# A model category of abelian categories?

Let $\mathcal{M}$ be the following category:

• The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels.
• The morphisms are functors that preserve the structure strictly.
• Composition and identities are the obvious ones.

If I'm not mistaken, $\mathcal{M}$ is a locally finitely presentable category. At any rate, the forgetful functor $U : \mathcal{M} \to \mathbf{Cat}$ preserves all limits and filtered colimits, so under the assumption that $\mathcal{M}$ is l.f.p., we have a left adjoint $F : \mathbf{Cat} \to \mathcal{M}$.

Question 1. Can the standard model structure on $\mathbf{Cat}$ be transferred to $\mathcal{M}$, i.e. does there exist a model structure on $\mathcal{M}$ where the weak equivalences and fibrations are created by $U : \mathcal{M} \to \mathbf{Cat}$?

Question 2. Assuming the desired model structure on $\mathcal{M}$ exists, is it compatible with the obvious $\mathbf{Cat}$-enrichment, i.e. do we have a model 2-category?

Question 3. Let $\mathfrak{M}$ be the 2-category whose objects and morphisms are as in $\mathcal{M}$ and whose 2-cells are natural transformations; and let $\mathfrak{A}$ be the 2-category whose objects are small abelian categories, whose morphisms are exact functors, and whose 2-cells are natural transformations. There is an evident 2-functor $\mathfrak{M} \to \mathfrak{A}$ which is surjective on objects, (locally) injective on morphisms, and (locally) bijective on 2-cells. Does this exhibit $\mathfrak{A}$ as the higher-categorical localisation of $\mathfrak{M}$ with respect to weak equivalences?

• Are you most interested in question 3 itself, or do you really want to go via questions 1 and 2? – David Roberts May 11 '15 at 23:59
• Question 3 is the most interesting part, and questions 1 and 2 seem like the natural way to proceed. – Zhen Lin May 12 '15 at 0:00
• I could probably jump direct to question 3, using Pronk's bicategorical localisation technology. I presume weak equivalences in $\mathcal{M}$ are ess. surj. fully faithful functors. If it is true that for any such $f\colon A \to B$, one can find a surjective on objects, ff functor $c\colon C\to B$ in $\mathcal{M}$ (say from a freely generated version of $B$ or something), a functor $C\to A$ and a nat iso in the resulting triangle, the one gets a bicategory of fractions. Then one can apply Pronk's comparison theorem (proposition 24 in numdam.org/item?id=CM_1996__102_3_243_0) ... – David Roberts May 12 '15 at 0:07
• ...to get the result. One only needs to check the condition that for every functor $F$ in $\mathfrak{A}$ there is a weak equivalence $W$ and a functor $G$ in $\mathfrak{M}$ such that $U(G) \simeq F\circ U(W)$, for $U\colon \mathfrak{M}\to \mathfrak{A}$ the functor you give. A similar result is true for either weak 2-functors or monoidal functors, so I expect something similar here. Makkai's anafunctors are also worth mentioning here. – David Roberts May 12 '15 at 0:11
• I would expect there to be a finitary 2-monad $T$ on Cat with $\mathfrak{M} = T\text{-Alg}_s$, so your first two questions would be answered in the affirmative by Steve Lack's Homotopy-theoretic aspects of 2-monads. – Alexander Campbell May 12 '15 at 3:09

It seems to me that you can answer this question without 2-monads. Perhaps 2-monads do apply in this example but they might not in similar ones so here is an outline not using them.

(1)Yes. More generally consider an adjunction $F\dashv U:M \leftrightarrows Cat$ with M locally presentable as a 2-category (just amounts to its underlying category being locally presentable and it has cotensors with the walking arrow 2 - equally it is complete as a 2-category) and U an accessible right 2-adjoint (just means its underlying functor is accessible right adjoint and preserves cotensors with 2, equally all limits).
Then you get a projectively lifted model structure in which the weak equivalences and fibrations are those whose image under U is one. Now M, being complete, has cotensors with the free isomorphism $I$. The objects $X^{I}$ provide functorial path objects for $M$: indeed you get a factorisation of $X \to X \times X$ into an internal equivalence followed by a (discrete) isofibration in any 2-category. So according to Proposition 1 on http://ncatlab.org/nlab/show/transferred+model+structure you get a transferred model structure.

(2) I haven't checked but I would strongly suspect so. Certainly the relevant condition is true for generating cofibrations and trivial cofibrations (by adjointness) and presumably you can extend from there by general nonsense?

(3) This can be seen in a couple of ways. As David Roberts says, you can use Pronk's work on bicategorical localization, and that might be the most direct route. The key point here is that given any weak map $f:A \rightsquigarrow B$ you can cover it by a span of strict maps $(p,q):A \leftarrow P_{f} \rightarrow B$ where $p$ is a trivial fibration and $q$ an isofibration such that $fp \cong q$. Here $P_{f}$ is the pseudolimit of the arrow $f$ - in the $F$-categorical sense of Lack and Shulman. It is just the full subcategory of the comma category $B/f$ consisting of the invertible arrows, with the evident structure lifted to $M$. In particular this shows that the inclusion $M \to \mathbf M$ is sufficiently surjective on 1-cells that Pronk's Proposition 24 may be applied. Though you would also need to verify the calculus of fractions stuff therein to apply that too.

Beyond this I would point out that a 2-functor (or pseudofunctor) $F:M \to C$ sends weak equivalences to equivalences iff it sends trivial fibrations to equivalences (use the above covering $(p,q)$ of a $U$-equivalence $f$ by a span of trivial fibrations, just like in Ken Brown's lemma). So $\mathbf M$ is equally the 2-categorical localisation at the trivial fibrations.

(3*) Although this property characterises $\mathbf M$ up to biequivalence, there is in fact a stronger property characterising it up to isomorphism. Namely, it is the (2)-category of weak maps for the algebraic weak factorisation system generated by the lifted generating cofibrations. You can see this using an argument identical similar to the proof of Theorem 16 of http://arxiv.org/abs/1412.6560

From that you get a Kleisli 2-adjunction $Q \dashv j:M \to \mathbf M$ whose counit $QA \to A$ is a trivial fibration. (Actually the awfs stuff only gives you the 1-adjunction, but since j preserves cotensors this extends to a unique 2-adjunction). And you could now use an argument identical to that of Theorem 4.15 of the paper of Steve Lack mentioned by Alexander Campbell to verify the 2-categorical localisation property of Q3.

• For clarification: you write "with M locally presentable as a 2-category (just amounts to its underlying category being locally presentable and it has cotensors with the walking arrow 2 - equally it is complete as a 2-category." Is the walking arrow the category $0\rightarrow 1$? Also is the property of being locally presentable and having cotensors with the walking arrow category equivalent to being complete as a 2-category? – user84563 Aug 4 '16 at 2:44
• Also, is the "free isomorphism" the groupoid with two objects? Sorry I'm not familiar with some terminology. – user84563 Aug 4 '16 at 3:00
• Oops - all that "walking" stuff is a bit confusing. Yes the walking arrow is as you mentioned; the free isomorphism is the groupoid with two objects and a single morphism connecting each pair of objects in it. – john Nov 8 '16 at 3:46

Expanding on my comment, there ought to be a finitary 2-monad $T$ on Cat with $\mathfrak{M} = T\text{-Alg}_s$ and $\mathfrak{A} = T\text{-Alg}$. If this is so, then all of your questions are answered in the affirmative by Steve Lack's paper Homotopy-theoretic aspects of 2-monads (arXiv link): Q1 and Q2 by Theorem 4.5, and Q3 by Theorem 4.15.

• But is there such a finitary 2-monad? I'm not even sure the underlying ordinary adjunction here is monadic. – Zhen Lin May 12 '15 at 8:24
• The final paragraph of section 6.4 of Blackwell-Kelly-Power suggests that there is, but I will ask the experts tomorrow. – Alexander Campbell May 12 '15 at 9:26
• I wonder if there's an actual proof in the literature somewhere? Anyway, Ignacio López Franco suggested to me an argument for reducing the monadicity of $\mathfrak{M}$ to the monadicity of categories with finite (co)limits, so I can more-or-less believe that there is a 2-monad of the desired form. But it would be nice to have some intuition for when a 2-category of categories-with-structure is 2-monadic. – Zhen Lin May 13 '15 at 11:18