Diagonal model structures - Semantic Scholar
Diagonal model structures J.F. Jardine Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada September 29, 2010
Introduction The original purpose of this note was to display a model structure for the category s2 Set of bisimplicial sets whose cofibrations are the monomorphisms and whose weak equivalences are the diagonal weak equivalences, and then show that it is cofibrantly generated in a very precise way. The project grew to include analogous model structures on categories of bisimplicial presheaves. These model structures are the diagonal model structures of the title. The fibrations for the diagonal model structure on bisimplicial sets are the Kan fibrations, which are defined by a lifting property with respect to the bisimplicial analogues of inclusions of horns in simplices. A horn can be viewed as the part of boundary ∂∆p,q of a bisimplex ∆p,q that one gets by removing a single cell of maximal total degree — the inclusions of all such horns are simple examples of anodyne extensions of bisimplicial sets. It is relatively painless to show that the diagonal model structures exist for all categories s2 Pre(C) of bisimplicial presheaves — this result is Theorem 4. It is also easy to show that the diagonal functor and its left adjoint d∗ define a Quillen equivalence d∗ : s Pre(C) s2 Pre(C) : d between the injective model structure on simplicial presheaves and the diagonal structure for bisimplicial presheaves; this equivalence appears here as Proposition 6. The Quillen equivalence d∗ : sSet s2 Set : d between the standard model structure on simplicial sets and the diagonal model structure on bisimplicial sets is an immediate consequence. The Moerdijk model structure for bisimplicial sets [7], [2] is induced from the standard model structure for simplicial sets by the diagonal functor — this was the first published example of a model structure for bisimplicial sets whose 1
weak equivalences are defined by the diagonal functor. We show that there is a plethora of such model structures intermediate between an analog of the Moerdijk structure for bisimplicial presheaves and the diagonal structure; the precise statement is Theorem 10. This result is analogous to a phenomenon observed in the relation between the projective and injective structures for simplicial presheaves in [6]. Bisimplicial sets come with their own particular fun. The problem of showing that the fibrations of the diagonal model structure for bisimplicial sets are precisely the Kan fibrations is technically interesting, and is the subject of the second section of this paper, leading to Theorem 24. This result mirrors well known results for simplicial sets and cubical sets [1], [5]. Roughly speaking, the overall idea of the proof is to produce the minimal fibrewise model structure for the category of bisimplicial set maps X → ∆p,q over the bisimplex ∆p,q in which the anodyne extensions are weak equivalences, and then show that this structure (here called the anodyne structure) coincides with the diagonal structure given by Theorem 4. The proof of this last statement amounts to a derivation of standard homotopy colimit results within the confines of fibrewise anodyne model structures. This is not the end — one finishes with an appeal to Quillen’s Theorem B. Overall, the methods and results of this paper are already known. Most of the main proof techniques were introduced in Cisinski’s thesis [1], and then described again in [5]. In particular, Theorem 4 is a special case of a general result for test categories which appears as Theorem 6.2 of [5], but the proof of this result which is given here is new and much more direct. Theorem 24 has been known in some form since at least 2003: in particular, Cisinski and Joyal-Tierney knew proofs of that result at that time. The proof of Theorem 10 is an almost word for word translation of the intermediate model structures story for simplicial presheaves. The theory described here has not been collected together in a public document up to now, and this paper was written to fill the gap, in anticipation of concrete applications.
1
Bisimplicial presheaves
Recall that a bisimplicial set X is a functor X : ∆op × ∆op → Set, and a morphism of bisimplicial sets is a natural transformation of such functors. Write Xp,q = X(p, q) for ordinal numbers p and q. Let s2 Set denote the category of bisimplicial sets. The bisimplicial set hom( (p, q)) which is represented by the pair of ordinal numbers (p, q) is denoted by ∆p,q , and is called a standard bisimplex. The bisimplices are the cells for the category of bisimplicial sets. As usual, the diagonal simplicial set d(X) is defined by d(X)p = Xp,p . 2
This construction defines the diagonal functor d : s2 Set → sSet. The diagonal functor has both a left adjoint d∗ and a right adjoint d∗ . The left adjoint d∗ is defined by extending the assignment d∗ ∆n = ∆n,n in a canonical way, while the right adjoint d∗ is defined by d∗ (Y )p,q = hom(∆p × ∆q , Y ), All functorial constructions on bisimplicial sets extend to presheaves of bisimplicial sets. Let C be a small Grothendieck site, and let s2 Pre(C) denote the category of functors X : C op → s2 Set and all natural transformations between them — this is the category of bisimplicial presheaves, or presheaves of bisimplicial sets on the site C. Say that a map f : X → Y of bisimplicial presheaves is a diagonal weak equivalence if the induced simplicial presheaf map d(X) → d(Y ) is a local weak equivalence in the usual sense [3], [4]. A monomorphism of bisimplicial presheaves is a cofibration. An injective fibration of bisimplicial presheaves is a morphism which has the right lifting property with respect to trivial cofibrations. Suppose that β is a cardinal number. A bisimplicial presheaf A is said to be β-bounded if |Ap,q (U )| < β for all p, q ≥ 0 and all objects U in C. Suppose that α is an infinite cardinal which is an upper bound for the site C in the sense that α > | Mor(C|. We have the following “bounded cofibration lemma” for bisimplicial presheaves: Lemma 1. Suppose that i : X → Y is a trivial cofibration of bisimplicial presheaves, and that A is an α-bounded subobject of Y . Then Y has an αbounded subobject B such that A ⊂ B and the cofibration B ∩ X → B is a diagonal weak equivalence. Proof. There is an induced diagram d(X) i∗ / d(Y )
d(A)
where i∗ is a trivial cofibration of simplicial presheaves and d(A) is an α-bounded subobject of d(Y ). The bounded cofibration lemma for simplicial presheaves (this result first appeared as Lemma 12 of [4]) implies that there is an α-bounded subobject D1 of d(Y ) such that d(A) ⊂ D1 and D1 ∩ d(X) → D1 is a local weak equivalence. Since D1 is α-bounded there is an α-bounded subobject A1 of the bisimplicial presheaf Y such that A ⊂ A1 and D1 ⊂ d(A1 ). Repeat this construction inductively to find an ascending families of α-bounded subobjects A ⊂ A1 ⊂ A2 ⊂ · · · ⊂ Y 3
and d(A) ⊂ D1 ⊂ D2 ⊂ · · · ⊂ d(Y ) such that Di ⊂ d(Ai+1 ) and the map Di ∩d(X) → Di is a local weak equivalence for all i. Set B = ∪i Ai . Then the map B ∩ X → B of bisimplicial presheaves is a diagonal weak equivalence. Corollary 2. A map p : X → Y is an injective fibration of bisimplicial presheaves if and only if it has the right lifting property with respect to all αbounded trivial cofibrations. The proof of this corollary is a standard Zorn’s lemma argument. Recall that every simplicial set K can be identified with a horizontally constant bisimplicial set having the same name in a standard way, with Kp,q = Kq . I also use the same notation for a bisimplicial set B and its associated constant simplicial presheaf, so that B(U ) = B for all objects U of C. Lemma 3. A map q : Z → Y is an injective fibration and a diagonal weak equivalence if and only if it has the right lifting property with respect to all α-bounded cofibrations. Proof. If q has the right lifting property with respect to all α-bounded cofibrations, then it has the right lifting property with respect to all cofibrations, by the usual Zorn’s lemma argument. In this case, q has a section σ : Y → Z, and the lifting exists in the diagram (σq,1)
Z tZ Z × ∆1
6/ Z q
/Z
pr
q
/Y
It follows that the induced map d(q) is a simplicial homotopy equivalence, and hence a local weak equivalence. Suppose that q is an injective fibration and a diagonal weak equivalence. Then q has a factorization i / Z@ X @@ @@ p q @@ Y such that p has the right lifting property with respect to all α-bounded cofibrations and i is a cofibration. Then p is a diagonal weak equivalence, so the cofibration i is a diagonal weak equivalence, and the lift exists in the diagram Z
1
q
i
X
/Z >
p
4
/Y
The map q is therefore a retract of the map p, and has the right lifting property with respect to all α-bounded cofibrations. The function complex hom(X, Y ) for bisimplicial sets X and Y is the simplicial set whose n-simplices are the bisimplicial set maps X × ∆n → Y . Theorem 4. Suppose that C is a small Grothendieck site. Then, with the definitions of cofibration, injective fibration and diagonal weak equivalence given above, the category s2 Pre(C) of bisimplicial sets has the structure of a cofibrantly generated closed simplicial model category. Properness for the model structure of Theorem 4 is proved in Corollary 8 below. Proof. The axioms CM1, CM2 and CM3 are easy to verify: in particular, CM2 and CM3 are straightforward consequences of the corresponding statements for the injective model structure on simplicial presheaves. Similarly, trivial cofibrations are closed under pushout, so that Corollary 2 and Lemma 3 imply the factorization axiom CM5. The lifting axiom CM4 also follows from Lemma 3. The cofibrant generation follows from Corollary 2 and Lemma 3. For the simplicial structure, we show that if i : A → B is a cofibration of bisimplicial presheaves and j : K → L is a cofibration of simplicial sets, then the cofibration (B × K) ∪ (A × L) → B × L is trivial if either i or j is trivial, but this is a consequence of the corresponding statement for simplicial presheaves. Remark 5. Methods of Cisinski [1], [5] can be used to give an alternative derivation of the model structure of Theorem 4. In particular, Theorem 6.2 of [5] gives a model structure on the bisimplicial presheaf category for which the cofibrations are the monomorphisms and a map f : X → Y is a weak equivalence if and only if the induced map f∗ : B(i∆2 X) → B(i∆2 Y ) is a local weak equivalence of simplicial presheaves. Here, the “cell category” i∆2 Z for a bisimplicial set Z has all bisimplicial set maps ∆p,q → Z for objects and all commutative diagrams ∆p,q N NNN NN' 7Z ppp ppp ∆r,s for morphisms. There is a weak equivalence B(i∆2 Z) ' d(Z) which is natural in bisimplicial sets Z — see Lemma 23 below. The model structure of Theorem 4 is the diagonal structure on the category of bisimplicial presheaves. This result specializes to give diagonal model structures
5
for all categories s2 SetI of small diagrams of simplicial sets and to the category s2 Set. In particular, a cofibration for the diagonal structure on bisimplicial sets is a monomorphism, a weak equivalences is a bisimplicial set map X → Y such that the induced map d(X) → d(Y ) is a weak equivalence of simplicial sets, and fibrations are defined by a right lifting property with respect to trivial cofibrations. The left adjoint d∗ : sSet → s2 Set of the diagonal functor d preserves cofibrations and takes trivial cofibrations to diagonally trivial cofibrations [2]. It follows that the functors d∗ and d define a Quillen adjunction between the standard model structure on simplicial sets and the diagonal structure on bisimplicial sets. The adjunction map η : ∆n → dd∗ (∆n ) can be identified up to isomorphism with the diagonal map ∆n → ∆n × ∆n , which map is a weak equivalence. The functors d and d∗ both preserve colimits, cofibrations and trivial cofibrations, so an induction on skeleta shows that the adjunction map η : X → dd∗ (X) is a weak equivalence for all simplicial sets X. A triangle identity argument then shows that the natural map : d∗ d(Y ) → Y is a diagonal equivalence for all bisimplicial sets Y . It follows that the left adjoint d∗ : s Pre(C) → s2 Pre(C) takes local weak equivalences to diagonal weak equivalences for simplicial presheaves on a Grothendieck site C. The functors d and d∗ therefore determine a Quillen adjunction between the injective model structure for simplicial presheaves and the diagonal model structure for bisimplicial presheaves. The adjunction map : d∗ d(Y ) → Y is a sectionwise diagonal equivalence for bisimplicial presheaves Y , and we then have the following result: Proposition 6. Suppose that C is a small Grothendieck site. Then the adjoint functors d∗ : s Pre(C) s2 Pre(C) : d define a Quillen equivalence between the injective model structure on simplicial presheaves and the diagonal structure on bisimplicial presheaves on the site C. Corollary 7. The adjoint functors d∗ : sSet s2 Set : d define a Quillen equivalence between the standard model structure on simplicial sets and the diagonal structure on bisimplicial sets. Corollary 8. The diagonal model structure on the category s2 Pre(C) is proper. Proof. All bisimplicial presheaves are cofibrant, so that pushouts of diagonal weak equivalences along cofibrations are diagonal weak equivalences [2, II.8.5].
6
The functor d∗ preserves cofibrations and weak equivalences, so that d preserves fibrations. The functor d also preserves pullbacks. Thus, right properness for the diagonal model structure bisimplicial presheaves follows from right properness for the injective structure on simplicial presheaves. Corollary 9. The diagonal model structure on the category s2 Set of bisimplicial sets is proper. The Moerdijk model structure is another well known example ([7], Section IV.3.3 of [2]) of a model structure on the category s2 Set of bisimplicial sets for which the weak equivalences are the diagonal weak equivalences. The Moerdijk structure is induced from the standard model structure on simplicial sets, in that a bisimplicial set map X → Y is a fibration (respectively weak equivalence) for this model structure if and only if the induced map d(X) → d(Y ) is a Kan fibration (respectively weak equivalence) of simplicial sets. The Moerdijk structure is also Quillen equivalent to the standard model structure on simplicial sets, via the diagonal functor and its left adjoint. Suppose that S is a set of cofibrations of bisimplicial presheaves which contains the set S0 of all maps d∗ A → d∗ B which are induced by α-bounded cofibrations A → B of simplicial presheaves. Suppose that S further satisfies the closure property that if the map C → D is in S, then so is the induced cofibration (D × ∂∆n ) ∪ (C × ∆n ) → D × ∆n , for all n ≥ 0. Let CS be the saturation of the set S in the class of all cofibrations (monomorphisms) of the bisimplicial set category. I say that CS is the class of S-cofibrations. Say that a bisimplicial presheaf map p : X → Y is an S-fibration if it has the right lifting property with respect to all S-cofibrations which are diagonal weak equivalences. The proof of the following result follows the outline established in [6]: Theorem 10. The category s2 Pre(C) of bisimplicial presheaves, together with the S-cofibrations, diagonal weak equivalences and S-fibrations satisfis the axioms for a proper closed simplicial model category. This model structure is cofibrantly generated. Proof. Every map f : X → Y has a factorization /Z X@ @@ @@ q f @@ Y j
where j is a member of CS and q has the right lifting property with respect to all members of CS . Then q∗ : d(Z) → d(Y ) is a trivial injective fibration of simplicial presheaves, so that q is a diagonal weak equivalence. The map q is an S-fibration. 7
The map f : X → Y also has a factorization /W XB BB BB p B f BB Y i
where i is a trivial cofibration and p is a fibration for the diagonal model structure of Theorem 4. The map p is an S-fibration. The cofibraton i has a factorization i = q · j as above, where j is an S-cofibration and q is an S-fibration and a diagonal equivalence. The map j is a diagonal equivalence, so that f has a factorization f = (p·q)·j such that pq˙ is an S-fibration and j is an S-cofibration and a diagonal equivalence. We have verified the model category axiom CM5. If p : X → Y is an Sfibration and a diagonal equivalence, then it is a retract of a map which has the right lifting property with respect to all S-cofibrations, giving CM4. The rest of the model category axioms are trivial. The simplicial model axiom SM7 is a consequence of the construction of the class CS and the instance of this axiom for the injective model structure on simplicial presheaves. The left properness of this structure is an easy consequence of left properness for the diagonal structure on s2 Pre(C), while right properness follows from right properness for the injective structure on s Pre(C). The cofibrant generation follows from what is now a familiar trick. Every α-bounded trivial cofibration β : A → B has a factorization jβ
/ Zβ A@ @@ @@ @ qβ β @@ B as in the first paragraph, where jβ is an S-cofibration and qβ has the right lifting property with respect to all S-cofibrations. Then both jβ and qβ are diagonal equivalences. One shows that if i : C → D is an α-bounded S-cofibration and there is a commutative diagram /X
C i
D
/Y
f
where f is a diagonal equivalence, then the diagram has a factorization C i
D
/A jβ
/ Zβ 8
/X f
/Y
for some β. Finally, if j : E → F is an S-cofibration and a diagonal equivalence, then j has a factorization i /V E@ @@ @@ p j @@ F where p has the right lifting property with respect to all jβ and i is in the saturation of the set of all maps jβ . But then j and p are diagonal equivalences, and the construction of the last paragraph shows that p has the right lifting property with respect to all members of CS , so that i is a retract of j. This means that the set of all maps jβ generates the class of trivial cofibrations in the model structure defined by the set of cofibrations S. Say that the model structure of Theorem 10 is the S-model structure on the category of bisimplicial presheaves. The S0 -model structure on bisimplicial sets (for whatever infinite cardinal α) is the Moerdijk structure, and the S0 -model structure for bisimplicial presheaves is a locally defined analogue of the Moerdijk structure. An obvious comparison with the various intermediate model structures for simplicial presheaves [6] says that the S0 -model structure for bisimplicial presheaves is a “projective” model structure, while the diagonal model structure of Theorem 4 is an “injective” model structure, and all S-model structures have classes of cofibrations lying between these two extremes.
2
Bisimplicial sets
˜ be the bisimplical set Suppose that K and L are simplicial sets, and let K ×L defined by ˜ p,q = Kp × Lq . (K ×L) ˜ is the external product of K and L. The bisimplicial set K ×L Examples: 1) The standard bisimplex ∆p,q has the form ˜ q. ∆p,q = ∆p ×∆ 2) Set q ˜ q ) ∪ (∆p ×∂∆ ˜ ˜ q = ∆p,q . ∂∆p,q = (∂∆p ×∆ ) ⊂ ∆p ×∆
Then the boundary ∂∆p,q of the bisimplex ∆p,q is generated as a subcomplex by the images of the maps (di , 1) : ∆p−1,q → ∆p,q and (1, dj ) : ∆p,q−1 → ∆p,q . The following statement about simplicial sets is well known. The argument for it which is given here may be a bit unfamiliar. Lemma 11. Suppose that x, y are non-degenerates simplices of X and s, t are ordinal number epimorphisms such that s∗ (x) = t∗ (y). Then x = y and s = t. 9
Proof. If dim(y) < dim(x) then x is degenerate: in effect, if d is a section of s then x = d∗ s∗ y = d∗ t∗ y is degenerate. It follows that x and y have the same simplicial dimension. If the ordinal number epis s and t have distinct sections, then there is a section d of s (say) which is not a section of t, in which case t · d is not an ordinal number epi, and so x = d∗ s∗ (x) = d∗ t∗ (y) is degenerate. It follows that s and t have the same sections, and therefore s = t. But then s∗ is an injective function, so that x = y. Suppose that X is a bisimplicial set and that x ∈ Xp,q . The number p + q is the total degree of x. Suppose that A is a subcomplex of a bisimplicial set X and that x ∈ Xp,q is a bisimplex of X − A of minimal total degree. Write x : ∆p,q → X for the classifying map of the bisimplex x. The bisimplices (di , 1)(x) and (1, dj )(x) have smaller total degree than x and are therefore in A, and it follows that there is a pullback diagram α /A ∂∆p,q i
∆p,q
x
/X
of bisimplicial set maps. Lemma 12. Suppose that A is a subcomplex of a bisimplicial set X and that x ∈ Xp,q is a bisimplex of X − A of minimal total degree. Form the pushout ∂∆p,q
α
i
∆p,q
/A
x
/B
Then the induced bisimplicial set map B → X is a monomorphism. Proof. If x = s(y) for some degeneracy s (vertical or horizontal), then y has smaller total degree, and so y ∈ A and x ∈ A. It follows that x is vertically and horizontally non-degenerate. There is a decomposition Br,s = Ar,s t {u × v : r × s → p × q, u, v epi}. in all bidegrees. If a ∈ Ar,s and u × v have the same image in X, then a = (u × v)∗ (x) is in A so that x ∈ A by applying a suitable section of u × v, which can’t happen. The restriction of Br,s → Xr,s to Ar,s is the monomorphism i : Ar,s → Xr,s .
10
Finally, if the epis u × v, u0 × v 0 : r × s → p × q have the same image in X, then (u × v)∗ (x) = (u0 × v 0 )∗ (x) in X. The bisimplex (1 × v)∗ (x) is horizontally non-degenerate. Otherwise, (1 × v)∗ (x) = (s × 1)∗ (y) for some y and non-trivial ordinal number epi s, and if d is a section of v then x = (1 × d)∗ (1 × v)∗ (x) = (1 × d)∗ (s × 1)∗ (y) = (s × 1)∗ (1 × d)∗ (y) so that x is horizontally degenerate. Similarly, (1 × v 0 )∗ (x) is horizontally nondegenerate, and so Lemma 11 and the relations (u × 1)∗ (1 × v)∗ (x) = (u0 × 1)∗ (1 × v 0 )∗ (x) together imply that u = u0 and (1 × v)∗ (x) = (1 × v 0 )∗ (x), so that v = v 0 Corollary 13. The set of inclusions ∂∆p,q ⊂ ∆p,q generates the class of cofibrations of s2 Set. The class A of anodyne extensions of s2 Set is the saturation of the set of bisimplicial set maps S, which consists of all morphisms s ˜ s ) ∪ (∆r ×∂∆ ˜ ˜ s = ∆r,s (Λrk ×∆ ) ⊂ ∆r ×∆
as well as all morphisms ˜ s ) ∪ (∆r ×Λ ˜ sj ) ⊂ ∆r ×∆ ˜ s = ∆r,s (∂∆r ×∆ The class A contains the set of all cofibrations ˜ ˜ ˜ (A×D) ∪ (B ×C) ⊂ B ×D induced by cofibrations A → B and C → D, where one of the two maps is a trivial cofibration of simplicial sets. The diagonal of such a map is the trivial cofibration (A × D) ∪ (B × C) ⊂ B × D. in simplicial sets. It follows that every anodyne extension is a diagonal weak equivalence. Say that a map p : X → Y of bisimplicial sets is a Kan fibration if it has the right lifting property with respect to all anodyne extensions. Every fibration for the diagonal model structure of Theorem 4 is a Kan fibration. The purpose of this section is to prove the converse assertion, so that the diagonal fibrations of bisimplicial sets are precisely the Kan fibrations. This statement appears as Theorem 24 below. The method of proof involves showing that a map p : X → ∆p,q is a diagonal fibration if and only if it is a Kan fibration. For this, we need a model structure on the slice category s2 Set/∆p,q which is defined by formally inverting the anodyne extensions. The fibrant objects in the resulting “anodyne model 11
structure” are precisely the Kan fibrations X → ∆p,q . Then the idea is to show that a map /Y X9 99 99 99 ∆p,q f
is a weak equivalence of the anodyne model structure if and only if the bisimplicial set map f : X → Y is a diagonal weak equivalence (Lemma 23). In this case, the anodyne model structure on the slice category s2 Set/∆p,q coincides with the model structure that category canonically inherits from the diagonal structure on bisimplicial sets, and then the two theories have the same fibrant objects. We begin by describing the anodyne model structures. Suppose that X is a bisimplicial set and that K is a simplicial set. Define a bisimplicial set X ⊗ K by the assignment X ⊗ K = X × K, where we have identified K wiht a bisimplicial set which is constant in the horizontal direction: Kp,q = Kq . It follows that there is a natural isomorphism d(X ⊗ K) ∼ = d(X) × K. The construction X ⊗ K preserves diagonal weak equivalences in bisimplicial sets X and weak equivalences in simplicial sets K. Lemma 14. Suppose that i : A → B is a cofibration of bisimplicial sets and that j : K → L is a cofibration of simplicial sets. Then the induced map (i, j)∗ : (B ⊗ K) ∪ (A ⊗ L) → B ⊗ L is a cofibration which is anodyne if either i or j is anodyne. Proof. The map (∆r,s ⊗ K) ∪ (∂∆r,s ⊗ L) → ∆r,s ⊗ L can be identified with the map s ˜ s × L)) ∪ (∆r ×((∂∆ ˜ ˜ s × L), (∂∆r ×(∆ × L) ∪ (∆s × K))) → ∆r ×(∆
which is a cofibration. It follows from Corollary 13 that the map (i, j)∗ is a cofibration in general. The simplicial set map (∂∆s × L) ∪ (∆s × K) → ∆s × L is anodyne if j is anodyne, so that (i, j)∗ is anodyne in general if j is anodyne. The remaining statements have similar proofs. 12
Suppose that Z is a fixed bisimplicial set, and write s2 Set/Z for the category whose objects are the bisimplicial set maps φ : X → Z and whose morphisms f : φ → ψ are the commutative diagrams f /Y X4 44
4
φ 44
ψ
Z
of bisimplicial set maps. The class AZ of anodyne extensions over Z is the saturation of the set of maps SZ in s2 Set/Z, where SZ consists of the morphisms s ˜ s ) ∪ (∆r ×∂∆ ˜ ˜ s = ∆r,s → Z (Λrk ×∆ ) ⊂ ∆r ×∆
as well as the morphisms ˜ s ) ∪ (∆r ×Λ ˜ sj ) ⊂ ∆r ×∆ ˜ s = ∆r,s → Z (∂∆r ×∆ The class AZ contains the set of all cofibrations ˜ ˜ ˜ →Z (A×D) ∪ (B ×C) ⊂ B ×D induced by cofibrations A → B and C → D, where one of the two maps is a trivial cofibration of simplicial sets. Let I = ∆1 be the standard interval in simplicial sets. One uses the interval theory which assigns to an object φ : X → Z the objects defined by the compositions pr
φ
X ⊗ I ×n −→ X − →Z along with the set SZ of generators for the class AZ of anodyne extensions to define a closed simplicial model structure on the category s2 Set/Z (Theorem 4.17 of [5]). The function complex hom(φ, ψ) has a standard construction: an n-simplex is a commutative diagram /Y
X × ∆n pr
X
ψ
φ
/Z
Quillen’s simplicial model axiom SM7 for this theory is a consequence of Lemma 14, but see also Lemma 15 below. A weak equivalence for this model structure, which will be called an anodyne weak equivalence over Z is a map f : φ → ψ such that the induced map π(ψ, γ) → π(φ, γ) in fibre homotopy classes of maps (defined by the interval 13
theory) is a bijection for all Kan fibrant objects γ : W → Z. A cofibration is a monomorphism of s2 Set/Z, and a map p : φ → ψ is a fibration, here called an anodyne fibration over Z, if it has the right lifting property with respect to all cofibrations which are anodyne weak equivalences over Z. This model structure is left proper since every object is cofibrant [2, II.8.5], and is cofibrantly generated by construction. It follows from Lemma 14 that a map f : X → Y → Z of s2 Set/Z is injective for this theory if and only if it has the right lifting property with respect to all members of AZ , and an object φ : X → Z is injective if the map φ → 1Z is injective. Equivalently the object φ is injective if and only if the underlying map φ : X → Z is a Kan fibration of bisimplicial sets. The classes of anodyne fibrant objects and injective objects over Z coincide, by standard nonsense (see Corollary 4.14 of [5]). The following simple result fixes ideas: f
i
Lemma 15. A map A → − B− → Z is an anodyne extension over Z if and only if the map i : A → B is an anodyne extension of bisimplicial sets. Proof. Suppose that i : A → B is an anodyne extension, and construct a factorization j /X A@ @@ @@ p i @@ B such that j is a filtered colimit of pushouts of morphisms of S, hence of morphisms of SZ , and p is a Kan fibration. Then the dotted arrow lifting exists in the diagram j /X A > p
i
B
1
/B @@ @@f @@ @ Z
i
f
j
fp
so that the map A → − B − → Z is a retract of the map A − → C −→ Z, and the latter map is an anodyne extension over Z. The converse is clear. We shall need some properties of homotopy colimits in bisimplicial sets and their relation to anodyne weak equivalences. Suppose first of all that X is a simplicial set. Let γn : ∆n × Xn → X
14
be the map which classifies the n-simplices of X. Then we have skn X = ∪i≤n im(γi ). Write s[r] Xn = ∪i≤r si (Xn ) ⊂ Xn+1 , and observe that s[n] Xn is the degenerate part of Xn+1 . Then there are natural pushout diagrams sr+1 / s[r] Xn s[r] Xn−1 (1) Xn
sr+1
/ s[r+1] Xn
and (∆n+1 × s[n] Xn ) ∪ (∂∆n+1 × Xn+1 )
/ skn X
∆n+1 × Xn+1
/ skn+1 X
(2)
The diagrams (1) and (2) are natural in simplicial sets X. Thus, if m 7→ Ym is a simplicial object in simplicial sets (aka. a bisimplicial set, where m is the horizontal variable), the simplicial set maps ∆n × Yn,m → Y∗,m induce bisimplicial set maps ˜ n → Y. γn : ∆n ×Y The bisimplicial set Y has a natural filtration skn Y by (horizontal) skeleta, and there is a natural pushout diagram s[r] Yn−1 Yn
sr+1
sr+1
/ s[r] Yn
(3)
/ s[r+1] Yn
in simplicial sets, and a pushout diagram ˜ n+1 ) ˜ [n] Yn ) ∪ (∂∆n+1 ×Y (∆n+1 ×s
/ skn Y
˜ n+1 ∆n+1 ×Y
/ skn+1 Y
in bisimplicial sets in which the vertical maps are cofibrations. 15
(4)
Lemma 16. Suppose that f : K → K 0 and g : L → L0 are weak equivalences of ˜ 0 → Z. simplicial sets, and suppose given a fixed map of bisimplicial sets L×L Then the induced map ˜ : K ×L ˜ → K 0 ×L ˜ 0→Z f ×g is an anodyne weak equivalence of bisimplicial sets over Z. Proof. We show that the map ˜ → K 0 ×L ˜ →Z f × 1 : K ×L is an anodyne weak equivalence. This is true if f is a trivial cofibration by Lemma 14, and is therefore true in general since all simplicial sets are cofibrant. Lemma 17. Suppose that f : X → Y → Z is a map of bisimplicial sets over Z such that the map f : Xn → Yn is a weak equivalence of simplicial sets in each horizontal degree n. Then the map f is an anodyne weak equivalence over Z. Proof. Use the natural diagrams (3) and (4) to approximate the map f by maps of skeleta over Z. One shows inductively that each map s[r] X → s[r] Y → Z and each skn X → skn Y → Z is an anodyne equivalence over Z by using Lemma 16. The induced diagram sk0 X
/ sk1 X
/ sk2 X
/ ...
sk0 Y
/ sk1 Y
/ sk2 Y
/ ...
defines a weak equivalence of projective cofibrant diagrams for the anodyne structure on s2 Set/Z, and therefore induces an anodyne weak equivalence X → Y → Z in the filtered colimit. If m 7→ Ym is a simplicial object in bisimplicial sets (aka. a trisimplicial set Ym,p,q , where m is the first variable), the simplicial set maps ∆n × Yn,p,q → Y∗,p,q induce trisimplicial set maps ˜ n → Y, γn : ∆n ×Y where the indicated external product has the obvious definition. The trisimplicial set Y has a natural filtration skn Y by (first variable) skeleta, and there is a pushout diagram sr+1 / s[r] Yn s[r] Yn−1 (5) Yn
sr+1
16
/ s[r+1] Yn
in bisimplicial sets, and a pushout diagram ˜ [n] Yn ) ∪ (∂∆n+1 ×Y ˜ n+1 ) (∆n+1 ×s
/ skn Y
˜ n+1 ∆n+1 ×Y
/ skn+1 Y
(6)
in trisimplicial sets. The diagrams (5) and (6) are natural in trisimplicial sets Y. Let s3 Set denote the category of trisimplicial sets. Define the partial diagonal functor d1,2 : s3 Set → s2 Set by setting d1,2 (Y )p,q = Yp,p,q for a trisimplicial set Y . This functor preserves all limits and colimits. If K is a simplicial set and X → Z is a bisimplicial set over Z, then there is a natural isomorphism ∼ =
˜ d1,2 (K ×X) CC CC CC CC !
Z
/ K ×X
of bisimplicial sets over Z. The bifunctor K × X takes weak equivalences in simplicial sets K and anodyne equivalences in bisimplicial sets X → Z to anodyne weak equivalences over Z, by the simplicial structure of the anodyne model structure over Z and the fact that all objects of s2 Set/Z are cofibrant. Lemma 18. Suppose that Z is a bisimplicial set and that the map f in the diagram f /Y X4 44
44
4
Z
of trisimplicial set maps is a sectionwise anodyne equivalence over Z in the sense that all bisimplicial set maps Xn → Yn → Z are anodyne equivalences over Z, where Z is identified with a trisimplicial set which is constant in the first variable. Then the induced diagram f∗ / d1,2 Y d1,2 X :: :: :: : Z
is an anodyne equivalence over Z. 17
Proof. Write d1,2 (Y )(p) = d1,2 (skp Y ), where the skeletal filtration skp Y of Y in the first variable is as defined above. Apply the partial diagonal d1,2 to instances of the diagrams (5) and (6) for X and Y , and recall that these diagrams are functorial. It follows by induction (and left properness) that all maps d1,2 (X)(p) AA AA AA A
f∗
Z
/ d1,2 (Y )(p) ~ ~~ ~ ~ ~~ ~
are anodyne weak equivalences of bisimplicial sets over Z. The proof is finished (as was the argument for Lemma 17) with a “telescope” argument. In effect, the diagram d1,2 (X)(0)
/ d1,2 (X)(1)
/ d1,2 (X)(2)
/ ...
d1,2 (Y )(0)
/ d1,2 (Y )(1)
/ d1,2 (Y )(2)
/ ...
defines a weak equivalence of projective cofibrant diagrams for the anodyne model structure on bisimplicial sets over Z. The map f therefore induces an anodyne weak equivalence d1,2 (X) → d1,2 (Y ) → Z in the filtered colimit. Suppose that X : I → s2 Set/Z is a diagram of bisimplicial sets over Z which is indexed on a small category I. Then there is a trisimplicial set holim −−−→ I X over Z which is specified in the first variable by the bisimplicial set maps G holim X = X(i0 ) → Z. I n −−−→ i0 →···→in
Write also holim −−−→ I X for the bisimplicial set d1,2 (holim −−−→ I X) Then we have the following: Corollary 19. Suppose that f : X → Y is a natural transformation of Idiagrams of bisimplicial sets over Z such that all maps X(i) → Y (i) → Z are anodyne weak equivalences of bisimplicial sets over Z. Then the induced map f∗
holim −−−→ I X −→ holim −−−→ I Y → Z is an anodyne weak equivalence of bisimplicial sets over Z. Proof. All maps G i0 →···→in
G
X(i0 ) →
Y (i0 ) → Z
i0 →···→in
are anodyne weak equivalences over Z. Apply Lemma 18. 18
Suppose that p : U → X is a projective cofibrant model of the diagram X over Z. In horizontal degrees, p consists of maps p : Un → Xn of I-diagrams of simplicial sets over Zn , for which each object Un is projective cofibrant. The following result is then a consequence of Corollary 17 and the corresponding fact for simplicial sets (Section 5 of [5]): Lemma 20. Suppose that the diagram U : I → s2 Set/Z is projective cofibrant. Then the canonical map U →Z holim −−−→ I U → lim −→ I
is an anodyne weak equivalence of bisimplicial sets over Z. The following result says that we have been using the right homotopy colimit construction: Corollary 21. Suppose that X : I → s2 Set/Z is a small diagram of bisimplicial sets over Z, and let p : U → X be a projective cofibrant model. Then the indicated maps ' / lim U holim −→I −−−→ I U p∗ '
holim −−−→ I X
/Z
are anodyne weak equivalences of bisimplicial sets over Z. Here’s the “regularity” statement [1], [5] for bisimplicial sets over a fixed base: Lemma 22. Suppose that X → Z is a bisimplicial set over Z. Then the canonical map p,q p,q φ:− holim −−→ ∆ →X ∆ → X → Z is an anodyne weak equivalence of bisimplicial sets over Z. Proof. The tricomplex map underlying φ consists of weak equivalences of simplicial sets p,q p,q φ:− holim −−→ ∆ →X ∆r,s → Xr,s . It follows that the induced map p,q p,q φ : d1,2 (holim −−−→ ∆ →X ∆r,s ) → X
of bisimplicial sets consists of weak equivalences of simplicial sets in each vertical degree. The map φ is therefore an anodyne weak equivalence of bisimplicial sets over Z, by Lemma 17.
19
Lemma 23. Suppose that the bisimplicial set map f in the diagram f /Y X4 44
44
4
Z
is a diagonal weak equivalence. Then the diagram is an anodyne weak equivalence of bisimplicial sets over Z. Proof. There are natural anodyne weak equivalences φ
p,q p,q p,q B(∆2 /X) × Z = − holim → X, −−→ ∆ →X ∗ ← holim −−−→ ∆ →X ∆ −
of bisimplicial sets over Z by Corollary 19 and Lemma 20. The simplicial set map B(∆2 /X) → B(∆2 /Y ) is a weak equivalence of simplicial sets since f : X → Y is a diagonal weak equivalence, so that the induced map pr
B(∆2 /X) × Z → B(∆2 /Y ) × Z −→ Z is an anodyne weak equivalence of bisimplicial sets over Z. Theorem 24. A map p : X → Y of bisimplicial sets is a fibration if and only if it is a Kan fibration. Proof. We show that every Kan fibration which is a diagonal weak equivalence has the right lifting property with respect to all cofibrations. Suppose that this is so, and let i : A → B be a cofibration which is a diagonal weak equivalence. Find a factorization /Z A@ @@ @@ p i @@ B j
such that j is anodyne and p is a Kan fibration. Then, subject to the claim of the first paragraph, the map p is a diagonal weak equivalence and the lifting exists in the diagram j /Z A > p
i
B
1
/B
Then i is a retract of j, and is therefore an anodyne extension. Thus, the classes of diagonal trivial cofibrations and anodyne extensions coincide, so the classes of fibrations (of Theorem 4) and Kan fibrations coincide. 20
We use the method of proof of Theorem 8.6 of [5] to finish the argument. Suppose that p : X → Y is a Kan fibration and a diagonal weak equivalence, and consider the pullbacks /X
∆p,q ×Y X p∗
p
/Y
∆p,q
Then all maps p∗ are fibrations by Lemma 23, since the induced diagonal and anodyne model structures for s2 Set/∆p,q coincide. The maps α∗ in all pullback diagrams α∗ / ∆p,q ×Y X ∆r,s ×Y X p∗
p∗
∆r,s
/ ∆p,q
α
are diagonal weak equivalences by properness of the diagonal structure, since the maps p∗ are fibrations. It follows from Quillen’s Theorem B [2, IV.5.6] that all induced diagrams of simplicial set maps / B(i∆2 X)
B(i∆2 (∆p,q ×Y X)) p∗
/ B(i∆2 Y )
B(i∆2 ∆p,q )
are homotopy cartesian. Here, i∆2 X is the category of bisimplices ∆k,l → X of the bisimplicial set X. The indicated map p∗ is therefore a weak equivalence, so Corollary 19 and Lemma 22 together imply that the map p∗ : ∆p,q ×Y X → ∆p,q is a diagonal weak equivalence. Thus, suppose given a commutative diagram ∂∆p,q
/X
∆p,q
/Y
where p is a Kan fibration and a diagonal weak equivalence. Then the indicated lifting exists in the diagram ∂∆p,q
/ ∆p,q ×Y X 8
/X
p∗
∆p,q
/ ∆p,q
since p∗ is a fibration and a diagonal equivalence. 21
p
/Y
References [1] Denis-Charles Cisinski. Les pr´efaisceaux comme mod`eles des types d’homotopie, volume 308 of Ast´erisque. Soci´et´e Math´ematique de France, Paris, 2006. [2] P. G. Goerss and J. F. Jardine. Simplicial Homotopy Theory, volume 174 of Progress in Mathematics. Birkh¨auser Verlag, Basel, 1999. [3] J. F. Jardine. Simplicial presheaves. J. Pure Appl. Algebra, 47(1):35–87, 1987. [4] J. F. Jardine. Boolean localization, in practice. Doc. Math., 1:No. 13, 245– 275 (electronic), 1996. [5] J. F. Jardine. Categorical homotopy theory. Homology, Homotopy Appl., 8(1):71–144 (electronic), 2006. [6] J. F. Jardine. Intermediate model structures for simplicial presheaves. Canad. Math. Bull., 49(3):407–413, 2006. [7] Ieke Moerdijk. Bisimplicial sets and the group-completion theorem. In Algebraic K-theory: connections with geometry and topology (Lake Louise, AB, 1987), volume 279 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 225–240. Kluwer Acad. Publ., Dordrecht, 1989.
22
Introduction The original purpose of this note was to display a model structure for the category s2 Set of bisimplicial sets whose cofibrations are the monomorphisms and whose weak equivalences are the diagonal weak equivalences, and then show that it is cofibrantly generated in a very precise way. The project grew to include analogous model structures on categories of bisimplicial presheaves. These model structures are the diagonal model structures of the title. The fibrations for the diagonal model structure on bisimplicial sets are the Kan fibrations, which are defined by a lifting property with respect to the bisimplicial analogues of inclusions of horns in simplices. A horn can be viewed as the part of boundary ∂∆p,q of a bisimplex ∆p,q that one gets by removing a single cell of maximal total degree — the inclusions of all such horns are simple examples of anodyne extensions of bisimplicial sets. It is relatively painless to show that the diagonal model structures exist for all categories s2 Pre(C) of bisimplicial presheaves — this result is Theorem 4. It is also easy to show that the diagonal functor and its left adjoint d∗ define a Quillen equivalence d∗ : s Pre(C) s2 Pre(C) : d between the injective model structure on simplicial presheaves and the diagonal structure for bisimplicial presheaves; this equivalence appears here as Proposition 6. The Quillen equivalence d∗ : sSet s2 Set : d between the standard model structure on simplicial sets and the diagonal model structure on bisimplicial sets is an immediate consequence. The Moerdijk model structure for bisimplicial sets [7], [2] is induced from the standard model structure for simplicial sets by the diagonal functor — this was the first published example of a model structure for bisimplicial sets whose 1
weak equivalences are defined by the diagonal functor. We show that there is a plethora of such model structures intermediate between an analog of the Moerdijk structure for bisimplicial presheaves and the diagonal structure; the precise statement is Theorem 10. This result is analogous to a phenomenon observed in the relation between the projective and injective structures for simplicial presheaves in [6]. Bisimplicial sets come with their own particular fun. The problem of showing that the fibrations of the diagonal model structure for bisimplicial sets are precisely the Kan fibrations is technically interesting, and is the subject of the second section of this paper, leading to Theorem 24. This result mirrors well known results for simplicial sets and cubical sets [1], [5]. Roughly speaking, the overall idea of the proof is to produce the minimal fibrewise model structure for the category of bisimplicial set maps X → ∆p,q over the bisimplex ∆p,q in which the anodyne extensions are weak equivalences, and then show that this structure (here called the anodyne structure) coincides with the diagonal structure given by Theorem 4. The proof of this last statement amounts to a derivation of standard homotopy colimit results within the confines of fibrewise anodyne model structures. This is not the end — one finishes with an appeal to Quillen’s Theorem B. Overall, the methods and results of this paper are already known. Most of the main proof techniques were introduced in Cisinski’s thesis [1], and then described again in [5]. In particular, Theorem 4 is a special case of a general result for test categories which appears as Theorem 6.2 of [5], but the proof of this result which is given here is new and much more direct. Theorem 24 has been known in some form since at least 2003: in particular, Cisinski and Joyal-Tierney knew proofs of that result at that time. The proof of Theorem 10 is an almost word for word translation of the intermediate model structures story for simplicial presheaves. The theory described here has not been collected together in a public document up to now, and this paper was written to fill the gap, in anticipation of concrete applications.
1
Bisimplicial presheaves
Recall that a bisimplicial set X is a functor X : ∆op × ∆op → Set, and a morphism of bisimplicial sets is a natural transformation of such functors. Write Xp,q = X(p, q) for ordinal numbers p and q. Let s2 Set denote the category of bisimplicial sets. The bisimplicial set hom( (p, q)) which is represented by the pair of ordinal numbers (p, q) is denoted by ∆p,q , and is called a standard bisimplex. The bisimplices are the cells for the category of bisimplicial sets. As usual, the diagonal simplicial set d(X) is defined by d(X)p = Xp,p . 2
This construction defines the diagonal functor d : s2 Set → sSet. The diagonal functor has both a left adjoint d∗ and a right adjoint d∗ . The left adjoint d∗ is defined by extending the assignment d∗ ∆n = ∆n,n in a canonical way, while the right adjoint d∗ is defined by d∗ (Y )p,q = hom(∆p × ∆q , Y ), All functorial constructions on bisimplicial sets extend to presheaves of bisimplicial sets. Let C be a small Grothendieck site, and let s2 Pre(C) denote the category of functors X : C op → s2 Set and all natural transformations between them — this is the category of bisimplicial presheaves, or presheaves of bisimplicial sets on the site C. Say that a map f : X → Y of bisimplicial presheaves is a diagonal weak equivalence if the induced simplicial presheaf map d(X) → d(Y ) is a local weak equivalence in the usual sense [3], [4]. A monomorphism of bisimplicial presheaves is a cofibration. An injective fibration of bisimplicial presheaves is a morphism which has the right lifting property with respect to trivial cofibrations. Suppose that β is a cardinal number. A bisimplicial presheaf A is said to be β-bounded if |Ap,q (U )| < β for all p, q ≥ 0 and all objects U in C. Suppose that α is an infinite cardinal which is an upper bound for the site C in the sense that α > | Mor(C|. We have the following “bounded cofibration lemma” for bisimplicial presheaves: Lemma 1. Suppose that i : X → Y is a trivial cofibration of bisimplicial presheaves, and that A is an α-bounded subobject of Y . Then Y has an αbounded subobject B such that A ⊂ B and the cofibration B ∩ X → B is a diagonal weak equivalence. Proof. There is an induced diagram d(X) i∗ / d(Y )
d(A)
where i∗ is a trivial cofibration of simplicial presheaves and d(A) is an α-bounded subobject of d(Y ). The bounded cofibration lemma for simplicial presheaves (this result first appeared as Lemma 12 of [4]) implies that there is an α-bounded subobject D1 of d(Y ) such that d(A) ⊂ D1 and D1 ∩ d(X) → D1 is a local weak equivalence. Since D1 is α-bounded there is an α-bounded subobject A1 of the bisimplicial presheaf Y such that A ⊂ A1 and D1 ⊂ d(A1 ). Repeat this construction inductively to find an ascending families of α-bounded subobjects A ⊂ A1 ⊂ A2 ⊂ · · · ⊂ Y 3
and d(A) ⊂ D1 ⊂ D2 ⊂ · · · ⊂ d(Y ) such that Di ⊂ d(Ai+1 ) and the map Di ∩d(X) → Di is a local weak equivalence for all i. Set B = ∪i Ai . Then the map B ∩ X → B of bisimplicial presheaves is a diagonal weak equivalence. Corollary 2. A map p : X → Y is an injective fibration of bisimplicial presheaves if and only if it has the right lifting property with respect to all αbounded trivial cofibrations. The proof of this corollary is a standard Zorn’s lemma argument. Recall that every simplicial set K can be identified with a horizontally constant bisimplicial set having the same name in a standard way, with Kp,q = Kq . I also use the same notation for a bisimplicial set B and its associated constant simplicial presheaf, so that B(U ) = B for all objects U of C. Lemma 3. A map q : Z → Y is an injective fibration and a diagonal weak equivalence if and only if it has the right lifting property with respect to all α-bounded cofibrations. Proof. If q has the right lifting property with respect to all α-bounded cofibrations, then it has the right lifting property with respect to all cofibrations, by the usual Zorn’s lemma argument. In this case, q has a section σ : Y → Z, and the lifting exists in the diagram (σq,1)
Z tZ Z × ∆1
6/ Z q
/Z
pr
q
/Y
It follows that the induced map d(q) is a simplicial homotopy equivalence, and hence a local weak equivalence. Suppose that q is an injective fibration and a diagonal weak equivalence. Then q has a factorization i / Z@ X @@ @@ p q @@ Y such that p has the right lifting property with respect to all α-bounded cofibrations and i is a cofibration. Then p is a diagonal weak equivalence, so the cofibration i is a diagonal weak equivalence, and the lift exists in the diagram Z
1
q
i
X
/Z >
p
4
/Y
The map q is therefore a retract of the map p, and has the right lifting property with respect to all α-bounded cofibrations. The function complex hom(X, Y ) for bisimplicial sets X and Y is the simplicial set whose n-simplices are the bisimplicial set maps X × ∆n → Y . Theorem 4. Suppose that C is a small Grothendieck site. Then, with the definitions of cofibration, injective fibration and diagonal weak equivalence given above, the category s2 Pre(C) of bisimplicial sets has the structure of a cofibrantly generated closed simplicial model category. Properness for the model structure of Theorem 4 is proved in Corollary 8 below. Proof. The axioms CM1, CM2 and CM3 are easy to verify: in particular, CM2 and CM3 are straightforward consequences of the corresponding statements for the injective model structure on simplicial presheaves. Similarly, trivial cofibrations are closed under pushout, so that Corollary 2 and Lemma 3 imply the factorization axiom CM5. The lifting axiom CM4 also follows from Lemma 3. The cofibrant generation follows from Corollary 2 and Lemma 3. For the simplicial structure, we show that if i : A → B is a cofibration of bisimplicial presheaves and j : K → L is a cofibration of simplicial sets, then the cofibration (B × K) ∪ (A × L) → B × L is trivial if either i or j is trivial, but this is a consequence of the corresponding statement for simplicial presheaves. Remark 5. Methods of Cisinski [1], [5] can be used to give an alternative derivation of the model structure of Theorem 4. In particular, Theorem 6.2 of [5] gives a model structure on the bisimplicial presheaf category for which the cofibrations are the monomorphisms and a map f : X → Y is a weak equivalence if and only if the induced map f∗ : B(i∆2 X) → B(i∆2 Y ) is a local weak equivalence of simplicial presheaves. Here, the “cell category” i∆2 Z for a bisimplicial set Z has all bisimplicial set maps ∆p,q → Z for objects and all commutative diagrams ∆p,q N NNN NN' 7Z ppp ppp ∆r,s for morphisms. There is a weak equivalence B(i∆2 Z) ' d(Z) which is natural in bisimplicial sets Z — see Lemma 23 below. The model structure of Theorem 4 is the diagonal structure on the category of bisimplicial presheaves. This result specializes to give diagonal model structures
5
for all categories s2 SetI of small diagrams of simplicial sets and to the category s2 Set. In particular, a cofibration for the diagonal structure on bisimplicial sets is a monomorphism, a weak equivalences is a bisimplicial set map X → Y such that the induced map d(X) → d(Y ) is a weak equivalence of simplicial sets, and fibrations are defined by a right lifting property with respect to trivial cofibrations. The left adjoint d∗ : sSet → s2 Set of the diagonal functor d preserves cofibrations and takes trivial cofibrations to diagonally trivial cofibrations [2]. It follows that the functors d∗ and d define a Quillen adjunction between the standard model structure on simplicial sets and the diagonal structure on bisimplicial sets. The adjunction map η : ∆n → dd∗ (∆n ) can be identified up to isomorphism with the diagonal map ∆n → ∆n × ∆n , which map is a weak equivalence. The functors d and d∗ both preserve colimits, cofibrations and trivial cofibrations, so an induction on skeleta shows that the adjunction map η : X → dd∗ (X) is a weak equivalence for all simplicial sets X. A triangle identity argument then shows that the natural map : d∗ d(Y ) → Y is a diagonal equivalence for all bisimplicial sets Y . It follows that the left adjoint d∗ : s Pre(C) → s2 Pre(C) takes local weak equivalences to diagonal weak equivalences for simplicial presheaves on a Grothendieck site C. The functors d and d∗ therefore determine a Quillen adjunction between the injective model structure for simplicial presheaves and the diagonal model structure for bisimplicial presheaves. The adjunction map : d∗ d(Y ) → Y is a sectionwise diagonal equivalence for bisimplicial presheaves Y , and we then have the following result: Proposition 6. Suppose that C is a small Grothendieck site. Then the adjoint functors d∗ : s Pre(C) s2 Pre(C) : d define a Quillen equivalence between the injective model structure on simplicial presheaves and the diagonal structure on bisimplicial presheaves on the site C. Corollary 7. The adjoint functors d∗ : sSet s2 Set : d define a Quillen equivalence between the standard model structure on simplicial sets and the diagonal structure on bisimplicial sets. Corollary 8. The diagonal model structure on the category s2 Pre(C) is proper. Proof. All bisimplicial presheaves are cofibrant, so that pushouts of diagonal weak equivalences along cofibrations are diagonal weak equivalences [2, II.8.5].
6
The functor d∗ preserves cofibrations and weak equivalences, so that d preserves fibrations. The functor d also preserves pullbacks. Thus, right properness for the diagonal model structure bisimplicial presheaves follows from right properness for the injective structure on simplicial presheaves. Corollary 9. The diagonal model structure on the category s2 Set of bisimplicial sets is proper. The Moerdijk model structure is another well known example ([7], Section IV.3.3 of [2]) of a model structure on the category s2 Set of bisimplicial sets for which the weak equivalences are the diagonal weak equivalences. The Moerdijk structure is induced from the standard model structure on simplicial sets, in that a bisimplicial set map X → Y is a fibration (respectively weak equivalence) for this model structure if and only if the induced map d(X) → d(Y ) is a Kan fibration (respectively weak equivalence) of simplicial sets. The Moerdijk structure is also Quillen equivalent to the standard model structure on simplicial sets, via the diagonal functor and its left adjoint. Suppose that S is a set of cofibrations of bisimplicial presheaves which contains the set S0 of all maps d∗ A → d∗ B which are induced by α-bounded cofibrations A → B of simplicial presheaves. Suppose that S further satisfies the closure property that if the map C → D is in S, then so is the induced cofibration (D × ∂∆n ) ∪ (C × ∆n ) → D × ∆n , for all n ≥ 0. Let CS be the saturation of the set S in the class of all cofibrations (monomorphisms) of the bisimplicial set category. I say that CS is the class of S-cofibrations. Say that a bisimplicial presheaf map p : X → Y is an S-fibration if it has the right lifting property with respect to all S-cofibrations which are diagonal weak equivalences. The proof of the following result follows the outline established in [6]: Theorem 10. The category s2 Pre(C) of bisimplicial presheaves, together with the S-cofibrations, diagonal weak equivalences and S-fibrations satisfis the axioms for a proper closed simplicial model category. This model structure is cofibrantly generated. Proof. Every map f : X → Y has a factorization /Z X@ @@ @@ q f @@ Y j
where j is a member of CS and q has the right lifting property with respect to all members of CS . Then q∗ : d(Z) → d(Y ) is a trivial injective fibration of simplicial presheaves, so that q is a diagonal weak equivalence. The map q is an S-fibration. 7
The map f : X → Y also has a factorization /W XB BB BB p B f BB Y i
where i is a trivial cofibration and p is a fibration for the diagonal model structure of Theorem 4. The map p is an S-fibration. The cofibraton i has a factorization i = q · j as above, where j is an S-cofibration and q is an S-fibration and a diagonal equivalence. The map j is a diagonal equivalence, so that f has a factorization f = (p·q)·j such that pq˙ is an S-fibration and j is an S-cofibration and a diagonal equivalence. We have verified the model category axiom CM5. If p : X → Y is an Sfibration and a diagonal equivalence, then it is a retract of a map which has the right lifting property with respect to all S-cofibrations, giving CM4. The rest of the model category axioms are trivial. The simplicial model axiom SM7 is a consequence of the construction of the class CS and the instance of this axiom for the injective model structure on simplicial presheaves. The left properness of this structure is an easy consequence of left properness for the diagonal structure on s2 Pre(C), while right properness follows from right properness for the injective structure on s Pre(C). The cofibrant generation follows from what is now a familiar trick. Every α-bounded trivial cofibration β : A → B has a factorization jβ
/ Zβ A@ @@ @@ @ qβ β @@ B as in the first paragraph, where jβ is an S-cofibration and qβ has the right lifting property with respect to all S-cofibrations. Then both jβ and qβ are diagonal equivalences. One shows that if i : C → D is an α-bounded S-cofibration and there is a commutative diagram /X
C i
D
/Y
f
where f is a diagonal equivalence, then the diagram has a factorization C i
D
/A jβ
/ Zβ 8
/X f
/Y
for some β. Finally, if j : E → F is an S-cofibration and a diagonal equivalence, then j has a factorization i /V E@ @@ @@ p j @@ F where p has the right lifting property with respect to all jβ and i is in the saturation of the set of all maps jβ . But then j and p are diagonal equivalences, and the construction of the last paragraph shows that p has the right lifting property with respect to all members of CS , so that i is a retract of j. This means that the set of all maps jβ generates the class of trivial cofibrations in the model structure defined by the set of cofibrations S. Say that the model structure of Theorem 10 is the S-model structure on the category of bisimplicial presheaves. The S0 -model structure on bisimplicial sets (for whatever infinite cardinal α) is the Moerdijk structure, and the S0 -model structure for bisimplicial presheaves is a locally defined analogue of the Moerdijk structure. An obvious comparison with the various intermediate model structures for simplicial presheaves [6] says that the S0 -model structure for bisimplicial presheaves is a “projective” model structure, while the diagonal model structure of Theorem 4 is an “injective” model structure, and all S-model structures have classes of cofibrations lying between these two extremes.
2
Bisimplicial sets
˜ be the bisimplical set Suppose that K and L are simplicial sets, and let K ×L defined by ˜ p,q = Kp × Lq . (K ×L) ˜ is the external product of K and L. The bisimplicial set K ×L Examples: 1) The standard bisimplex ∆p,q has the form ˜ q. ∆p,q = ∆p ×∆ 2) Set q ˜ q ) ∪ (∆p ×∂∆ ˜ ˜ q = ∆p,q . ∂∆p,q = (∂∆p ×∆ ) ⊂ ∆p ×∆
Then the boundary ∂∆p,q of the bisimplex ∆p,q is generated as a subcomplex by the images of the maps (di , 1) : ∆p−1,q → ∆p,q and (1, dj ) : ∆p,q−1 → ∆p,q . The following statement about simplicial sets is well known. The argument for it which is given here may be a bit unfamiliar. Lemma 11. Suppose that x, y are non-degenerates simplices of X and s, t are ordinal number epimorphisms such that s∗ (x) = t∗ (y). Then x = y and s = t. 9
Proof. If dim(y) < dim(x) then x is degenerate: in effect, if d is a section of s then x = d∗ s∗ y = d∗ t∗ y is degenerate. It follows that x and y have the same simplicial dimension. If the ordinal number epis s and t have distinct sections, then there is a section d of s (say) which is not a section of t, in which case t · d is not an ordinal number epi, and so x = d∗ s∗ (x) = d∗ t∗ (y) is degenerate. It follows that s and t have the same sections, and therefore s = t. But then s∗ is an injective function, so that x = y. Suppose that X is a bisimplicial set and that x ∈ Xp,q . The number p + q is the total degree of x. Suppose that A is a subcomplex of a bisimplicial set X and that x ∈ Xp,q is a bisimplex of X − A of minimal total degree. Write x : ∆p,q → X for the classifying map of the bisimplex x. The bisimplices (di , 1)(x) and (1, dj )(x) have smaller total degree than x and are therefore in A, and it follows that there is a pullback diagram α /A ∂∆p,q i
∆p,q
x
/X
of bisimplicial set maps. Lemma 12. Suppose that A is a subcomplex of a bisimplicial set X and that x ∈ Xp,q is a bisimplex of X − A of minimal total degree. Form the pushout ∂∆p,q
α
i
∆p,q
/A
x
/B
Then the induced bisimplicial set map B → X is a monomorphism. Proof. If x = s(y) for some degeneracy s (vertical or horizontal), then y has smaller total degree, and so y ∈ A and x ∈ A. It follows that x is vertically and horizontally non-degenerate. There is a decomposition Br,s = Ar,s t {u × v : r × s → p × q, u, v epi}. in all bidegrees. If a ∈ Ar,s and u × v have the same image in X, then a = (u × v)∗ (x) is in A so that x ∈ A by applying a suitable section of u × v, which can’t happen. The restriction of Br,s → Xr,s to Ar,s is the monomorphism i : Ar,s → Xr,s .
10
Finally, if the epis u × v, u0 × v 0 : r × s → p × q have the same image in X, then (u × v)∗ (x) = (u0 × v 0 )∗ (x) in X. The bisimplex (1 × v)∗ (x) is horizontally non-degenerate. Otherwise, (1 × v)∗ (x) = (s × 1)∗ (y) for some y and non-trivial ordinal number epi s, and if d is a section of v then x = (1 × d)∗ (1 × v)∗ (x) = (1 × d)∗ (s × 1)∗ (y) = (s × 1)∗ (1 × d)∗ (y) so that x is horizontally degenerate. Similarly, (1 × v 0 )∗ (x) is horizontally nondegenerate, and so Lemma 11 and the relations (u × 1)∗ (1 × v)∗ (x) = (u0 × 1)∗ (1 × v 0 )∗ (x) together imply that u = u0 and (1 × v)∗ (x) = (1 × v 0 )∗ (x), so that v = v 0 Corollary 13. The set of inclusions ∂∆p,q ⊂ ∆p,q generates the class of cofibrations of s2 Set. The class A of anodyne extensions of s2 Set is the saturation of the set of bisimplicial set maps S, which consists of all morphisms s ˜ s ) ∪ (∆r ×∂∆ ˜ ˜ s = ∆r,s (Λrk ×∆ ) ⊂ ∆r ×∆
as well as all morphisms ˜ s ) ∪ (∆r ×Λ ˜ sj ) ⊂ ∆r ×∆ ˜ s = ∆r,s (∂∆r ×∆ The class A contains the set of all cofibrations ˜ ˜ ˜ (A×D) ∪ (B ×C) ⊂ B ×D induced by cofibrations A → B and C → D, where one of the two maps is a trivial cofibration of simplicial sets. The diagonal of such a map is the trivial cofibration (A × D) ∪ (B × C) ⊂ B × D. in simplicial sets. It follows that every anodyne extension is a diagonal weak equivalence. Say that a map p : X → Y of bisimplicial sets is a Kan fibration if it has the right lifting property with respect to all anodyne extensions. Every fibration for the diagonal model structure of Theorem 4 is a Kan fibration. The purpose of this section is to prove the converse assertion, so that the diagonal fibrations of bisimplicial sets are precisely the Kan fibrations. This statement appears as Theorem 24 below. The method of proof involves showing that a map p : X → ∆p,q is a diagonal fibration if and only if it is a Kan fibration. For this, we need a model structure on the slice category s2 Set/∆p,q which is defined by formally inverting the anodyne extensions. The fibrant objects in the resulting “anodyne model 11
structure” are precisely the Kan fibrations X → ∆p,q . Then the idea is to show that a map /Y X9 99 99 99 ∆p,q f
is a weak equivalence of the anodyne model structure if and only if the bisimplicial set map f : X → Y is a diagonal weak equivalence (Lemma 23). In this case, the anodyne model structure on the slice category s2 Set/∆p,q coincides with the model structure that category canonically inherits from the diagonal structure on bisimplicial sets, and then the two theories have the same fibrant objects. We begin by describing the anodyne model structures. Suppose that X is a bisimplicial set and that K is a simplicial set. Define a bisimplicial set X ⊗ K by the assignment X ⊗ K = X × K, where we have identified K wiht a bisimplicial set which is constant in the horizontal direction: Kp,q = Kq . It follows that there is a natural isomorphism d(X ⊗ K) ∼ = d(X) × K. The construction X ⊗ K preserves diagonal weak equivalences in bisimplicial sets X and weak equivalences in simplicial sets K. Lemma 14. Suppose that i : A → B is a cofibration of bisimplicial sets and that j : K → L is a cofibration of simplicial sets. Then the induced map (i, j)∗ : (B ⊗ K) ∪ (A ⊗ L) → B ⊗ L is a cofibration which is anodyne if either i or j is anodyne. Proof. The map (∆r,s ⊗ K) ∪ (∂∆r,s ⊗ L) → ∆r,s ⊗ L can be identified with the map s ˜ s × L)) ∪ (∆r ×((∂∆ ˜ ˜ s × L), (∂∆r ×(∆ × L) ∪ (∆s × K))) → ∆r ×(∆
which is a cofibration. It follows from Corollary 13 that the map (i, j)∗ is a cofibration in general. The simplicial set map (∂∆s × L) ∪ (∆s × K) → ∆s × L is anodyne if j is anodyne, so that (i, j)∗ is anodyne in general if j is anodyne. The remaining statements have similar proofs. 12
Suppose that Z is a fixed bisimplicial set, and write s2 Set/Z for the category whose objects are the bisimplicial set maps φ : X → Z and whose morphisms f : φ → ψ are the commutative diagrams f /Y X4 44
4
φ 44
ψ
Z
of bisimplicial set maps. The class AZ of anodyne extensions over Z is the saturation of the set of maps SZ in s2 Set/Z, where SZ consists of the morphisms s ˜ s ) ∪ (∆r ×∂∆ ˜ ˜ s = ∆r,s → Z (Λrk ×∆ ) ⊂ ∆r ×∆
as well as the morphisms ˜ s ) ∪ (∆r ×Λ ˜ sj ) ⊂ ∆r ×∆ ˜ s = ∆r,s → Z (∂∆r ×∆ The class AZ contains the set of all cofibrations ˜ ˜ ˜ →Z (A×D) ∪ (B ×C) ⊂ B ×D induced by cofibrations A → B and C → D, where one of the two maps is a trivial cofibration of simplicial sets. Let I = ∆1 be the standard interval in simplicial sets. One uses the interval theory which assigns to an object φ : X → Z the objects defined by the compositions pr
φ
X ⊗ I ×n −→ X − →Z along with the set SZ of generators for the class AZ of anodyne extensions to define a closed simplicial model structure on the category s2 Set/Z (Theorem 4.17 of [5]). The function complex hom(φ, ψ) has a standard construction: an n-simplex is a commutative diagram /Y
X × ∆n pr
X
ψ
φ
/Z
Quillen’s simplicial model axiom SM7 for this theory is a consequence of Lemma 14, but see also Lemma 15 below. A weak equivalence for this model structure, which will be called an anodyne weak equivalence over Z is a map f : φ → ψ such that the induced map π(ψ, γ) → π(φ, γ) in fibre homotopy classes of maps (defined by the interval 13
theory) is a bijection for all Kan fibrant objects γ : W → Z. A cofibration is a monomorphism of s2 Set/Z, and a map p : φ → ψ is a fibration, here called an anodyne fibration over Z, if it has the right lifting property with respect to all cofibrations which are anodyne weak equivalences over Z. This model structure is left proper since every object is cofibrant [2, II.8.5], and is cofibrantly generated by construction. It follows from Lemma 14 that a map f : X → Y → Z of s2 Set/Z is injective for this theory if and only if it has the right lifting property with respect to all members of AZ , and an object φ : X → Z is injective if the map φ → 1Z is injective. Equivalently the object φ is injective if and only if the underlying map φ : X → Z is a Kan fibration of bisimplicial sets. The classes of anodyne fibrant objects and injective objects over Z coincide, by standard nonsense (see Corollary 4.14 of [5]). The following simple result fixes ideas: f
i
Lemma 15. A map A → − B− → Z is an anodyne extension over Z if and only if the map i : A → B is an anodyne extension of bisimplicial sets. Proof. Suppose that i : A → B is an anodyne extension, and construct a factorization j /X A@ @@ @@ p i @@ B such that j is a filtered colimit of pushouts of morphisms of S, hence of morphisms of SZ , and p is a Kan fibration. Then the dotted arrow lifting exists in the diagram j /X A > p
i
B
1
/B @@ @@f @@ @ Z
i
f
j
fp
so that the map A → − B − → Z is a retract of the map A − → C −→ Z, and the latter map is an anodyne extension over Z. The converse is clear. We shall need some properties of homotopy colimits in bisimplicial sets and their relation to anodyne weak equivalences. Suppose first of all that X is a simplicial set. Let γn : ∆n × Xn → X
14
be the map which classifies the n-simplices of X. Then we have skn X = ∪i≤n im(γi ). Write s[r] Xn = ∪i≤r si (Xn ) ⊂ Xn+1 , and observe that s[n] Xn is the degenerate part of Xn+1 . Then there are natural pushout diagrams sr+1 / s[r] Xn s[r] Xn−1 (1) Xn
sr+1
/ s[r+1] Xn
and (∆n+1 × s[n] Xn ) ∪ (∂∆n+1 × Xn+1 )
/ skn X
∆n+1 × Xn+1
/ skn+1 X
(2)
The diagrams (1) and (2) are natural in simplicial sets X. Thus, if m 7→ Ym is a simplicial object in simplicial sets (aka. a bisimplicial set, where m is the horizontal variable), the simplicial set maps ∆n × Yn,m → Y∗,m induce bisimplicial set maps ˜ n → Y. γn : ∆n ×Y The bisimplicial set Y has a natural filtration skn Y by (horizontal) skeleta, and there is a natural pushout diagram s[r] Yn−1 Yn
sr+1
sr+1
/ s[r] Yn
(3)
/ s[r+1] Yn
in simplicial sets, and a pushout diagram ˜ n+1 ) ˜ [n] Yn ) ∪ (∂∆n+1 ×Y (∆n+1 ×s
/ skn Y
˜ n+1 ∆n+1 ×Y
/ skn+1 Y
in bisimplicial sets in which the vertical maps are cofibrations. 15
(4)
Lemma 16. Suppose that f : K → K 0 and g : L → L0 are weak equivalences of ˜ 0 → Z. simplicial sets, and suppose given a fixed map of bisimplicial sets L×L Then the induced map ˜ : K ×L ˜ → K 0 ×L ˜ 0→Z f ×g is an anodyne weak equivalence of bisimplicial sets over Z. Proof. We show that the map ˜ → K 0 ×L ˜ →Z f × 1 : K ×L is an anodyne weak equivalence. This is true if f is a trivial cofibration by Lemma 14, and is therefore true in general since all simplicial sets are cofibrant. Lemma 17. Suppose that f : X → Y → Z is a map of bisimplicial sets over Z such that the map f : Xn → Yn is a weak equivalence of simplicial sets in each horizontal degree n. Then the map f is an anodyne weak equivalence over Z. Proof. Use the natural diagrams (3) and (4) to approximate the map f by maps of skeleta over Z. One shows inductively that each map s[r] X → s[r] Y → Z and each skn X → skn Y → Z is an anodyne equivalence over Z by using Lemma 16. The induced diagram sk0 X
/ sk1 X
/ sk2 X
/ ...
sk0 Y
/ sk1 Y
/ sk2 Y
/ ...
defines a weak equivalence of projective cofibrant diagrams for the anodyne structure on s2 Set/Z, and therefore induces an anodyne weak equivalence X → Y → Z in the filtered colimit. If m 7→ Ym is a simplicial object in bisimplicial sets (aka. a trisimplicial set Ym,p,q , where m is the first variable), the simplicial set maps ∆n × Yn,p,q → Y∗,p,q induce trisimplicial set maps ˜ n → Y, γn : ∆n ×Y where the indicated external product has the obvious definition. The trisimplicial set Y has a natural filtration skn Y by (first variable) skeleta, and there is a pushout diagram sr+1 / s[r] Yn s[r] Yn−1 (5) Yn
sr+1
16
/ s[r+1] Yn
in bisimplicial sets, and a pushout diagram ˜ [n] Yn ) ∪ (∂∆n+1 ×Y ˜ n+1 ) (∆n+1 ×s
/ skn Y
˜ n+1 ∆n+1 ×Y
/ skn+1 Y
(6)
in trisimplicial sets. The diagrams (5) and (6) are natural in trisimplicial sets Y. Let s3 Set denote the category of trisimplicial sets. Define the partial diagonal functor d1,2 : s3 Set → s2 Set by setting d1,2 (Y )p,q = Yp,p,q for a trisimplicial set Y . This functor preserves all limits and colimits. If K is a simplicial set and X → Z is a bisimplicial set over Z, then there is a natural isomorphism ∼ =
˜ d1,2 (K ×X) CC CC CC CC !
Z
/ K ×X
of bisimplicial sets over Z. The bifunctor K × X takes weak equivalences in simplicial sets K and anodyne equivalences in bisimplicial sets X → Z to anodyne weak equivalences over Z, by the simplicial structure of the anodyne model structure over Z and the fact that all objects of s2 Set/Z are cofibrant. Lemma 18. Suppose that Z is a bisimplicial set and that the map f in the diagram f /Y X4 44
44
4
Z
of trisimplicial set maps is a sectionwise anodyne equivalence over Z in the sense that all bisimplicial set maps Xn → Yn → Z are anodyne equivalences over Z, where Z is identified with a trisimplicial set which is constant in the first variable. Then the induced diagram f∗ / d1,2 Y d1,2 X :: :: :: : Z
is an anodyne equivalence over Z. 17
Proof. Write d1,2 (Y )(p) = d1,2 (skp Y ), where the skeletal filtration skp Y of Y in the first variable is as defined above. Apply the partial diagonal d1,2 to instances of the diagrams (5) and (6) for X and Y , and recall that these diagrams are functorial. It follows by induction (and left properness) that all maps d1,2 (X)(p) AA AA AA A
f∗
Z
/ d1,2 (Y )(p) ~ ~~ ~ ~ ~~ ~
are anodyne weak equivalences of bisimplicial sets over Z. The proof is finished (as was the argument for Lemma 17) with a “telescope” argument. In effect, the diagram d1,2 (X)(0)
/ d1,2 (X)(1)
/ d1,2 (X)(2)
/ ...
d1,2 (Y )(0)
/ d1,2 (Y )(1)
/ d1,2 (Y )(2)
/ ...
defines a weak equivalence of projective cofibrant diagrams for the anodyne model structure on bisimplicial sets over Z. The map f therefore induces an anodyne weak equivalence d1,2 (X) → d1,2 (Y ) → Z in the filtered colimit. Suppose that X : I → s2 Set/Z is a diagram of bisimplicial sets over Z which is indexed on a small category I. Then there is a trisimplicial set holim −−−→ I X over Z which is specified in the first variable by the bisimplicial set maps G holim X = X(i0 ) → Z. I n −−−→ i0 →···→in
Write also holim −−−→ I X for the bisimplicial set d1,2 (holim −−−→ I X) Then we have the following: Corollary 19. Suppose that f : X → Y is a natural transformation of Idiagrams of bisimplicial sets over Z such that all maps X(i) → Y (i) → Z are anodyne weak equivalences of bisimplicial sets over Z. Then the induced map f∗
holim −−−→ I X −→ holim −−−→ I Y → Z is an anodyne weak equivalence of bisimplicial sets over Z. Proof. All maps G i0 →···→in
G
X(i0 ) →
Y (i0 ) → Z
i0 →···→in
are anodyne weak equivalences over Z. Apply Lemma 18. 18
Suppose that p : U → X is a projective cofibrant model of the diagram X over Z. In horizontal degrees, p consists of maps p : Un → Xn of I-diagrams of simplicial sets over Zn , for which each object Un is projective cofibrant. The following result is then a consequence of Corollary 17 and the corresponding fact for simplicial sets (Section 5 of [5]): Lemma 20. Suppose that the diagram U : I → s2 Set/Z is projective cofibrant. Then the canonical map U →Z holim −−−→ I U → lim −→ I
is an anodyne weak equivalence of bisimplicial sets over Z. The following result says that we have been using the right homotopy colimit construction: Corollary 21. Suppose that X : I → s2 Set/Z is a small diagram of bisimplicial sets over Z, and let p : U → X be a projective cofibrant model. Then the indicated maps ' / lim U holim −→I −−−→ I U p∗ '
holim −−−→ I X
/Z
are anodyne weak equivalences of bisimplicial sets over Z. Here’s the “regularity” statement [1], [5] for bisimplicial sets over a fixed base: Lemma 22. Suppose that X → Z is a bisimplicial set over Z. Then the canonical map p,q p,q φ:− holim −−→ ∆ →X ∆ → X → Z is an anodyne weak equivalence of bisimplicial sets over Z. Proof. The tricomplex map underlying φ consists of weak equivalences of simplicial sets p,q p,q φ:− holim −−→ ∆ →X ∆r,s → Xr,s . It follows that the induced map p,q p,q φ : d1,2 (holim −−−→ ∆ →X ∆r,s ) → X
of bisimplicial sets consists of weak equivalences of simplicial sets in each vertical degree. The map φ is therefore an anodyne weak equivalence of bisimplicial sets over Z, by Lemma 17.
19
Lemma 23. Suppose that the bisimplicial set map f in the diagram f /Y X4 44
44
4
Z
is a diagonal weak equivalence. Then the diagram is an anodyne weak equivalence of bisimplicial sets over Z. Proof. There are natural anodyne weak equivalences φ
p,q p,q p,q B(∆2 /X) × Z = − holim → X, −−→ ∆ →X ∗ ← holim −−−→ ∆ →X ∆ −
of bisimplicial sets over Z by Corollary 19 and Lemma 20. The simplicial set map B(∆2 /X) → B(∆2 /Y ) is a weak equivalence of simplicial sets since f : X → Y is a diagonal weak equivalence, so that the induced map pr
B(∆2 /X) × Z → B(∆2 /Y ) × Z −→ Z is an anodyne weak equivalence of bisimplicial sets over Z. Theorem 24. A map p : X → Y of bisimplicial sets is a fibration if and only if it is a Kan fibration. Proof. We show that every Kan fibration which is a diagonal weak equivalence has the right lifting property with respect to all cofibrations. Suppose that this is so, and let i : A → B be a cofibration which is a diagonal weak equivalence. Find a factorization /Z A@ @@ @@ p i @@ B j
such that j is anodyne and p is a Kan fibration. Then, subject to the claim of the first paragraph, the map p is a diagonal weak equivalence and the lifting exists in the diagram j /Z A > p
i
B
1
/B
Then i is a retract of j, and is therefore an anodyne extension. Thus, the classes of diagonal trivial cofibrations and anodyne extensions coincide, so the classes of fibrations (of Theorem 4) and Kan fibrations coincide. 20
We use the method of proof of Theorem 8.6 of [5] to finish the argument. Suppose that p : X → Y is a Kan fibration and a diagonal weak equivalence, and consider the pullbacks /X
∆p,q ×Y X p∗
p
/Y
∆p,q
Then all maps p∗ are fibrations by Lemma 23, since the induced diagonal and anodyne model structures for s2 Set/∆p,q coincide. The maps α∗ in all pullback diagrams α∗ / ∆p,q ×Y X ∆r,s ×Y X p∗
p∗
∆r,s
/ ∆p,q
α
are diagonal weak equivalences by properness of the diagonal structure, since the maps p∗ are fibrations. It follows from Quillen’s Theorem B [2, IV.5.6] that all induced diagrams of simplicial set maps / B(i∆2 X)
B(i∆2 (∆p,q ×Y X)) p∗
/ B(i∆2 Y )
B(i∆2 ∆p,q )
are homotopy cartesian. Here, i∆2 X is the category of bisimplices ∆k,l → X of the bisimplicial set X. The indicated map p∗ is therefore a weak equivalence, so Corollary 19 and Lemma 22 together imply that the map p∗ : ∆p,q ×Y X → ∆p,q is a diagonal weak equivalence. Thus, suppose given a commutative diagram ∂∆p,q
/X
∆p,q
/Y
where p is a Kan fibration and a diagonal weak equivalence. Then the indicated lifting exists in the diagram ∂∆p,q
/ ∆p,q ×Y X 8
/X
p∗
∆p,q
/ ∆p,q
since p∗ is a fibration and a diagonal equivalence. 21
p
/Y
References [1] Denis-Charles Cisinski. Les pr´efaisceaux comme mod`eles des types d’homotopie, volume 308 of Ast´erisque. Soci´et´e Math´ematique de France, Paris, 2006. [2] P. G. Goerss and J. F. Jardine. Simplicial Homotopy Theory, volume 174 of Progress in Mathematics. Birkh¨auser Verlag, Basel, 1999. [3] J. F. Jardine. Simplicial presheaves. J. Pure Appl. Algebra, 47(1):35–87, 1987. [4] J. F. Jardine. Boolean localization, in practice. Doc. Math., 1:No. 13, 245– 275 (electronic), 1996. [5] J. F. Jardine. Categorical homotopy theory. Homology, Homotopy Appl., 8(1):71–144 (electronic), 2006. [6] J. F. Jardine. Intermediate model structures for simplicial presheaves. Canad. Math. Bull., 49(3):407–413, 2006. [7] Ieke Moerdijk. Bisimplicial sets and the group-completion theorem. In Algebraic K-theory: connections with geometry and topology (Lake Louise, AB, 1987), volume 279 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 225–240. Kluwer Acad. Publ., Dordrecht, 1989.
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