on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
on strict ∞-categories?
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 $Cat$ 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 $Cat$ 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.
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 $C_0$ for the set of objects of a small category $C$. Define a functor to be:
a weak equivalence if it is an equivalence of categories, or equivalently if it is fully faithful and essentially surjective.
a cofibration if it is injective on objects, i.e. an isocofibration.
a fibration if it is an isofibration.
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
in which $i$ is a cofibration and $p$ a fibration.
Suppose first that $p$ 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 $h_0\colon B_0\to C_0$ filling the square, and then full-faithfulness of $p$ gives a unique definition of $h$ on arrows.
Now suppose that $i$ is acyclic, so that $i_0 \colon A_0 \to B_0$ is injective. Since $i$ is essentially surjective, we can choose, for each $b\in B_0 \setminus A_0$, an isomorphism $\phi_b\colon i(a_b) \cong b$. We then have $g(\phi_b)\colon p(f(a_b)) = g(i(a_b)) \cong g(b)$, so since $p$ is an isofibration, we can also choose, for each $b\in B_0$, an isomorphism $\psi_b\colon f(a_b) \cong c_b$ such that $p(\psi_b) = g(\phi_b)$. Define $h_0\colon B_0\to C_0$ to be $f_0$ on the image of $i_0$ and to take $b\in B_0 \setminus A_0$ to $c_b$. We can define $h$ on arrows by composing with the isomorphisms $\psi$ to make it a lifting.
It remains to prove the factorization axioms. Suppose given a functor $f\colon A\to B$. First, define $C_0 = A_0 + B_0$, and make $C$ into a category in the unique way such that the map $C\to B$ induced by $f$ and $1_B$ is fully faithful. Since it is surjective on objects, it is an acyclic fibration, and clearly the induced map $A\to C$ is injective on objects, i.e. a cofibration.
Next, define $D$ to be the category of triples $(a,b,\phi)$ where $\phi\colon f(a)\cong b$ is an isomorphism in $B$. In other words, it is the strict iso-comma category $(f/_\cong 1_b)$. The projection $D\to B$ is easily shown to be an isofibration, while the functor $A\to D$ defined by $a \mapsto (a, f(a), 1_{f(a)})$ 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).
Recall that Rezk’s classifying diagram for a (small) category $C$ is the bisimplicial set $N (C)$ defined by $N (C)_{n, m} = Fun ([n] \times \mathbf{I}[m], C)$, where $[n]$ is the standard $n$-simplex considered as a category and $\mathbf{I}$ is the groupoid completion functor (i.e. $D \mapsto D [D^{-1}]$). There is then an adjunction
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 $ssSet$.
A remarkable and perhaps surprising result (and surprisingly not better known!) is that there is just one model structure on $Cat$^{1} 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).
Let $\mathbf{M}$ denote any model structure on $Cat$ whose weak equivalences are categorical equivalences. We will prove that $\mathbf{M}$-fibrations are exactly canonical fibrations and that $\mathbf{M}$-cofibrations are exactly canonical cofibrations.
The terminal object $1$ is $\mathbf{M}$-cofibrant, i.e., the inclusion $0 \to 1$ in $Cat$ is an $\mathbf{M}$-cofibration.
Let $C$ be any noninitial category; by a standard result of model category theory, there is an $\mathbf{M}$-cofibrant replacement $\tilde{C} \to C$, a weak equivalence such that $0 \to \tilde{C}$ is a cofibration. This $\tilde{C}$ is noninitial and therefore has $1$ as a retract; thus $1$ is $\mathbf{M}$-cofibrant since cofibrant objects are closed under retracts.
Each acyclic $\mathbf{M}$-fibration is a canonical acyclic fibration.
Each acyclic $\mathbf{M}$-fibration $f: E \to X$ has the right lifting property with respect to $\mathbf{M}$-cofibrations. The right lifting property with respect to the $\mathbf{M}$-cofibration $0 \to 1$ is exactly the condition of being surjective on objects. Thus acyclic $\mathbf{M}$-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 $c: A \to B$ and $f: c \to d$ are functors, and define $c \perp f$ if for every morphism from $c$ to $f$ in the arrow category $Cat^\mathbf{2}$, i.e., for every commutative diagram
of functors, there is a lifting $B \to C$ filling in to make two commutative triangles. As any relation does, this lifting relation $\perp$ gives a Galois connection on subclasses of $Mor(Cat)$. General facts about Galois connections may then be applied.
Every canonical cofibration is an $\mathbf{M}$-cofibration. Every $\mathbf{M}$-fibration is a canonical fibration.
By the Galois connection induced by the lifting relation, Proposition 1 implies that canonical cofibrations form a subset of $\mathbf{M}$-cofibrations, and therefore that canonical acyclic cofibrations are a subset of acyclic $\mathbf{M}$-cofibrations. Again by the Galois connection, this in turn implies that $\mathbf{M}$-fibrations form a subset of canonical fibrations.
At this point, we would like to show conversely that every $\mathbf{M}$-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 $\mathbf{M}$-fibration, and we would be done. Let us suppose otherwise, that there exists an $\mathbf{M}$-cofibration that is not injective on objects, and derive a contradiction.
For a set $S$, let $K(S)$ be the category whose objects are the elements of $S$, with exactly one morphism $x \to y$ for any $x, y \in S$. This gives the codiscrete (or chaotic) functor $K: Set \to Cat$, which is right adjoint to the forgetful functor $U: Cat \to Set$ that takes a category to its underlying set of objects. For each inhabited set $S$, we have that $K(S)$ is equivalent to $1$, and conversely any category equivalent to $1$ is isomorphic to some $K(S)$.
If there is any $\mathbf{M}$-cofibration $f: A \to B$ that is not injective on objects, then the map $K(2) \to 1$ ($2 = \{0, 1\}$) is an (acyclic) $\mathbf{M}$-cofibration.
First we observe that for any category $E$, the unit map $\eta_E: E \to K U(E)$ for the adjunction $U \dashv K$ is an $\mathbf{M}$-cofibration. For, the map $\eta_E$ is an isomorphism on objects and therefore a canonical cofibration; it is an $\mathbf{M}$-cofibration by Corollary 1.
By hypothesis, $f$ maps two objects $a, a'$ of $A$ to the same object $b$ of $B$, so there is a commutative diagram
Let $r: U A \to 2$ be a retraction of the injection $2 \to U A$. By the adjunction $U \dashv K$, the map $r$ corresponds to a map $s: A \to K(2)$. We form a pushout square
where $g$ is an $\mathbf{M}$-cofibration (being the pushout of a cofibration $f$). Thus we have a composite cofibration $t \coloneqq \eta_E \circ g: K(2) \to K U(E)$. It may be verified that $K(2) \to 1$ is a retract of $t$, i.e., there is a commutative square
where $j = \eta_E \circ h \circ b$; this diagram commutes on objects by construction, and it commutes on morphisms because all diagrams commute in $K U(E)$. Thus $K(2) \to 1$, being a retract of an $\mathbf{M}$-cofibration, is also an $\mathbf{M}$-cofibration.
The conclusion of Proposition 2 now leads to a contradiction:
If $K(2) \to 1$ is an acyclic $\mathbf{M}$-cofibration, then for any category $C$, every automorphism of $C$ is an identity (which is absurd!).
The object $C$ of $Cat$ has an $\mathbf{M}$-fibrant replacement $\hat{C}$ equivalent to $C$. For any isomorphism $\phi$ of $\hat{C}$, let $e: K(2) \to \hat{C}$ be the unique functor taking $0 \to 1$ in $K(2)$ to $\phi$. Then we have a commutative diagram
and the existence of a lift $1 \to \hat{C}$ filling in this diagram means that $\phi$ is an identity. In particular, every automorphism of $\hat{C}$ is an identity; since $C \simeq \hat{C}$, the same is true of $C$.
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 $Cat$ 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 $A\leftarrow F \to B$ of functors in which $F\to A$ 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.
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 $f\colon A\to B$ to be:
a weak equivalence if it is a weak equivalence of categories, i.e. fully faithful and essentially surjective on objects.
a cofibration if it is injective on objects, and $B_0\setminus f(A_0)$ is a projective object in $Set$.
a fibration if it is an isofibration.
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 $f\colon A\to B$ into a cofibration followed by an acyclic fibration is given by first factoring $f_0$ as $A_0 \to A_0 + B_0' \to B_0$, where $B_0'\to B_0$ 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 $B_0\setminus i(A_0)$, which we have assumed to be projective when $i\colon A\to B$ 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 $Ana(C,D)$ is essentially small, being in fact equivalent to the category $Fun(C',D)$ of ordinary functors where $C'$ is a cofibrant replacement for $C$.
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 $Set$. (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 $Ana(C,D)$ is essentially small) sufficient for the existence of an “injective” model structure on Cat?
canonical model structure on $Cat$