### 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 $Chu(Cat,Set)$. 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 $Adj$ embeds in $Chu(Cat, Set)$)? 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 $Model$ 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 $C$ equipped with a family of distinguished cones. I won’t require $C$ to be small, but I will require the cones to be small. We regard a cone as a functor $F : A \to C$, where $A$ is a category with an initial object (the cone vertex). Thus, for instance, for a cone over a cospan, $A$ 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 $LSk$.

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

Let’s equip $LSk$ with the following closed symmetric monoidal structure. Given two limit sketches $C,D$, let $C\otimes D$ be the category $C\times D$ equipped with the distinguished cones $F\times G : A\times B \to C\times D$, for all distinguished cones $F:A\to C$ and $G:B\to D$. (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 $[C,D]$ is the functor category $D^C$ equipped with the set of all cones $F:A\to D^C$ such that for any distinguished cone $G:B\to C$, the cone $A\times B \xrightarrow{F\times G} D^C \times C \xrightarrow{ev} D$ is distinguished. This will be the monoidal (2-)category to which we apply the (2-)Chu construction.

Now, suppose $C$ is a complete category equipped with a class of maps $\mathcal{R}$. We make it a limit sketch by distinguishing every cone $F:A\to C$ over a small diagram $F' : A' \to C$ such that the induced map $F(0) \to \lim(F')$ 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 $\mathbf{2}\to C$ is distinguished. Moreover, if $H:C\to D$ 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 $C$ is cocomplete and has a class $\mathcal{L}$, we make $C^{op}$ a limit sketch in a dual way. Equivalently, this means we make $C$ a *colimit* sketch in which the distinguished cocones are those $F:A\to C$ *under* a diagram $F':A'\to C$ such that the induced map $colim(F') \to F(1)$ is in $\mathcal{L}$.

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

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

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

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

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

A morphism $(C^{op},C,\hom_C) \to (D^{op},D,\hom_D)$ in $Chu(LSk,Set)$ is just a morphism in $Chu(Cat,Set)$, i.e. an adjunction between $C$ and $D$, in which the left adjoint is a morphism of limit sketches $C^{op}\to D^{op}$ and the right adjoint is a morphism of limit sketches $D\to C$. 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 $(C^{op},C,\hom_C) \otimes (D^{op},D,\hom_D) \to (E^{op},E,\hom_E)$ in $Chu(LSk,Set)$ is a morphism in $Chu(Cat,Set)$, i.e. a two-variable adjunction $(C,D) \to E$, in which the constituent functors $H:C^{op}\times D^{op} \to E^{op}$, $K:C^{op}\times E\to D$, and $L:D^{op}\times E\to C$ are morphisms of limit sketches. By the above claim again, this is equivalent to saying that $H$ has the pushout product property for $\mathcal{L}$-maps, while $K$ and $L$ 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 $C$ and $D$ are limit bisketches, let $C\otimes D$ be $C\times D$ in which a cone $F\times G: A\times B \to C\times D$ is distinguished if $F$ and $G$ are distinguished, and acyclic if in addition at least one of $F$ and $G$ is acyclic. We then get a closed symmetric monoidal (2-)category $LBiSk$.

Make $Set$ 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 $C$, we make $C$ a limit bisketch in which the distinguished and acyclic cones arise respectively from the fibrations and acyclic fibrations as above, while we make $C^{op}$ a limit bisketch in which the distinguished and acyclic cones arise respectively from the cofibrations and acyclic cofibrations. This ensures that $\hom_C : C^{op}\otimes C \to Set$ is a morphism of limit bisketches, hence $(C^{op},C,\hom_C)$ is an object of $Chu(LBiSk,Set)$.

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

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

It’s interesting to ponder whether the objects of $Chu(LBiSk,Set)$ 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 $Chu(LBiSk,Set)$ in a similar way to how the monoidal category of locally presentable categories is related to $Chu(Cat,Set)$. Barton’s model structure for premodel categories is probably similar. (**Edited** to give better citations.)

These examples still have underlying objects of $Chu(Cat,Set)$ of the form $(C^{op},C,\hom_C)$, 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 $\mathbf{2}\to C$ 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 $Chu(LBiSk,Set)$ 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.

## 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 $C$, to make $(C^{op},C,\hom_C)$ an object of $Chu(LSk,Set)$ is precisely to equip $C$ with an analogously-defined

weak cylinder factorization system. (We would get strict/orthogonal cylinder factorization systems if we instead took the dualizing object $Set$ to be a limit sketch with only its actual limit cones rather than its weak limit cones.) Similarly, an object of $Chu(LBiSk,Set)$ of the form $(C^{op},C,\hom_C)$ 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.