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April 28, 2020

Model Categories as a Chu Construction

Posted by Mike Shulman

A couple years ago I blogged about the polycategory of multivariable adjunctions and how it embeds in the 2-Chu construction Chu(Cat,Set)Chu(Cat,Set). After my talk at the ACT@UCR seminar this week, some folks were hanging out at the category theory zulip, and Reid Barton asked

The theory of model categories is self-dual. Is there some useful way to embed {model categories + Quillen adjunctions} in Chu(something) (similarly to how AdjAdj embeds in Chu(Cat,Set)Chu(Cat, Set))? Where “something” would be some flavor of “half a model category structure”, e.g., a cofibration category?

At first I thought the answer was “no”, but now I think it is “yes”. I don’t know whether the construction is good for anything, but I found it amusing, so I thought I would share.

First of all, let me burst any potential bubbles by saying that our Chu construction isn’t going to know anything about weak equivalences, only about weak factorization systems. A model category is a category with two weak factorization systems satisfying some additional property, so we’ll just impose that extra property as part of defining the sub-polycategory ModelModel of our Chu construction. In fact it’s even worse than that: our Chu construction won’t know anything about the existence of factorizations, only about lifting properties; thus its objects will be a generalization of categories equipped with two “weak pre-factorization systems”. But I still think it’s interesting.

Recall that a limit sketch is a category CC equipped with a family of distinguished cones. I won’t require CC to be small, but I will require the cones to be small. We regard a cone as a functor F:ACF : A \to C, where AA is a category with an initial object (the cone vertex). Thus, for instance, for a cone over a cospan, AA is a commutative square.

A morphism of limit sketches is of course a functor that takes distinguished cones to distinguished cones. So we get a (2-)category LSkLSk.

Note that if AA and BB are categories with initial objects, so is A×BA\times B. But importantly, if AA and BB represent cones over diagrams of shape AA' and BB' respectively (so that AA is AA' with a free initial object adjoined, and similarly for BB and BB'), then A×BA\times B does not represent a cone over a diagram of shape A×BA'\times B'. For instance, if A=B=2A = B = \mathbf{2} are the free-living arrow, representing cones over single-object diagrams, then A×BA\times B is a commutative square, representing a cone over a cospan.

Let’s equip LSkLSk with the following closed symmetric monoidal structure. Given two limit sketches C,DC,D, let CDC\otimes D be the category C×DC\times D equipped with the distinguished cones F×G:A×BC×DF\times G : A\times B \to C\times D, for all distinguished cones F:ACF:A\to C and G:BDG:B\to D. (To make this coherently associative, we may need to pull some trick like strictifying ×\times on the domain categories, or assuming that the cones in a limit sketch are closed under precomposition with isomorphisms.) The internal-hom [C,D][C,D] is the functor category D CD^C equipped with the set of all cones F:AD CF:A\to D^C such that for any distinguished cone G:BCG:B\to C, the cone A×BF×GD C×CevDA\times B \xrightarrow{F\times G} D^C \times C \xrightarrow{ev} D is distinguished. This will be the monoidal (2-)category to which we apply the (2-)Chu construction.

Now, suppose CC is a complete category equipped with a class of maps \mathcal{R}. We make it a limit sketch by distinguishing every cone F:ACF:A\to C over a small diagram F:ACF' : A' \to C such that the induced map F(0)lim(F)F(0) \to \lim(F') is in \mathcal{R}. Note in particular that this sketch remembers \mathcal{R}, since a map is in \mathcal{R} if and only if the corresponding cone 2C\mathbf{2}\to C is distinguished. Moreover, if H:CDH:C\to D is a limit-preserving functor between two such categories, then it is a morphism of limit sketches if and only if it preserves \mathcal{R}-maps.

Dually, if CC is cocomplete and has a class \mathcal{L}, we make C opC^{op} a limit sketch in a dual way. Equivalently, this means we make CC a colimit sketch in which the distinguished cocones are those F:ACF:A\to C under a diagram F:ACF':A'\to C such that the induced map colim(F)F(1)colim(F') \to F(1) is in \mathcal{L}.

Now let C,D,EC,D,E be complete and equipped with classes \mathcal{R}, and suppose H:C×DEH:C\times D \to E is a functor that preserves limits in each variable separately. I claim that HH is a morphism of limit sketches if and only if for any \mathcal{R}-maps f:xyf:x\to y in CC and g:uvg:u\to v in DD, the induced map H(x,u)H(x,v)× H(y,v)H(y,u)H(x,u) \to H(x,v) \times_{H(y,v)} H(y,u) is an \mathcal{R}-map in EE.

For “only if”, represent ff and gg as distinguished cones 2C\mathbf{2}\to C and 2D\mathbf{2}\to D; then by assumption 2×2C×DHE\mathbf{2}\times \mathbf{2} \to C\times D \xrightarrow{H} E is a distinguished cone in EE, which is the desired statement.

For “if”, let F:ACF:A\to C and G:BDG:B\to D be distinguished cones, i.e. F(0)lim(F)F(0) \to \lim(F') and G(0)lim(G)G(0) \to \lim(G') are in \mathcal{R}. Then H(F×G):A×BEH\circ (F\times G) : A\times B \to E is a cone over a diagram (H(F×G))(H\circ (F\times G))' whose limit can be equivalently computed as the pullback of lim bBH(F(0),G(b))\lim_{b\in B'} H(F(0),G(b)) and lim aAH(F(a),G(0))\lim_{a\in A'} H(F(a),G(0)) over lim aA,bBH(F(a),G(b))\lim_{a\in A',b\in B'} H(F(a),G(b)). But since HH preserves limits in each variable separately, this is the same as the pullback of H(F(0),lim(G))H(F(0),\lim(G')) and H(lim(F),G(0))H(\lim(F'), G(0)) over H(lim(F),lim(G))H(\lim(F'),\lim(G')). So by assumption on HH and on F,GF,G, the map from H(F(0),G(0))H(F(0),G(0)) to this pullback is an \mathcal{R}-map in EE.

Define the \mathcal{R}-maps in SetSet to be the surjections, and regard it as a limit sketch in the above way. Now suppose that CC is complete and cocomplete and has both classes \mathcal{L} and \mathcal{R}, and use them to make CC and C opC^{op} into limit sketches in the above ways. Then the above claim shows that hom C:C opCSet\hom_C : C^{op}\otimes C \to Set is a morphism of limit sketches if and only if for any \mathcal{L}-map f:xyf:x\to y and \mathcal{R}-map g:uvg:u\to v in CC, the “Leibniz” pullback corner map C(y,u)C(x,u)× C(x,v)C(y,v)C(y,u) \to C(x,u) \times_{C(x,v)} C(y,v) is a surjection — which is another way of saying that (,)(\mathcal{L},\mathcal{R}) has the lifting property.

In sum, any bicomplete category CC equipped with a pair (,)(\mathcal{L},\mathcal{R}) satisfying the lifting property can be regarded as an object (C op,C,hom C)(C^{op},C,\hom_C) of Chu(LSk,Set)Chu(LSk,Set).

A morphism (C op,C,hom C)(D op,D,hom D)(C^{op},C,\hom_C) \to (D^{op},D,\hom_D) in Chu(LSk,Set)Chu(LSk,Set) is just a morphism in Chu(Cat,Set)Chu(Cat,Set), i.e. an adjunction between CC and DD, in which the left adjoint is a morphism of limit sketches C opD opC^{op}\to D^{op} and the right adjoint is a morphism of limit sketches DCD\to C. Since these two functors preserve limits, this means that the left adjoint preserves \mathcal{L}-maps and the right adjoint preserves \mathcal{R}-maps. (Of course, if the pairs (,)(\mathcal{L},\mathcal{R}) are weak factorization systems, either of these conditions implies the other.)

Similarly, a morphism (C op,C,hom C)(D op,D,hom D)(E op,E,hom E)(C^{op},C,\hom_C) \otimes (D^{op},D,\hom_D) \to (E^{op},E,\hom_E) in Chu(LSk,Set)Chu(LSk,Set) is a morphism in Chu(Cat,Set)Chu(Cat,Set), i.e. a two-variable adjunction (C,D)E(C,D) \to E, in which the constituent functors H:C op×D opE opH:C^{op}\times D^{op} \to E^{op}, K:C op×EDK:C^{op}\times E\to D, and L:D op×ECL:D^{op}\times E\to C are morphisms of limit sketches. By the above claim again, this is equivalent to saying that HH has the pushout product property for \mathcal{L}-maps, while KK and LL have the pullback corner property relating \mathcal{L}-maps in their contravariant domain to \mathcal{R}-maps in their covariant domain and their codomain. If the pairs (,)(\mathcal{L},\mathcal{R}) are weak factorization systems, any of these conditions implies the other two and says that we have a “two-variable adjunction of wfs”.

Finally, to deal with model categories rather than single weak factorization systems, we duplicate everything. Define a limit bisketch to be a category with one family of “distinguished cones” and a separate family of “acyclic cones” such that every acyclic cone is distinguished. Define a morphism of such to preserve the two classes individually. If CC and DD are limit bisketches, let CDC\otimes D be C×DC\times D in which a cone F×G:A×BC×DF\times G: A\times B \to C\times D is distinguished if FF and GG are distinguished, and acyclic if in addition at least one of FF and GG is acyclic. We then get a closed symmetric monoidal (2-)category LBiSkLBiSk.

Make SetSet a limit bisketch in which all cones are distinguished, while the acyclic cones are induced by the surjections as before. Then for any model category CC, we make CC a limit bisketch in which the distinguished and acyclic cones arise respectively from the fibrations and acyclic fibrations as above, while we make C opC^{op} a limit bisketch in which the distinguished and acyclic cones arise respectively from the cofibrations and acyclic cofibrations. This ensures that hom C:C opCSet\hom_C : C^{op}\otimes C \to Set is a morphism of limit bisketches, hence (C op,C,hom C)(C^{op},C,\hom_C) is an object of Chu(LBiSk,Set)Chu(LBiSk,Set).

A morphism in Chu(LBiSk,Set)Chu(LBiSk,Set) between model categories is a morphism in Chu(Cat,Set)Chu(Cat,Set), i.e. an adjunction, whose left adjoint is a morphism of limit bisketches C opD opC^{op}\to D^{op} (i.e. preserves cofibrations and acyclic cofibrations) and whose right adjoint is a morphism of limit bisketches DCD\to C (i.e. preserves fibrations and acyclic fibrations) — in other words, a Quillen adjunction. Similarly, a morphism CDEC\otimes D \to E between model categories in Chu(LBiSk,Set)Chu(LBiSk,Set) is a Quillen two-variable adjunction. In this way, we have a full sub-polycategory of Chu(LBiSk,Set)Chu(LBiSk,Set) consisting of model categories and Quillen multivariable adjunctions.

Note that the same setup handles enriched model categories as well. For any monoidal model category VV, we can consider VV-enriched limit bisketches, and replace SetSet by VV whose distinguished and acyclic cones arise respectively from its fibrations and acyclic fibrations.

It’s interesting to ponder whether the objects of Chu(LBiSk,Set)Chu(LBiSk,Set) that aren’t model categories have any homotopy-theoretic meaning. They certainly include various known weakenings of model categories, such as left and right semi-model categories, weak model categories, and premodel categories. In fact, Henry’s model structure for model categories lives on a monoidal category of locally presentable categories equipped with two cofibrantly generated weak factorization systems, which seems vaguely related to Chu(LBiSk,Set)Chu(LBiSk,Set) in a similar way to how the monoidal category of locally presentable categories is related to Chu(Cat,Set)Chu(Cat,Set). Barton’s model structure for premodel categories is probably similar. (Edited to give better citations.)

These examples still have underlying objects of Chu(Cat,Set)Chu(Cat,Set) of the form (C op,C,hom C)(C^{op},C,\hom_C), and limit bisketch structures induced by classes of maps (,)(\mathcal{L},\mathcal{R}). What about the more general case? I could imagine imposing closure conditions on a limit bisketch ensuring that it behaves something like a fibration category. One might hope that the morphisms whose classifying cone 2C\mathbf{2}\to C is distinguished or acyclic behave like fibrations or acyclic fibrations, respectively, and that the additional distinguished and acyclic cones might carry more useful information about limits and colimits. (In particular, we could relax the assumption about which (co)limits are required to exist in the first place.) A general object of Chu(LBiSk,Set)Chu(LBiSk,Set) with such conditions on both components would be something like a cofibration category related to a fibration category by a profunctor. It would be tempting to call such a thing a “polarized model category”. Probably the categories of cofibrant and fibrant objects in an ambient model category would be an example. But I don’t know whether there would still be interesting homotopy theory visible.

Posted at April 28, 2020 12:11 AM UTC

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Re: Model Categories as a Chu Construction

I just made an interesting connection. In Garner’s paper The Isbell monad he defined a notion of “cylinder factorization system” involving orthogonality between a set of cones (over diagrams) and cocones (under diagrams). I believe that for a category CC, to make (C op,C,hom C)(C^{op},C,\hom_C) an object of Chu(LSk,Set)Chu(LSk,Set) is precisely to equip CC with an analogously-defined weak cylinder factorization system. (We would get strict/orthogonal cylinder factorization systems if we instead took the dualizing object SetSet to be a limit sketch with only its actual limit cones rather than its weak limit cones.) Similarly, an object of Chu(LBiSk,Set)Chu(LBiSk,Set) of the form (C op,C,hom C)(C^{op},C,\hom_C) should be a category with two weak cylinder factorization systems.

That gives an interesting gloss on the sense in which these objects, at least, generalize model categories: any weak factorization system yields a weak cylinder factorization system, and any weak cylinder factorization system has an underlying weak factorization system, but not every weak cylinder factorization system is determined by its underlying weak factorization system.

Posted by: Mike Shulman on May 1, 2020 6:18 AM | Permalink | Reply to this

Re: Model Categories as a Chu Construction

make (C op,C,hom C)(C^{op},C,\hom_C) an object of Chu(LSk,Set)Chu(LSk,Set) is precisely to equip CC with an analogously-defined weak cylinder factorization system

I must have meant, a weak cylinder prefactorization system. It doesn’t encode any factorizations, only lifting properties.

Posted by: Mike Shulman on May 7, 2021 5:27 AM | Permalink | Reply to this

Re: Model Categories as a Chu Construction

Hi Mike ! great post, I don’t why I’m only discovering this now ! So, I don’t quite know what to do with it, but at some point I thought about building a framework that would encompass both my “weak model categories” and the kind of model categories considered in Cisinski’s “Catégorie Derivables” or in Radulescu-Banu’s work, and the answer I came up with was exactly the data of a (slightly weakened) fibration category and a cofibration category paired through a profonctor (satisfying some conditions). I don’t think these are exactly the objects of a Chu construction, but they definitely live inside one similar to what you are describing.

Posted by: Simon Henry on April 2, 2021 4:51 PM | Permalink | Reply to this

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