nLab
canonical model structure on Cat

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

2-Category theory

Contents

The canonical model structure on CatCat

The canonical model structure on Cat is a model structure which encapsulates part of category theory as a version of homotopy theory. It is a special case of the general notion of canonical model structure on categorical structures, and is also called the trivial model structure or the categorical model structure. Its weak equivalences are the equivalences of categories and its homotopy category is Ho(Cat), the category obtained from the 1-category CatCat by identifying naturally isomorphic functors. See the Catlab for the theory of this structure.

Assuming the axiom of choice, the canonical model is the unique model structure on CatCat such that the weak equivalences are categorical equivalences (thus justifying the word ‘canonical’).

On this page we give a concise construction of the canonical model structure, as well as two variants that make sense in the absence of the full axiom of choice.

The classical version

For purposes of this page, Cat will denote the 1-category of small categories and functors, and our categories are all strict categories as in ordinary set-theoretic foundations. We write C 0C_0 for the set of objects of a small category CC. Define a functor to be:

We claim that this defines a model structure. It is easy to verify that the weak equivalences satisfy the 2-out-of-3 property; thus it remains to show that (acyclic cofibrations, fibrations) and (cofibrations, acyclic fibrations) are weak factorization systems.

Suppose given a square

A f C i p B g D\array{A & \overset{f}{\to} & C\\ ^i\downarrow && \downarrow^p\\ B& \underset{g}{\to} & D}

in which ii is a cofibration and pp a fibration.

Suppose first that pp is acyclic. It is easy to see that the acyclic fibrations are precisely the equivalences of categories that are surjective on objects. Thus, since (mono, epi) is a weak factorization system on Set, we can define h 0:B 0C 0h_0\colon B_0\to C_0 filling the square, and then full-faithfulness of pp gives a unique definition of hh on arrows.

Now suppose that ii is acyclic, so that i 0:A 0B 0i_0 \colon A_0 \to B_0 is injective. Since ii is essentially surjective, we can choose, for each bB 0A 0b\in B_0 \setminus A_0, an isomorphism ϕ b:i(a b)b\phi_b\colon i(a_b) \cong b. We then have g(ϕ b):p(f(a b))=g(i(a b))g(b)g(\phi_b)\colon p(f(a_b)) = g(i(a_b)) \cong g(b), so since pp is an isofibration, we can also choose, for each bB 0b\in B_0, an isomorphism ψ b:f(a b)c b\psi_b\colon f(a_b) \cong c_b such that p(ψ b)=g(ϕ b)p(\psi_b) = g(\phi_b). Define h 0:B 0C 0h_0\colon B_0\to C_0 to be f 0f_0 on the image of i 0i_0 and to take bB 0A 0b\in B_0 \setminus A_0 to c bc_b. We can define hh on arrows by composing with the isomorphisms ψ\psi to make it a lifting.

It remains to prove the factorization axioms. Suppose given a functor f:ABf\colon A\to B. First, define C 0=A 0+B 0C_0 = A_0 + B_0, and make CC into a category in the unique way such that the map CBC\to B induced by ff and 1 B1_B is fully faithful. Since it is surjective on objects, it is an acyclic fibration, and clearly the induced map ACA\to C is injective on objects, i.e. a cofibration.

Next, define DD to be the category of triples (a,b,ϕ)(a,b,\phi) where ϕ:f(a)b\phi\colon f(a)\cong b is an isomorphism in BB. In other words, it is the strict iso-comma category (f/ 1 b)(f/_\cong 1_b). The projection DBD\to B is easily shown to be an isofibration, while the functor ADA\to D defined by a(a,f(a),1 f(a))a \mapsto (a, f(a), 1_{f(a)}) is an injective equivalence.

This completes the proof. Note that the two factorizations constructed above are in fact functorial. This model structure is easily seen to be cofibrantly generated, although the above factorizations are not those constructed from the small object argument (though they are closely related to the algebraic weak factorization systems produced from Richard Garner’s modified small object argument).

Relation with bisimplicial sets

Recall that Rezk’s classifying diagram for a (small) category CC is the bisimplicial set N(C)N (C) defined by N(C) n,m=Fun([n]×I[m],C)N (C)_{n, m} = Fun ([n] \times \mathbf{I}[m], C), where [n][n] is the standard nn-simplex considered as a category and I\mathbf{I} is the groupoid completion functor (i.e. DD[D 1]D \mapsto D [D^{-1}]). There is then an adjunction

τ 1N:CatssSet\tau_1 \dashv N : Cat \to ssSet

and it can be shown (see the following subsection) that the canonical model structure on Cat is the model structure obtained by transferring the projective model structure on ssSetssSet.

Uniqueness of model structure

A remarkable and perhaps surprising result (and surprisingly not better known!) is that there is just one model structure on CatCat1 whose equivalences are the usual categorical equivalences. This result justifies the term “canonical”.

The proof we present below is adapted (with minor changes) from a proof given by Chris Schommer-Pries. See also this MathOverflow thread, particularly the answer given by Steve Lack (with a pertinent comment by Denis-Charles Cisinski).

Proof

Let M\mathbf{M} denote any model structure on CatCat whose weak equivalences are categorical equivalences. We will prove that M\mathbf{M}-fibrations are exactly canonical fibrations and that M\mathbf{M}-cofibrations are exactly canonical cofibrations.

Lemma

The terminal object 11 is M\mathbf{M}-cofibrant, i.e., the inclusion 010 \to 1 in CatCat is an M\mathbf{M}-cofibration.

Proof

Let CC be any noninitial category; by a standard result of model category theory, there is an M\mathbf{M}-cofibrant replacement C˜C\tilde{C} \to C, a weak equivalence such that 0C˜0 \to \tilde{C} is a cofibration. This C˜\tilde{C} is noninitial and therefore has 11 as a retract; thus 11 is M\mathbf{M}-cofibrant since cofibrant objects are closed under retracts.

Proposition

Each acyclic M\mathbf{M}-fibration is a canonical acyclic fibration.

Each acyclic M\mathbf{M}-fibration f:EXf: E \to X has the right lifting property with respect to M\mathbf{M}-cofibrations. The right lifting property with respect to the M\mathbf{M}-cofibration 010 \to 1 is exactly the condition of being surjective on objects. Thus acyclic M\mathbf{M}-fibrations are necessarily categorical equivalences that are surjective on objects, i.e., are necessarily canonical acyclic fibrations.

Before giving the next result, we recall that the lifting relation on morphisms gives a Galois connection. Specifically, suppose c:ABc: A \to B and f:cdf: c \to d are functors, and define cfc \perp f if for every morphism from cc to ff in the arrow category Cat 2Cat^\mathbf{2}, i.e., for every commutative diagram

A C c f B D\array{ A & \to & C \\ _\mathllap{c} \downarrow & & \downarrow_\mathrlap{f} \\ B & \to & D }

of functors, there is a lifting BCB \to C filling in to make two commutative triangles. As any relation does, this lifting relation \perp gives a Galois connection on subclasses of Mor(Cat)Mor(Cat). General facts about Galois connections may then be applied.

Corollary

Every canonical cofibration is an M\mathbf{M}-cofibration. Every M\mathbf{M}-fibration is a canonical fibration.

Proof

By the Galois connection induced by the lifting relation, Proposition 1 implies that canonical cofibrations form a subset of M\mathbf{M}-cofibrations, and therefore that canonical acyclic cofibrations are a subset of acyclic M\mathbf{M}-cofibrations. Again by the Galois connection, this in turn implies that M\mathbf{M}-fibrations form a subset of canonical fibrations.

At this point, we would like to show conversely that every M\mathbf{M}-cofibration is a canonical cofibration (i.e., is injective on objects); another appeal to Galois connections would then allow us to deduce that every canonical fibration is an M\mathbf{M}-fibration, and we would be done. Let us suppose otherwise, that there exists an M\mathbf{M}-cofibration that is not injective on objects, and derive a contradiction.

For a set SS, let K(S)K(S) be the category whose objects are the elements of SS, with exactly one morphism xyx \to y for any x,ySx, y \in S. This gives the codiscrete (or chaotic) functor K:SetCatK: Set \to Cat, which is right adjoint to the forgetful functor U:CatSetU: Cat \to Set that takes a category to its underlying set of objects. For each inhabited set SS, we have that K(S)K(S) is equivalent to 11, and conversely any category equivalent to 11 is isomorphic to some K(S)K(S).

Proposition

If there is any M\mathbf{M}-cofibration f:ABf: A \to B that is not injective on objects, then the map K(2)1K(2) \to 1 (2={0,1}2 = \{0, 1\}) is an (acyclic) M\mathbf{M}-cofibration.

Proof

First we observe that for any category EE, the unit map η E:EKU(E)\eta_E: E \to K U(E) for the adjunction UKU \dashv K is an M\mathbf{M}-cofibration. For, the map η E\eta_E is an isomorphism on objects and therefore a canonical cofibration; it is an M\mathbf{M}-cofibration by Corollary 1.

By hypothesis, ff maps two objects a,aa, a' of AA to the same object bb of BB, so there is a commutative diagram

2 (a,a) UA f 1 b UB.\array{ 2 & \stackrel{(a, a')}{\hookrightarrow} & U A \\ \downarrow & & \downarrow_\mathrlap{f} \\ 1 & \underset{b}{\to} & U B. }

Let r:UA2r: U A \to 2 be a retraction of the injection 2UA2 \to U A. By the adjunction UKU \dashv K, the map rr corresponds to a map s:AK(2)s: A \to K(2). We form a pushout square

A s K(2) f g B h E η E KU(E)\array{ A & \stackrel{s}{\to} & K(2) & & \\ _\mathllap{f} \downarrow & & \downarrow_\mathrlap{g} & & \\ B & \underset{h}{\to} & E & \underset{\eta_E}{\to} & K U(E) }

where gg is an M\mathbf{M}-cofibration (being the pushout of a cofibration ff). Thus we have a composite cofibration tη Eg:K(2)KU(E)t \coloneqq \eta_E \circ g: K(2) \to K U(E). It may be verified that K(2)1K(2) \to 1 is a retract of tt, i.e., there is a commutative square

K(2) id K(2) t KU(E) j 1\array{ K(2) & \stackrel{id}{\to} & K(2) \\ _\mathllap{t} \downarrow & & \downarrow \\ K U(E) & \underset{j}{\leftarrow} & 1 }

where j=η Ehbj = \eta_E \circ h \circ b; this diagram commutes on objects by construction, and it commutes on morphisms because all diagrams commute in KU(E)K U(E). Thus K(2)1K(2) \to 1, being a retract of an M\mathbf{M}-cofibration, is also an M\mathbf{M}-cofibration.

The conclusion of Proposition 2 now leads to a contradiction:

Proposition

If K(2)1K(2) \to 1 is an acyclic M\mathbf{M}-cofibration, then for any category CC, every automorphism of CC is an identity (which is absurd!).

Proof

The object CC of CatCat has an M\mathbf{M}-fibrant replacement C^\hat{C} equivalent to CC. For any isomorphism ϕ\phi of C^\hat{C}, let e:K(2)C^e: K(2) \to \hat{C} be the unique functor taking 010 \to 1 in K(2)K(2) to ϕ\phi. Then we have a commutative diagram

K(2) e C^ acyc.cof. fib. 1 1\array{ K(2) & \stackrel{e}{\to} & \hat{C} \\ _\mathllap{acyc.\; cof.} \downarrow & & \downarrow _\mathrlap{fib.} \\ 1 & \to & 1 }

and the existence of a lift 1C^1 \to \hat{C} filling in this diagram means that ϕ\phi is an identity. In particular, every automorphism of C^\hat{C} is an identity; since CC^C \simeq \hat{C}, the same is true of CC.

Without choice

In the absence of the axiom of choice, one must distinguish between strong equivalences of categories, which come with an inverse up to isomorphism, and weak equivalences of categories, which are merely fully faithful and essentially surjective on objects. Since weak equivalences of categories still “preserve all categorical information,” we might hope to find a model structure on CatCat whose weak equivalences are the weak equivalences of categories. The notion of anafunctor also suggests such an approach, since an anafunctor (the “right” replacement for a functor in the absence of choice) is a particular sort of generalized morphism?: a span AFBA\leftarrow F \to B of functors in which FAF\to A is a surjective equivalence.

If there is to be such a model structure, however, then since generalized morphisms between fibrant-and-cofibrant objects are all represented by ordinary ones, there must exist “cofibrant categories” and “fibrant categories” such that every anafunctor between fibrant-and-cofibrant categories is equivalent to an honest functor, and every category can be replaced by a fibrant and cofibrant one. It seems unlikely that this would be true without any choice-like axioms, but notably weaker axioms than full AC do suffice.

The projective model structure

For the existence of the model structure in this case, we assume COSHEP, aka the “presentation axiom,” namely that the category Set has enough projective objects. Define a functor f:ABf\colon A\to B to be:

As before, the acyclic fibrations are precisely the weak equivalences that are literally surjective on objects. Now recall that assuming COSHEP, (monics with projective complement, epics) is a weak factorization system on Set. This supplies the lifting of cofibrations against acyclic fibrations. Likewise, the factorization of f:ABf\colon A\to B into a cofibration followed by an acyclic fibration is given by first factoring f 0f_0 as A 0A 0+B 0B 0A_0 \to A_0 + B_0' \to B_0, where B 0B 0B_0'\to B_0 is a projective cover.

The other factorization works exactly as before, while for lifting acyclic cofibrations against fibrations, we notice that in the original proof, we only needed to apply choice for sets indexed by B 0i(A 0)B_0\setminus i(A_0), which we have assumed to be projective when i:ABi\colon A\to B is a cofibration.

Weak equivalences of categories are easily seen to satisfy the 2-out-of-3 property, so we have a model category. Note that all categories are fibrant in this model structure, while the cofibrant categories are those whose set of objects is projective.

The existence of this model structure implies, in particular, that under COSHEP the category Ana(C,D)Ana(C,D) is essentially small, being in fact equivalent to the category Fun(C,D)Fun(C',D) of ordinary functors where CC' is a cofibrant replacement for CC.

The injective model structure

Is there a dual model structure in which all categories are cofibrant? This seemingly has to do with stack completion: the fibrant objects would be stacks for the regular coverage of SetSet. (Without AC, not all small categories are stacks.) Is Makkai’s axiom of small cardinality selection (which he uses, instead of COSHEP, to prove that Ana(C,D)Ana(C,D) is essentially small) sufficient for the existence of an “injective” model structure on Cat?


  1. Along similar lines, one can prove (assuming AC) that there are nine – count ‘em, nine – Quillen model structures on SetSet.

Revised on July 2, 2014 12:03:46 by Todd Trimble (98.208.182.196)