆ τ . \begin{aligned}
B &= \{\sigma: \sigma\text{ is a nonempty chain in }P\},\\
A &= \{\sigma\in B : \sigma\subseteq P_K\},\qquad
\sigma\le\tau\Longleftrightarrow\sigma\subseteq\tau.
\end{aligned} B A = { σ : σ is a nonempty chain in P } , = { σ ∈ B : σ ⊆ P K } , σ ≤ τ ⟺ σ ⊆ τ . This proves the claimed identifications as simplicial sets, including degeneracies, and identifies their inclusion.
Every nonempty subchain of a chain in A A A A A A A A A B B B
Z = { σ ∈ B : σ ∩ P K ≠ ∅ } . Z = \{\sigma\in B : \sigma\cap P_K \ne\varnothing\}. Z = { σ ∈ B : σ ∩ P K = ∅ } . For the nontrivial implication in this equality, a face F ∈ σ ∩ P K F \in\sigma\cap P_K F ∈ σ ∩ P K { F } ∈ A \{F\} \in A { F } ∈ A σ \sigma σ
r : Z ⟶ A , r ( σ ) = σ ∩ P K . r: Z \longrightarrow A,\qquad r(\sigma) = \sigma\cap P_K. r : Z ⟶ A , r ( σ ) = σ ∩ P K . This is a nonempty chain, is monotone under inclusion, and fixes A A A α ∈ A \alpha\in A α ∈ A σ ∈ Z \sigma\in Z σ ∈ Z
α ⊆ σ ⟺ α ⊆ σ ∩ P K = r ( σ ) . \alpha\subseteq\sigma\Longleftrightarrow\alpha\subseteq\sigma\cap P_K = r(\sigma). α ⊆ σ ⟺ α ⊆ σ ∩ P K = r ( σ ) . This is precisely the right adjunction to A ↪ Z A \hookrightarrow Z A ↪ Z
When m = 0 m = 0 m = 0 K = ∂ Δ [ 0 ] K = \partial\Delta[0] K = ∂ Δ [ 0 ] B B B A = Z = ∅ A = Z = \varnothing A = Z = ∅ P K = { { k } } P_K = \{\{k\}\} P K = {{ k }} A A A Z Z Z { k } \{k\} { k }
Remark 6.3. For the zero-dimensional boundary, A = Z = T = ∅ A = Z = T = \varnothing A = Z = T = ∅ B = C B = C B = C V Z ( X ) = X V_Z(X) = X V Z ( X ) = X γ Z , X = id \gamma_{Z,X} = \operatorname{id} γ Z , X = id V B ( X ) = X ⨿ U B V_B(X) = X \amalg U_B V B ( X ) = X ⨿ U B
N n X ⨿ Δ [ 0 ] ≅ N n ( X ⨿ U B ) . N_nX \amalg\Delta[0] \cong N_n(X \amalg U_B). N n X ⨿ Δ [ 0 ] ≅ N n ( X ⨿ U B ) . Corollary 6.4 (Horn attachments are acyclic). For every horn inclusion d ∈ J d \in J d ∈ J L n d L_nd L n d W n W_n W n
Proof. Use the pair A ⊂ B A \subset B A ⊂ B L n d L_nd L n d U A → U B U_A \to U_B U A → U B N A → N B NA \to NB N A → N B Sd → id \operatorname{Sd} \to\operatorname{id} Sd → id [4 Theorem 4.5] . Naturality of the adjunction unit gives a square
N A → N B ↓ ↓ N n U A → N n U B . \begin{CD}
NA @>>> NB \\
@VVV @VVV \\
N_nU_A @>>> N_nU_B.
\end{CD} N A ↓ ⏐ N n U A N B ↓ ⏐ N n U B . The vertical maps are weak equivalences by Proposition 5.3. Two-out-of-three makes the lower map a weak equivalence, and it is monic by Proposition 6.1. Thus it is a simplicial trivial cofibration. For any attaching functor U A → X U_A \to X U A → X N n X → P B ( X ) N_nX \to P_B(X) N n X → P B ( X )
N n X ⟶ P B ( X ) → γ B , X N n V B ( X ) N_nX \longrightarrow P_B(X) \xrightarrow{\gamma_{B,X}} N_nV_B(X) N n X ⟶ P B ( X ) γ B , X N n V B ( X ) is the nerve of the attachment X → V B ( X ) X \to V_B(X) X → V B ( X ) W n W_n W n
Corollary 6.5 (Invariance under attachment). Let d ∈ I ∪ J d \in I \cup J d ∈ I ∪ J L n d L_nd L n d U A → U B U_A \to U_B U A → U B f : X → X ′ f : X \to X' f : X → X ′ W n W_n W n a : U A → X a : U_A \to X a : U A → X
X ⨿ U A U B ⟶ X ′ ⨿ U A U B , X \amalg_{U_A} U_B \longrightarrow X' \amalg_{U_A} U_B, X ⨿ U A U B ⟶ X ′ ⨿ U A U B , where the second attaching functor is f a fa f a W n W_n W n
Proof. The map N n X → P B ( X ) N_nX \to P_B(X) N n X → P B ( X ) N n U A ↪ N n U B N_nU_A \hookrightarrow N_nU_B N n U A ↪ N n U B
P B ( X ′ ) = N n X ′ ⨿ N n X P B ( X ) . P_B(X') = N_nX' \amalg_{N_nX} P_B(X). P B ( X ′ ) = N n X ′ ⨿ N n X P B ( X ) . Left properness of simplicial sets therefore makes P B ( X ) → P B ( X ′ ) P_B(X) \to P_B(X') P B ( X ) → P B ( X ′ ) N n f N_nf N n f (6.6) , both horizontal maps are weak equivalences by Proposition 6.1. Two-out-of-three implies that its right vertical map is a weak equivalence. This is the nerve of the asserted functor.
The model structure and the Quillen equivalence
Fix n ∈ { 1 , 2 , … , ∞ } n \in\{1, 2, \ldots, \infty\} n ∈ { 1 , 2 , … , ∞ }
L n = c n Sd 2 , R n = Ex 2 N n , L n ⊣ R n . L_n = c_n\operatorname{Sd}^{2}, \qquad R_n = \operatorname{Ex}^{2}N_n, \qquad L_n \dashv R_n. L n = c n Sd 2 , R n = Ex 2 N n , L n ⊣ R n . We use the Kan–Quillen model structure on sSet, with weak equivalences W K Q W_{\mathrm{KQ}} W KQ F i b K Q \mathrm{Fib}_{\mathrm{KQ}} Fib KQ
I = { ∂ Δ [ m ] ↪ Δ [ m ] ∣ m ≥ 0 } , I = \{\partial\Delta[m] \hookrightarrow\Delta[m] \mid m \ge0\}, I = { ∂ Δ [ m ] ↪ Δ [ m ] ∣ m ≥ 0 } , J = { Λ k [ m ] ↪ Δ [ m ] ∣ m ≥ 1 , 0 ≤ k ≤ m } . J = \{\Lambda^k[m] \hookrightarrow\Delta[m] \mid m \ge1,\ 0 \le k \le m\}. J = { Λ k [ m ] ↪ Δ [ m ] ∣ m ≥ 1 , 0 ≤ k ≤ m } . In particular, ∂ Δ [ 0 ] = ∅ \partial\Delta[0] = \varnothing ∂ Δ [ 0 ] = ∅
We first record the classical simplicial facts needed to identify the weak equivalences. The last-vertex transformation ℓ : S d ⟶ i d \ell: \mathrm{Sd} \longrightarrow\mathrm{id} ℓ : Sd ⟶ id [4 Theorem 4.5] . Its mate j : i d ⟶ E x j : \mathrm{id} \longrightarrow\mathrm{Ex} j : id ⟶ Ex [10 Theorem III.4.6] . The latter assertion is a theorem about subdivision and extension, and is not a formal consequence of being a mate. Both statements apply to every simplicial set independently of the model structure being constructed.
For a map f : K → K ′ f : K \to K' f : K → K ′
j K ′ f = ( E x f ) j K . j_{K'}f = (\mathrm{Ex}f)j_K. j K ′ f = ( Ex f ) j K . Two-out-of-three therefore proves that E x \mathrm{Ex} Ex ℓ \ell ℓ S d \mathrm{Sd} Sd
W n = N n − 1 ( W K Q ) = R n − 1 ( W K Q ) , F n = R n − 1 ( F i b K Q ) , C n = ⊥ ( F n ∩ W n ) . (7.1) \begin{aligned}
W_n &= N_n^{-1}(W_{\mathrm{KQ}}) = R_n^{-1}(W_{\mathrm{KQ}}),\\
F_n &= R_n^{-1}(\mathrm{Fib}_{\mathrm{KQ}}), \qquad C_n = {}^{\perp}(F_n \cap W_n). \tag*{(7.1)}
\end{aligned} W n F n = N n − 1 ( W KQ ) = R n − 1 ( W KQ ) , = R n − 1 ( Fib KQ ) , C n = ⊥ ( F n ∩ W n ) . (7.1) Here ⊥ \perp ⊥ W n W_n W n
For a set S \mathcal{S} S S \mathcal{S} S S \mathcal{S} S S \mathcal{S} S = ⊥ ( S -inj ) ={}^{\perp}(\mathcal{S}\text{-inj}) = ⊥ ( S -inj ) S \mathcal{S} S S \mathcal{S} S
Lemma 7.1 (Transfinite horn attachments). Every relative L n J L_nJ L n J W n W_n W n
Proof. Corollary 6.4 proves the assertion for one attachment, with an arbitrary attaching target. A set of simultaneous attachments may be well ordered and attached one at a time: at each stage use the given map from its source into the original target, followed by the map to the current stage. The universal property of the resulting colimit is the universal property of the pushout of the coproduct. Thus it suffices to consider a continuous ordinal sequence ( X α ) α ≤ λ (X_\alpha)_{\alpha\le\lambda} ( X α ) α ≤ λ
By transfinite induction, X 0 → X α X_0 \to X_\alpha X 0 → X α W n W_n W n α \alpha α β \beta β α < β \alpha< \beta α < β X 0 → X α X_0 \to X_\alpha X 0 → X α X 0 → colim α < β X α = X β X_0 \to\operatorname{colim}_{\alpha< \beta} X_\alpha= X_\beta X 0 → colim α < β X α = X β X 0 X_0 X 0
Proposition 7.2 (The transferred model structure). The classes ( C n , W n , F n ) (C_n,W_n,F_n) ( C n , W n , F n ) (7.1) form a combinatorial model structure on n C a t n\mathrm{Cat} n Cat ( L n I , L n J ) (L_nI,L_nJ) ( L n I , L n J )
Proof. The underlying category is locally presentable, and hence complete and cocomplete, by Lemma 2.4. We describe the small object argument to make clear that the factorizations have no restriction on their sources. For either set S = L n I \mathcal{S}=L_nI S = L n I S = L n J \mathcal{S}=L_nJ S = L n J κ \kappa κ S \mathcal{S} S κ \kappa κ S \mathcal{S} S
Given any map X → Y X \to Y X → Y X 0 = X X_0 = X X 0 = X α + 1 \alpha+ 1 α + 1 X α X_\alpha X α S \mathcal{S} S X α → Y X_\alpha\to Y X α → Y
X → i X κ → p Y . X \xrightarrow{i} X_\kappa\xrightarrow{p} Y. X i X κ p Y . Here i i i S \mathcal{S} S p ∈ S p \in\mathcal{S} p ∈ S X κ X_\kappa X κ X α X_\alpha X α κ \kappa κ κ \kappa κ p p p α + 1 \alpha+ 1 α + 1
Pushouts, transfinite composites, and retracts of maps with a fixed left lifting property retain that property. Consequently every relative S \mathcal{S} S S \mathcal{S} S u : A → B u : A \to B u : A → B S \mathcal{S} S A → i Z → p B A \xrightarrow{i} Z \xrightarrow{p} B A i Z p B i i i u u u p p p id B \operatorname{id}_B id B s : B → Z s : B \to Z s : B → Z s u = i su = i s u = i p s = id B ps = \operatorname{id}_B p s = id B u u u i i i A A A
( S -cof , S -inj ) , (\mathcal{S}\text{-cof},\mathcal{S}\text{-inj}), ( S -cof , S -inj ) , and its left class consists exactly of retracts of relative S \mathcal{S} S
The adjunction identifies lifting squares and their fillers. The Kan–Quillen lifting characterizations therefore give exactly
( L n I ) -inj = R n − 1 ( I -inj ) = F n ∩ W n , (7.2) (L_n I)\text{-inj} = R_n^{-1}(I\text{-inj}) = F_n \cap W_n, \tag*{(7.2)} ( L n I ) -inj = R n − 1 ( I -inj ) = F n ∩ W n , (7.2) ( L n J ) -inj = R n − 1 ( J -inj ) = F n . (7.3) (L_n J)\text{-inj} = R_n^{-1}(J\text{-inj}) = F_n. \tag*{(7.3)} ( L n J ) -inj = R n − 1 ( J -inj ) = F n . (7.3) In the first equality of classes on the right of (7.2) , we used the equality of the two definitions of W n W_n W n ( C n , F n ∩ W n ) (C_n, F_n \cap W_n) ( C n , F n ∩ W n )
Let D = ( L n J ) \mathcal{D} = (L_n J) D = ( L n J ) F n ∩ W n ⊆ F n F_n \cap W_n \subseteq F_n F n ∩ W n ⊆ F n D ⊆ C n \mathcal{D} \subseteq C_n D ⊆ C n D ⊆ W n \mathcal{D} \subseteq W_n D ⊆ W n u : A → B u : A \to B u : A → B C n ∩ W n C_n \cap W_n C n ∩ W n A → i Z → p B A \xrightarrow{i} Z \xrightarrow{p} B A i Z p B i ∈ D i \in \mathcal{D} i ∈ D p ∈ F n p \in F_n p ∈ F n i i i u = p i u = pi u = p i W n W_n W n p ∈ F n ∩ W n p \in F_n \cap W_n p ∈ F n ∩ W n u ∈ C n u \in C_n u ∈ C n s : B → Z s : B \to Z s : B → Z s u = i su = i s u = i p s = id B ps = \operatorname{id}_B p s = id B u u u D \mathcal{D} D
D = C n ∩ W n . \mathcal{D} = C_n \cap W_n. D = C n ∩ W n . Thus the second weak factorization system is ( C n ∩ W n , F n ) (C_n \cap W_n, F_n) ( C n ∩ W n , F n ) W n W_n W n
Proposition 7.3 (Properness). The model structure of Proposition 7.2 is left and right proper.
Proof. For right properness, let p : Y → Z p : Y \to Z p : Y → Z F n F_n F n w : Z ′ → Z w : Z' \to Z w : Z ′ → Z W n W_n W n R n R_n R n Y × Z Z ′ Y \times_Z Z' Y × Z Z ′ R n Y R_nY R n Y R n p R_np R n p R n w R_nw R n w W n W_n W n Y × Z Z ′ → Y Y \times_Z Z' \to Y Y × Z Z ′ → Y W n W_n W n
For left properness, first let i : X → Y i : X \to Y i : X → Y L n I L_n I L n I w : X → X ′ w : X \to X' w : X → X ′ W n W_n W n i i i X α X_\alpha X α X X X X α ′ = X ′ ⨿ X X α X'_{\alpha}=X'\amalg_X X_{\alpha} X α ′ = X ′ ⨿ X X α X α → X α ′ X_{\alpha}\to X'_{\alpha} X α → X α ′ W n W_n W n w w w X α → X α ′ X_{\alpha}\to X'_{\alpha} X α → X α ′ Y → X ′ ⨿ X Y Y\to X'\amalg_X Y Y → X ′ ⨿ X Y W n W_n W n
For a general i : X → Y i:X\to Y i : X → Y C n C_n C n i i i L n I L_nI L n I j : X → Z j:X\to Z j : X → Z s : Y → Z s:Y\to Z s : Y → Z p : Z → Y p:Z\to Y p : Z → Y s i = j si=j s i = j p j = i pj=i p j = i p s = id Y ps=\operatorname{id}_Y p s = id Y w w w s ′ s' s ′ p ′ p' p ′ X ′ ⨿ X Y X'\amalg_X Y X ′ ⨿ X Y X ′ ⨿ X Z X'\amalg_X Z X ′ ⨿ X Z p ′ s ′ = id p's'=\operatorname{id} p ′ s ′ = id Y → X ′ ⨿ X Y Y\to X'\amalg_X Y Y → X ′ ⨿ X Y Z → X ′ ⨿ X Z Z\to X'\amalg_X Z Z → X ′ ⨿ X Z W n W_n W n
Let η T : T → N n c n T \eta_T:T\to N_n c_nT η T : T → N n c n T c n ⊣ N n c_n\dashv N_n c n ⊣ N n T = N E T=NE T = N E
Proposition 7.4 (The unit after two subdivisions). For every simplicial set K K K
θ K = η Sd 2 K : Sd 2 K ⟶ N n L n K (7.4) \theta_K=\eta_{\operatorname{Sd}^2K}:\operatorname{Sd}^2K\longrightarrow N_nL_nK \tag*{(7.4)} θ K = η Sd 2 K : Sd 2 K ⟶ N n L n K (7.4) is a weak equivalence.
Proof. For K = Δ [ m ] K=\Delta[m] K = Δ [ m ] K = ∂ Δ [ m ] K=\partial\Delta[m] K = ∂ Δ [ m ] Sd 2 K \operatorname{Sd}^2K Sd 2 K c n ∅ c_n\varnothing c n ∅
Every simplicial set is obtained from the empty set by a continuous transfinite sequence of boundary attachments. One construction orders its nondegenerate simplices by dimension and well orders those in each dimension. The boundary of a simplex then maps into the stage already constructed. That attaching map may identify vertices or faces, or send faces to degenerate simplices. The construction still recovers K K K
Consider one successor step
K + = K 0 ⨿ ∂ Δ [ m ] Δ [ m ] K_+=K_0\amalg_{\partial\Delta[m]}\Delta[m] K + = K 0 ⨿ ∂ Δ [ m ] Δ [ m ] and assume the assertion for K 0 K_0 K 0
V = Sd 2 K 0 , A = Sd 2 ∂ Δ [ m ] , B = Sd 2 Δ [ m ] . V=\operatorname{Sd}^2K_0,\qquad A=\operatorname{Sd}^2\partial\Delta[m],\qquad B=\operatorname{Sd}^2\Delta[m]. V = Sd 2 K 0 , A = Sd 2 ∂ Δ [ m ] , B = Sd 2 Δ [ m ] . Here A , B , V A,B,V A , B , V A , B A,B A , B E , F E,F E , F Sd 2 K + = V ⨿ A B \operatorname{Sd}^2K_+=V\amalg_A B Sd 2 K + = V ⨿ A B η \eta η
q : V ⨿ A B ⟶ N n c n V ⨿ N n c n A N n c n B (7.5) q:V\amalg_A B\longrightarrow N_nc_nV\amalg_{N_nc_nA}N_nc_nB \tag*{(7.5)} q : V ⨿ A B ⟶ N n c n V ⨿ N n c n A N n c n B (7.5) The three maps on the objects of these spans are weak equivalences: on V V V A , B A,B A , B A → B A\to B A → B A = N E A=NE A = N E B = N F B=NF B = N F E ⊆ F E\subseteq F E ⊆ F c n A → c n B c_nA\to c_nB c n A → c n B N n c n A → N n c n B N_nc_nA\to N_nc_nB N n c n A → N n c n B q q q A → V A\to V A → V
Apply Proposition 6.1 to the arbitrary attaching functor c n A → c n V c_nA\to c_nV c n A → c n V
t : N n c n V ⨿ N n c n A N n c n B ⟶ N n ( c n V ⨿ c n A c n B ) (7.6) t:N_nc_nV\amalg_{N_nc_nA}N_nc_nB\longrightarrow N_n(c_nV\amalg_{c_nA}c_nB) \tag*{(7.6)} t : N n c n V ⨿ N n c n A N n c n B ⟶ N n ( c n V ⨿ c n A c n B ) (7.6) is a weak equivalence. Because c n c_n c n N n c n ( V ⨿ A B ) N_nc_n(V \amalg_A B) N n c n ( V ⨿ A B ) t q tq tq η V ⨿ A B \eta_{V \amalg_A B} η V ⨿ A B V V V B B B η \eta η θ K + \theta_{K_+} θ K +
At a nonzero limit ordinal, Sd 2 \operatorname{Sd}^2 Sd 2 L n L_n L n N n N_n N n (7.4) for every K K K
Theorem 7.5 (Quillen equivalence). The adjunction
L n : s S e t K Q ⇄ ( n C a t , C n , W n , F n ) : R n L_n : \mathrm{sSet}_{KQ} \rightleftarrows(\mathrm{nCat}, C_n, W_n, F_n) : R_n L n : sSet K Q ⇄ ( nCat , C n , W n , F n ) : R n is a Quillen equivalence.
Proof. Equations (7.2) and (7.3) show that R n R_n R n
Set S = Sd S = \operatorname{Sd} S = Sd T = Ex T = \operatorname{Ex} T = Ex a K : K → T S K a_K : K \to TSK a K : K → T S K S ⊣ T S \dashv T S ⊣ T
j K = T ( ℓ K ) a K : K ⟶ T K . j_K = T(\ell_K)a_K : K \longrightarrow TK. j K = T ( ℓ K ) a K : K ⟶ T K . The independently cited theorem above makes j K j_K j K ℓ K \ell_K ℓ K T T T T ( ℓ K ) : T S K → T K T(\ell_K) : TSK \to TK T ( ℓ K ) : T S K → T K a K a_K a K K K K a S K : S K → T S 2 K a_{SK} : SK \to TS^2K a S K : S K → T S 2 K S 2 ⊣ T 2 S^2 \dashv T^2 S 2 ⊣ T 2
b K = T ( a S K ) a K : K ⟶ T 2 S 2 K , T ( a S K ) : T S K ⟶ T 2 S 2 K . b_K = T(a_{SK})a_K : K \longrightarrow T^2S^2K,\qquad T(a_{SK}) : TSK \longrightarrow T^2S^2K. b K = T ( a S K ) a K : K ⟶ T 2 S 2 K , T ( a S K ) : T S K ⟶ T 2 S 2 K . Thus b K b_K b K T T T
The unit of the composite adjunction L n ⊣ R n L_n \dashv R_n L n ⊣ R n
u K = T 2 ( θ K ) b K : K ⟶ T 2 N n c n S 2 K = R n L n K . u_K = T^2(\theta_K)b_K : K \longrightarrow T^2N_nc_nS^2K = R_nL_nK. u K = T 2 ( θ K ) b K : K ⟶ T 2 N n c n S 2 K = R n L n K . Proposition 7.4, preservation of weak equivalences by T T T u K u_K u K K K K f : L n K → Y f : L_nK \to Y f : L n K → Y
f ♯ = R n ( f ) u K : K ⟶ R n Y . f^\sharp= R_n(f)u_K : K \longrightarrow R_nY. f ♯ = R n ( f ) u K : K ⟶ R n Y . Using the definition of W n W_n W n
f ∈ W n ⟺ R n ( f ) ∈ W K Q ⟺ f ♯ ∈ W K Q . f \in W_n \Longleftrightarrow R_n(f) \in W_{KQ} \Longleftrightarrow f^\sharp\in W_{KQ}. f ∈ W n ⟺ R n ( f ) ∈ W K Q ⟺ f ♯ ∈ W K Q . This equivalence holds for every K K K Y Y Y K K K Y Y Y n ≥ 1 n \ge1 n ≥ 1 n = ∞ n = \infty n = ∞
Presentations and local finite presentability
We give a direct proof of the presentability assertion used in Lemma 2.4. The point requiring care is that an equation between boundary cells can create new composable pairs. Presentations must therefore generate defined terms and equations together; taking an ordinary quotient of the cells already present would not suffice. All sets and constructions below lie in the fixed universe.
Operations and their domains
Fix n ∈ { 1 , 2 , … , ∞ } n \in\{1,2,\ldots,\infty\} n ∈ { 1 , 2 , … , ∞ } ∞ \infty ∞ ω \omega ω C d C_d C d 0 ≤ d ≤ n 0 \le d \le n 0 ≤ d ≤ n d d d n = ∞ n=\infty n = ∞
s d , t d : C d ⟶ C d − 1 ( d > 0 ) , e d : C d ⟶ C d + 1 ( d < n ) . s_d,t_d:C_d \longrightarrow C_{d-1}\quad(d>0),\qquad e_d:C_d \longrightarrow C_{d+1}\quad(d<n). s d , t d : C d ⟶ C d − 1 ( d > 0 ) , e d : C d ⟶ C d + 1 ( d < n ) . For 0 ≤ k < d 0 \le k < d 0 ≤ k < d s k d , t k d s_k^d,t_k^d s k d , t k d e k d e_k^d e k d k k k d d d ∗ k d *_k^d ∗ k d C d C_d C d
s k d ( x ) = t k d ( y ) . (A.1) s_k^d(x)=t_k^d(y).\tag{A.1} s k d ( x ) = t k d ( y ) . ( A.1 ) Thus x ∗ k d y x *_k^d y x ∗ k d y x x x y y y
For clarity, the equations imposed are the usual strict globular category equations. Globularity and the identity boundary equations are
s d − 1 s d = s d − 1 t d , t d − 1 s d = t d − 1 t d , s d + 1 e d = id = t d + 1 e d . s_{d-1}s_d=s_{d-1}t_d,\qquad t_{d-1}s_d=t_{d-1}t_d,\qquad s_{d+1}e_d=\operatorname{id}=t_{d+1}e_d. s d − 1 s d = s d − 1 t d , t d − 1 s d = t d − 1 t d , s d + 1 e d = id = t d + 1 e d . The globularity equations apply for 2 ≤ d ≤ n 2 \le d \le n 2 ≤ d ≤ n 0 ≤ d < n 0 \le d<n 0 ≤ d < n n = ∞ n=\infty n = ∞ (A.1) , the composition boundary equations are
s d ( x ∗ d − 1 y ) = s d ( y ) , t d ( x ∗ d − 1 y ) = t d ( x ) , s_d(x*_{d-1}y)=s_d(y),\qquad t_d(x*_{d-1}y)=t_d(x), s d ( x ∗ d − 1 y ) = s d ( y ) , t d ( x ∗ d − 1 y ) = t d ( x ) , s d ( x ∗ k y ) = s d ( x ) ∗ k s d ( y ) , t d ( x ∗ k y ) = t d ( x ) ∗ k t d ( y ) ( k < d − 1 ) . s_d(x*_k y)=s_d(x)*_k s_d(y),\qquad t_d(x*_k y)=t_d(x)*_k t_d(y)\quad(k<d-1). s d ( x ∗ k y ) = s d ( x ) ∗ k s d ( y ) , t d ( x ∗ k y ) = t d ( x ) ∗ k t d ( y ) ( k < d − 1 ) . For each k k k s k ( x ) = t k ( y ) s_k(x)=t_k(y) s k ( x ) = t k ( y ) s k ( y ) = t k ( z ) s_k(y)=t_k(z) s k ( y ) = t k ( z )
x ∗ k e k d ( s k d x ) = x , e k d ( t k d x ) ∗ k x = x . x*_k e_k^d(s_k^d x)=x,\qquad e_k^d(t_k^d x)*_k x=x. x ∗ k e k d ( s k d x ) = x , e k d ( t k d x ) ∗ k x = x . Identities preserve every lower composition: e d ( x ∗ k d y ) = e d ( x ) ∗ k d + 1 e d ( y ) e_d(x*_k^d y)=e_d(x)*_k^{d+1}e_d(y) e d ( x ∗ k d y ) = e d ( x ) ∗ k d + 1 e d ( y ) d < n d<n d < n
Interchange, for k < l < d k<l<d k < l < d
( x ∗ l y ) ∗ k ( z ∗ l w ) = ( x ∗ k z ) ∗ l ( y ∗ k w ) . (A.2) (x*_l y)*_k(z*_l w)=(x*_k z)*_l(y*_k w).\tag{A.2} ( x ∗ l y ) ∗ k ( z ∗ l w ) = ( x ∗ k z ) ∗ l ( y ∗ k w ) . ( A.2 ) Its guard includes all four inner matching equations:
s l d ( x ) = t l d ( y ) , s l d ( z ) = t l d ( w ) , s_l^d(x)=t_l^d(y),\qquad s_l^d(z)=t_l^d(w), s l d ( x ) = t l d ( y ) , s l d ( z ) = t l d ( w ) , s k d ( x ) = t k d ( z ) , s k d ( y ) = t k d ( w ) . (A.3) s_k^d(x)=t_k^d(z),\qquad s_k^d(y)=t_k^d(w).\tag{A.3} s k d ( x ) = t k d ( z ) , s k d ( y ) = t k d ( w ) . ( A.3 ) These equations also imply the two outer matching conditions. For the left side, globularity gives
s k d ( x ∗ l y ) = s k d ( x ) = s k d ( y ) , t k d ( z ∗ l w ) = t k d ( z ) = t k d ( w ) . s_k^d(x*_l y)=s_k^d(x)=s_k^d(y),\qquad t_k^d(z*_l w)=t_k^d(z)=t_k^d(w). s k d ( x ∗ l y ) = s k d ( x ) = s k d ( y ) , t k d ( z ∗ l w ) = t k d ( z ) = t k d ( w ) . For the right side, the iterated boundary formulas give
s l d ( x ∗ k z ) = s l d ( x ) ∗ k l s l d ( z ) , s_l^d(x*_k z)=s_l^d(x)*_k^l s_l^d(z), s l d ( x ∗ k z ) = s l d ( x ) ∗ k l s l d ( z ) , t l d ( y ∗ k w ) = t l d ( y ) ∗ k l t l d ( w ) . t_l^d(y*_k w)=t_l^d(y)*_k^l t_l^d(w). t l d ( y ∗ k w ) = t l d ( y ) ∗ k l t l d ( w ) . The lower-dimensional composites in these formulas are defined by the last two equations of (A.3) and globularity. The first two equations then make their values equal. Thus the guard describes an ordinary composable array, not an additional restriction on interchange. Definedness of the right-hand side alone is not used to infer the guard or definedness of the left-hand side.
These are a set of conditional equations with finite terms and finitely many premises. In the ω \omega ω n n n n n n n n n
Defined terms and presentations
A sorted set of generators G G G T ( G ) T(G) T ( G ) T ( G ) T(G) T ( G )
Let R R R R R R strong equation : its interpretation requires both terms to be defined and their values to agree. In particular, t = t t=t t = t
Lemma A.1 (Strict-category presentations). For every sorted set G G G R R R n n n P ( G , R ) P(G,R) P ( G , R ) Y Y Y P ( G , R ) → Y P(G,R) \to Y P ( G , R ) → Y Y Y Y R R R
Proof. Generate simultaneously judgments D ( t ) \mathrm{D}(t) D ( t ) t = u t=u t = u t t t t , u t,u t , u
(i) Every generator is defined, and each member of R R R
(ii) An equation t = u t=u t = u D ( t ) \mathrm{D}(t) D ( t ) D ( u ) \mathrm{D}(u) D ( u ) x ∗ k y x *_k y x ∗ k y (A.1) .
(iii) Total operations applied to defined arguments give defined terms. Defined arguments satisfying (A.1) give a defined composite.
(iv) Equality is symmetric and transitive. Reflexivity is the rule D ( t ) ⇒ t = t \mathrm{D}(t) \Rightarrow t=t D ( t ) ⇒ t = t
(v) Equal arguments transport definedness and equality through an operation. Precisely, from t i = u i t_i=u_i t i = u i D ( f ( t 1 , … , t r ) ) \mathrm{D}(f(t_1,\ldots,t_r)) D ( f ( t 1 , … , t r )) D ( f ( u 1 , … , u r ) ) \mathrm{D}(f(u_1,\ldots,u_r)) D ( f ( u 1 , … , u r ))
(vi) Impose every ground instance of the strict category equations above when its arguments are defined and its stated matching conditions hold. The conclusion is equality of the two sides and therefore definedness of both. In particular, interchange retains all four conditions in (A.3) .
Globularity, identity boundary and unit equations require just definedness of their variables. Composition boundary equations and preservation of composition by identities require (A.1) ; associativity requires its two matching equations. Thus every rule has finitely many premises. All rules are sound in a strict category, including the definedness conclusions in the last rule.
Start with the assumptions and repeatedly add the conclusions of all applicable rules. The union of the finite stages is the least closed set of judgments: any rule’s finitely many premises occur together at some stage. In particular, every resulting judgment has a finite derivation. This construction involves a set of judgments, since the signature and the set of raw finite terms are sets.
Let P ( G , R ) d P(G,R)_d P ( G , R ) d d d d [ x ] [x] [ x ] [ y ] [y] [ y ] s k ( x ) = t k ( y ) s_k(x)=t_k(y) s k ( x ) = t k ( y ) x ∗ k y x *_k y x ∗ k y
This construction does not depend on the representatives. Indeed, if x = x ′ x=x' x = x ′ y = y ′ y=y' y = y ′
s k ( x ) = s k ( x ′ ) , t k ( y ) = t k ( y ′ ) . s_k(x)=s_k(x'), \qquad t_k(y)=t_k(y'). s k ( x ) = s k ( x ′ ) , t k ( y ) = t k ( y ′ ) . The matching equality for x , y x,y x , y x ′ , y ′ x',y' x ′ , y ′
Now let an assignment G → Y G \to Y G → Y R R R Y Y Y P ( G , R ) → Y P(G,R) \to Y P ( G , R ) → Y
The constructed object is small. There is no inconsistency obstruction: the strict category with one cell in each sort satisfies every strong equation under the unique assignment. If G G G
Remark A.2. The simultaneous construction is necessary even for ordinary categories. If the source and target of the free arrow are identified, its presentation must create the square of the arrow and all higher powers. These are not obtained by merely identifying members of the original set of cells. Rule (iii) creates them as soon as their boundary equalities become derivable.
Limits, colimits, and finite presentations
Proposition A.3 (Local finite presentability). For every 1 ≤ n ≤ ∞ 1 \le n \le\infty 1 ≤ n ≤ ∞ n C a t n\mathrm{Cat} n Cat
Proof. Colimits. For a small diagram X : I → n C a t X:I \to n\mathrm{Cat} X : I → n Cat [ i , a ] [i,a] [ i , a ] a a a X i X_i X i
s ( [ i , a ] ) = [ i , s ( a ) ] , t ( [ i , a ] ) = [ i , t ( a ) ] , s([i,a])=[i,s(a)], \qquad t([i,a])=[i,t(a)], s ([ i , a ]) = [ i , s ( a )] , t ([ i , a ]) = [ i , t ( a )] , e ( [ i , a ] ) = [ i , e ( a ) ] , [ i , a ] ∗ k [ i , b ] = [ i , a ∗ k b ] e([i,a])=[i,e(a)], \qquad[i,a]\ast_k[i,b]=[i,a\ast_k b] e ([ i , a ]) = [ i , e ( a )] , [ i , a ] ∗ k [ i , b ] = [ i , a ∗ k b ] where the last equation is included for every defined composite. For each arrow f : i → j f:i \to j f : i → j [ i , a ] = [ j , X ( f ) ( a ) ] [i,a]=[j,X(f)(a)] [ i , a ] = [ j , X ( f ) ( a )] Y Y Y X i → Y X_i \to Y X i → Y
Limits and local smallness. In each dimension, take the set of compatible families in the diagram. Source, target and identity are coordinatewise. Two families have matching boundaries exactly when their coordinates do, so coordinatewise composition is defined on exactly the required domain and again gives a compatible family. All laws and the limit universal property hold coordinatewise. This includes the empty limit, with a singleton in each sort. Strict functors form a subset of the product of the sets of cell maps, so the category is locally small.
Filtered colimits and finite witnesses. For a filtered diagram, form the colimit of each set of cells. Finitely many representatives occur at a common stage. If a matching equation holds in the colimit, it holds at a later common stage, where the composite can be formed. A further common stage proves independence of choices. Each category law holds at a stage containing its finite tuple and its matching equations. This constructs the colimit with its universal property, without any injectivity assumption on transition maps.
We will also use the finite-term version of this argument. If finitely many assigned generator values make a finite collection of terms defined and satisfy specified equations in the colimit, these facts hold at a common stage. Induct on the finitely many subterms: first lift the arguments, then the boundary equality needed for a partial composite, and then the composite. Finally lift the required equalities. Each step uses the equality criterion for filtered colimits of sets.
Finite presentations. If G G G R R R P ( G , R ) → c o l i m i X i P(G,R) \to\mathop{\mathrm{colim}}\limits_i X_i P ( G , R ) → i colim X i G G G
c o l i m i Hom ( P ( G , R ) , X i ) ≅ Hom ( P ( G , R ) , c o l i m i X i ) . \mathop{\mathrm{colim}}\limits_i \operatorname{Hom}(P(G,R),X_i) \cong\operatorname{Hom}\left(P(G,R),\mathop{\mathrm{colim}}\limits_i X_i\right). i colim Hom ( P ( G , R ) , X i ) ≅ Hom ( P ( G , R ) , i colim X i ) . Hence every finite presentation is finitely presentable. This does not assert that it has finitely many cells.
A generating set of finite presentations. Every strict category X X X g a g_a g a X X X
Write this presentation as ( G X , R X ) (G_X,R_X) ( G X , R X ) ( G 0 , R 0 ) (G_0,R_0) ( G 0 , R 0 ) G 0 ⊆ G X G_0 \subseteq G_X G 0 ⊆ G X R 0 ⊆ R X R_0 \subseteq R_X R 0 ⊆ R X R 0 R_0 R 0 G 0 G_0 G 0 P ( G X , R X ) P(G_X,R_X) P ( G X , R X ) P ( G X , R X ) P(G_X,R_X) P ( G X , R X )
The finite pairs need not be closed under all operations. Finite presentations themselves supply the required closures. There is a set of finite presentation types: use finite numbered generator sets with specified sorts and finite sets of raw term equations. Thus every object is a filtered colimit of objects from a set of finitely presentable objects. Together with cocompleteness, this is local finite presentability. The same reasoning applies to the countable signature in the ω \omega ω
A common regular bound. More generally, if G G G R R R κ \kappa κ P ( G , R ) P(G,R) P ( G , R ) κ \kappa κ κ \kappa κ κ \kappa κ κ \kappa κ κ \kappa κ κ \kappa κ
The full cell presentation of X X X
max { ℵ 0 , ∣ ∐ d X d ∣ } . \max\left\{\aleph_0,\left|\coprod_d X_d\right|\right\}. max { ℵ 0 , d ∐ X d } . because the signature is countable and operations have arity at most two. Choose a larger infinite regular cardinal. The preceding argument makes X X X
Relation to the weak-groupoid expansion method
The elementary-expansion theorem of [13 ] concerns models of a fixed Grothendieck coherator in the Ara–Henry convention. Its Theorem 1.1 asserts that, for a cellular model X X X m m m ( m + 1 ) (m+1) ( m + 1 ) n n n
The construction in [13 Sections 3–4] first reflects a model to one with unique fillers above a fixed dimension q q q q q q
There is a formal strict analogue of the cutoff: retain the cells of a strict ω \omega ω q q q q q q q q q τ q \tau_q τ q 0 ⟶ 1 ⟵ 2 0 \longrightarrow1 \longleftarrow2 0 ⟶ 1 ⟵ 2
In the present directed setting, the corresponding role is played by the positive relative prism of Lemma 3.1. It is stationary on the old sieve, and its chain formula gives compatible strict maps on whole right Gray cylinders. Lemma 2.1 passes these cylinders to finite n n n [13 ] is a premise of this proof.
References 
