### Cellularity in Algebraic Model Structures

#### Posted by Mike Shulman

*(Guest post by Emily Riehl)*

The most substantial difference between Quillen’s original definition of a model category and the one in use today (which he called **closed**) is that, e.g., the *cofibrations* were defined to be any class of maps for which the usual factorizations and lifts exist — in particular, it is not necessary that all maps which lift against the trivial fibrations in the sense of the diagram
$\array{ \quad\cdot & \to & \cdot\qquad \\ {}^{\mathrm{cof}}\downarrow & {}^{\exists}\nearrow & \downarrow^{\mathrm{triv}}{}^{\mathrm{fib}} \\ \quad\cdot & \to & \cdot\qquad}$
are cofibrations.

For instance, there is a (non-closed) model structure on spaces given by relative cell complexes, Serre fibrations, and weak homotopy equivalences. There is a model structure on chain complexes of modules over a commutative ring given by injections with free cokernel, surjections, and quasi-isomorphisms. But any retract of a map with a particular lifting property inherits that same lifting property, so for these examples to be model categories as presently understood, we must enlarge the class of cofibrations to include all retracts of relative cell complexes in the first case and all injections with *projective* cokernel in the second. Quillen uses this “closure” to show that the homotopy category of a model category is **saturated**, meaning every map that becomes an isomorphism was originally a weak equivalence.

Despite the overwhelming popularity of the closed definition of model categories, even in the modern literature, the distinction between cellular cofibrations (e.g., the relative cell complexes above) and generic ones is still maintained — at least in the case where this class is *cofibrantly generated*. In what follows, we’ll present several unexpectedly strong existence results for certain maps between cofibrantly generated algebraic model categories that hold precisely when certain cofibrations are **cellular** and not merely retracts of cellular cofibrations, providing a new justification for the classically-held distinction.

### $I$-cell vs $I$-cof

Many (henceforth always closed) model categories are constructed using Quillen’s small object argument, which, given a set of *generating cofibrations* $I$, constructs a functorial factorization of any morphism
$\array{ \text{dom} f & & \stackrel{f}{\to} && \text{cod} f \\ & {}_{\mathllap{{C}{f}}}\searrow & & \nearrow_{\mathrlap{F_t f}} & \\ & & Qf & &}$
in such a way that the right factor $F_{t}f$ lifts against the maps in $I$, i.e., $F_{t}f$ is an element of the class $I$-fib, while the left factor ${C}{f}$ is a transfinite composite of pushouts of coproducts of maps in $I$, i.e., ${C}{f}$ is an element of the class typically denoted $I$-cell. It is easy to see that maps in $I$-cell lift on the left against those maps that lift on the right against $I$. But retracts of maps in $I$-cell also have this property. Indeed, the retract closure of $I$-cell, commonly denoted $I$-cof, is the class of all maps that lift on the left against all maps that lift on the right against $I$. In the setting of a model category, $I$-cof is the class of **cofibrations** and $I$-fib is the class of **trivial fibrations**. If $J$ is a set of generating trivial cofibrations, then $J$-cof is the class of **trivial cofibrations** and $J$-fib is the class of **fibrations**.

When constructing functorial factorizations from a set of generators, I prefer to use Richard Garner’s small object argument in place of Quillen’s. Assuming certain hypotheses on the ambient category, he shows that for any generating set (in fact, for any small category!) of arrows, a construction similar to Quillen’s *converges* to produce functorial factorizations, as displayed above in such a way that the functor $C$ is a comonad and $F_t$ is a monad on the arrow category, and this monad and comonad are compatible in a particular way. A functorial factorization with these properties forms the backbone of an algebraic weak factorization system.

Furthermore, every trivial fibration becomes an algebra for the monad $\mathbf{F}_t$ (with many choices of algebra structure in general) and conversely all algebras are trivial fibrations; hence we call an $\mathbf{F}_t$-algebra an **algebraic trivial fibration**. For example, when $I$ is the usual set of inclusions of spheres bounding disks in each dimension, an $\mathbf{F}_t$-algebra structure for a map $X \to Y$ amounts to a choice of lifted contracting homotopy for any sphere in $X$ that is contractible in $Y$. Similarly, all $\mathbf{C}$-coalgebras are cofibrations, but not all cofibrations admit coalgebra structures. However — as you might have guessed — the elements of $I$-cell always do and in a canonical way: As part of Garner’s construction, the generating maps in $I$ are given canonical $\mathbf{C}$-coalgebra structures. Coproducts, pushouts, and transfinite composites of coalgebras for the comonad of any algebraic weak factorization system all inherit canonical coalgebra structures.

With this in mind, we define the **cellular** cofibrations and trivial cofibrations in an algebraic model structure to be those maps admitting $\mathbf{C}$-coalgebra structures or $\mathbf{C}_t$-coalgebra structures, as appropriate. We’ll now describe a number of theorems which say that various complicated structures exist just when certain maps are cellular, and furthermore, are completely determined by the choice of coalgebra structures in this case.

### Algebraic model structures and algebraic Quillen adjunctions

Briefly, an **algebraic model category** is an ordinary model category in which the functorial factorizations take the form described above and such that there is also a natural transformation comparing the two functorial factorizations of any map
$\array{ & \text{dom} f & \\ {}^{C_{t}f}\swarrow & & \searrow {}^{{C}{f}} \\ Rf & \stackrel{\xi_f}{\to} & Qf \\ {}_{{F}{f}}\searrow & & \swarrow {}_{F_{t}f} \\ & \text{cod} f & }$
that defines a morphism between the monads and also the comonads, in the usual sense. In particular, $\xi$ defines functors from the category of algebraic trivial fibrations to the categories of algebraic fibrations and from the category of algebraic trivial cofibrations to the category of algebraic cofibrations. I’ve written about algebraic model categories before, so I won’t proselytize here, except to recall the main existence result:

** Theorem.** A model category with generating cofibrations $I$ and generating trivial cofibrations $J$ admits an algebraic model structure with the same generators if and only if the elements of $J$ are cellular cofibrations.

Furthermore, the natural transformation $\xi$ comparing the functorial factorizations is determined by the choice of $\mathbf{C}$-coalgebra structures assigned to the elements of $J$. (Actually, it’s always possible to get put an algebraic model structure on a cofibrantly generated model category at the cost of changing one of the generating sets; cf section 3 of this paper.)

The cellular cofibrations really shine when we consider maps between algebraic model categories. In the classical theory, a particularly useful notion of morphism is a Quillen adjunction. In the algebraic context, we ask for something stricter: an **algebraic Quillen adjunction** is a Quillen adjunction in which the left adjoint lifts to (commuting) functors between the categories of algebraic (trivial) cofibrations, the right adjoint lifts to (commuting) functors between the categories of algebraic (trivial) fibrations, and the lifted left and right adjoints somehow determine each other. One form of this last condition is that that the mates of the natural transformations characterizing the lifts of the left adjoint characterize the lifts of the right adjoint. Another equivalent condition is that the lifted right adjoints are actually double functors between suitably defined categories of algebraic (trivial) fibrations; such a thing turns out to define analogous lifted double functors between the double categories of algebraic (trivial) cofibrations (h/t Richard Garner). Or the most compact definition is that an algebraic Quillen adjunction is a doctrinal adjunction between algebraic model categories, for a particular 2-monad on CAT whose algebras are categories equipped with a pair of algebraic weak factorization systems together with a comparison map $\xi$ as above (h/t Mike Shulman).

If you unpack any one of these definitions, you’ll see that these requirements are quite strict, to the point that I originally feared that there would be no examples. However, when the algebraic model category at the domain of the left adjoint is cofibrantly generated, there is a simple *cellularity condition* that produces algebraic Quillen adjunctions from ordinary ones.

** Theorem.** A Quillen adjunction $T \dashv S$ between algebraic model categories is a (weak) algebraic Quillen adjunction if and only if $T$ maps the generating cofibrations and trivial cofibrations to cellular cofibrations and cellular trivial cofibrations. Furthermore, the lifted functors are determined by the coalgebra structures assigned to the images of the generating (trivial) cofibrations.

The slight awkwardness in the phrasing of the theorem above disappears if I state the result for each algebraic weak factorization system separately: an adjunction between categories equipped with algebraic weak factorization systems is an **adjunction of algebraic weak factorization systems** if and only if the left adjoint maps the generators for the former to cellular arrows for the latter. So if the arrows in the set ${T}{I}$ are cellular, this means that the left adjoint lifts to a functor between the categories of algebraic cofibrations and the right adjoint lifts to a functor between the categories of algebraic trivial fibrations, and these lifts determine each other.

Many examples of Quillen adjunctions arise when one model structure is lifted along the adjunction to produce another; we show that algebraic model structures can be produced in this way as well. Because of these examples, it seemed reasonable to define an algebraic Quillen adjunction to include the requirement that the lifted functors between the categories of algebraic (trivial) cofibrations commute with the functors produced by $\xi$; an equivalent statement exists for the lifts of the right adjoint. This stronger condition holds just when two canonical methods for assigning $\mathbf{C}$-coalgebra structures to the images of the generating trivial cofibrations agree. But perhaps a weaker definition without this compatibility requirement is preferable.

### Monoidal algebraic model structures

My favorite cellularity theorem establishes the theory of monoidal algebraic model structures for part II of my Ph.D. thesis.

As one might hope, in a monoidal algebraic model category, the internal hom from an algebraic cofibrant object to an algebraic fibrant object is canonically an algebraic fibrant object. Dually, the tensor product of two algebraic cofibrant objects is again (canonically) algebraic cofibrant. These results should be interpreted as (special cases of) an algebraization of the usual pushout-product axiom.

The pushout-product axiom asks for the tensor product $-\otimes-$ to be a left Quillen bifunctor. When the monoidal structure is closed, an equivalent axiom states that the internal hom is a right Quillen bifunctor. Together these functors form a **Quillen two-variable adjunction**.

For a few reasons, the problem of sorting out the appropriate algebraization of this notion was much less-tractible than for the theory of algebraic Quillen adjunctions: An ordinary functor is also a double functor between the double categories of arrows, but the associated pushout-product bifunctor on arrow categories is not a double functor. Neither was there a pre-existing theory of mates for 2-variable adjunctions. (The basic aspects of the theory of **parameterized mates** appears in section 2 of my preprint; a more robust development of the theory of **parametrised mates** for $n$-variable adjunctions will appear in a forthcoming paper with Eugenia Cheng and Nick Gurski.)

The correct definition is easier to state when we consider a two-variable adjunction $(-\otimes -, hom_{l}, \hom_{r}) \colon M \times N \to P$ between categories equipped with a single algebraic weak factorization system each. In this setting, a **two-variable adjunction of algebraic weak factorization systems** consists of a lift of each adjoint to a bifunctor between the appropriate categories of (co)algebras such that particular natural transformations characterizing these lifted functors are parameterized mates; in particular, any lifted bifunctor determines the other two.

An **algebraic Quillen two-variable adjunction** consists of three two-variable adjunctions of algebraic weak factorization systems, corresponding to the usual three parts of the pushout-product axiom. Once again, I worried that this definition would be prohibitively strong; however, a cellularity result that guarantees many examples:

** Theorem.** Assuming the usual unit condition is satisfied, a cofibrantly generated algebraic model category with a monoidal structure is a **monoidal algebraic model category** if and only if the pushout-products of the generating (trivial) cofibrations are cellular with respect to the appropriate comonads. In other words, it suffices to show that the pushout-products of pairs of maps in $I$ are cellular cofibrations and the pushout-products of a map in $I$ with a map in $J$ are cellular trivial cofibrations.

Furthermore, the lifted bifunctors are entirely determined by the choice of coalgebra structures given to the pushout-products of the generators. If we replace the two-variable adjunction exhibiting a closed monoidal structure with the tensor-cotensor-hom adjunction of an enriched category, an analogous theorem would produce **algebraically-enriched algebraic model categories**. I did not include this definition in my preprint because I have’t yet had the time to explore examples, but now that the theory is in place, it should be straightforward to work out.

I want to close with one final comment. It is not reasonable to hope that all of the lifted bifunctors in the algebraic Quillen two-variable adjunction of a monoidal algebraic model category will commute, though in certain cases this does hold (eg for the folk model structure on categories). Let me illustrate by means of an example:

In Quillen’s model structure on simplicial sets, all cofibrations are cellular (and indeed admit a unique $\mathbf{C}$-coalgebra structure) so it remains to consider the pushout-product of a generating cofibration (monomorphism) with a generating trivial cofibration (anodyne extension). To prove that these are cellular trivial cofibrations, we must show that each such map can be factored into a sequence of “horn-fillings.” Fortunately, this is precisely what is done in the old combinatorial proofs that the simplicial set of maps from a simplicial set to a Kan complex is again a Kan complex (cf May’s *Simplicial objects in algebraic topology*). Choosing such a factorization for each pair of generators will canonically determine a $\mathbf{C}_{t}$-coalgebra structure for each pushout-product of a cofibration and a trivial cofibration.

However, the $\mathbf{C}_{t}$-coalgebra structure assigned to the pushout-product of a pair of trivial cofibrations depends on which one is treated as a cellular cofibration and which one is treated as a cellular trivial cofibration. For instance, consider the pushout-product of the inclusion of the $\Lambda^{1}_{0}$-horn into the 1-simplex with the inclusion of the $\Lambda^{2}_{1}$-horn into the 2-simplex. Pictorially, this looks like the inclusion of a hollow “trough” with one triangle edge also missing into the solid triangular prism. One choice gives rise to a cellular decomposition that first fills the missing end triangle and then fills the trough. The other choice first fills in the missing top rectangular face before filling the interior in such a way that this missing end triangle appears in the last step.

Much more about this, including a detailed treatment of this example, is available in various forms on my website. For those in the Boston area, I’ll also be giving a talk on this subject on Monday at MIT.

## Re: Cellularity in algebraic model structures

Emily, here’s a general question: is there any reason these days not to do everything in terms of algebraic model categories, and forget ordinary model categories altogether?