## December 9, 2011

### 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.

Posted at December 9, 2011 4:14 PM UTC

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### 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?

Posted by: Tom Leinster on December 10, 2011 2:53 AM | Permalink | Reply to this

### Re: Cellularity in algebraic model structures

The emphasis the algebraic theory places on the generating (trivial) cofibrations can be a disadvantage for model structures obtained using Jeff Smith’s cardinality arguments, where the generating trivial cofibrations are known to exist but aren’t given explicitly.

The is also a procedure for mixing two model structures to obtain a third due to Cole that is useful in spaces, chain complexes, and so on, that I don’t know how to algebraicize.

This said, I think any construction that appeals to Quillen’s small object argument should just use Richard’s version instead, and the discussion of morphisms between categories equipped with algebraic weak factorization systems explains the universal properties of the resulting (co)monads.

Posted by: Emily Riehl on December 10, 2011 4:17 PM | Permalink | Reply to this

### Re: Cellularity in algebraic model structures

is there any reason these days not to do everything in terms of algebraic model categories, and forget ordinary model categories altogether?

I think this question is a bit analogous to a question of the following kind:

“Is there any reason these days not to make every trip in a car with power steering, and forget about cars without power steering altogether?”

Already a bare model structure is just an extra tool for handling and studying the $(\infty,1)$-category presented by a category equipped with weak equivalences.

Sometimes this extra toolset is available, and often it is useful to have it. But sometimes it is available and not needed for a given purpose, in which case it is useless to carry it around.

Or it is not available. Or it is available, and still not useful enough. In which case we may look for yet stronger tools to help us deal with what the underlying category with weak equivalences presents.

In this case we may want to appeal to the even stronger toolset of an algebraic model structure. Again, if it exists, often it is useful. But sometimes it is available but not needed for a given purpose, in which case it is not necessary to carry it around.

Posted by: Urs Schreiber on December 10, 2011 7:48 PM | Permalink | Reply to this

### Re: Cellularity in algebraic model structures

Hi Emily! I’m looking forward to your talk on Monday.

I was just reading through your previous June post which you linked to. There you give a plea for more applications in order to help justify the theory. I was wondering if you have found any of these new applications since then?

Posted by: Chris Schommer-Pries on December 11, 2011 12:02 PM | Permalink | Reply to this

### Re: Cellularity in algebraic model structures

There are applications of particular aspects of this theory, but I don’t know of one that takes advantage of the theory as a whole. For instance, when constructing derived functors it’s useful to have a cofibrant
behaved; Andrew Blumberg and I are working on a paper that discusses some
applications. Another advantage, which I believe I mentioned in the
previous post, has to do with detecting when colimits of cofibrations are
cofibrations. And another nice feature, is that certain model structures
which are not cofibrantly generated (as classically understood) are
generated by categories of arrows. This is helpful for constructing
projective model structures, for instance.

But on the other hand, an algebraic model category is in particular an
ordinary model category, so if we’re happy to use ordinary model
categories to present homotopy theories then in some sense this structure
is superfluous. I guess what I’m hoping to find is context for which
it’s really natural to think about the objects algebraically. Because algebra structures give rise to canonical solutions to lifting problems that then commute with maps of algebras, then the familiar lifting constructions one typically does in a model category would all be natural with regard to the algebras. Presumably this is useful?

Posted by: Emily Riehl on December 12, 2011 1:05 AM | Permalink | Reply to this

### Re: Cellularity in algebraic model structures

Peter Lumsdaine and I have found an application of algebraic weak factorization systems to constructing (higher) inductive types in homotopy type theory. (We weren’t looking for an application of awfs — we just suddenly realized they were exactly what we needed.)

I hope to write more about this soon, but here’s the general idea. In categorical models of type theory, inductive types are weakly initial algebras for endofunctors, with an additional “dependent eliminator” which makes them strictly initial algebras (in extensional type theory) or up-to-homotopy initial algebras (in intensional/homotopy type theory). In a locally presentable category, initial algebras for endofunctors are easy to construct by way of the free monad on an endofunctor, and to model extensional type theory this is all you need.

In the homotopy case, however, all types must be fibrant objects, and there is no reason why the initial algebra for an endofunctor should be fibrant. If we fibrantly replace it in a naive way, the result need no longer even be an algebra for the same endofunctor. But if we combine the free monad on the endofunctor with the algebraic fibrant replacement monad (which is coincidentally also free on an endofunctor), we can obtain a fibrant object which is also weakly initial among fibrant algebras. Furthermore, to obtain the dependent eliminator, we needed to use the fact that if $f$ and $g$ are algebraic fibrations, then $g f$ acquires an algebraic fibration structure such that $(f,1)\colon g f \to g$ is a map of algebraic fibrations.

There is also a more nebulous connection to type theory, which I don’t understand that well yet. In some approaches to categorical models of type theory, one gives a subfibration of the codomain (Grothendieck) fibration whose objects are the “display maps” (fibrations in the category) which model the dependent types. Other approaches don’t require the fibration of display maps to be a subfibration of the codomain fibration, only equipped with a map to it. To the extent that these latter approaches are important, they could be said to be a context in which it is natural to think of “being a fibration” as structure rather than a property. I don’t know to what extent this connection can be made precise, but I think Peter has thought about it more than I have.

I think my current take on algebraic wfs and model structures is that they are one more tool in our toolkit. The ordinary small object argument is a fairly coarse bludgeon, but surprisingly it is usually very effective. (Its efficacy does seem to depend to a significant degree, however, on that other very coarse bludgeon, the axiom of choice.) Algebraic wfs and Garner’s small object argument are more precise tools which give us more control when we happen to need it. I also find them quite aesthetically pleasing and helpful to understanding, but I can understand how an algebraic topologist might not want to think about the extra categorical complexity.

Posted by: Mike Shulman on December 12, 2011 6:15 AM | Permalink | Reply to this

### Re: Cellularity in Algebraic Model Structures

Hi, I have a little question (about homotopy type theory in a category with an algebraic weak factorisation system).

In “Homotopy theoretic models of identity types”, Steve Awodey and Michael Warren have constructed a model of the identity types in any category with a weak factorisation system “nice enough”. In particular, they require a functorial choice of path objects for the category and all of its slices, which is stable under substitution, and a technical coherence condition (it’s called (4.1) in the paper).

Are these conditions verified if we have an algebraic weak factorisation system instead? I guess that the functorial choice of path objects in the slices stable under substitution will be easy, but I’m not sure about the coherence condition.

Thanks!

Posted by: Guillaume Brunerie on December 13, 2011 12:31 AM | Permalink | Reply to this

### Re: Cellularity in Algebraic Model Structures

A functorial choice of path objects is easy with an awfs, but I don’t think that algebraicity per se helps with pullback-stability or coherence.

Posted by: Mike Shulman on December 13, 2011 6:31 AM | Permalink | Reply to this

### Re: Cellularity in Algebraic Model Structures

yes, in principle, coherence should follow from awfs: that notion is itself an extension of what used to be called a “natural weak factorization system”, which was originally introduced by Grandis and Tholen. Richard Garner showed how a nwfs gives the coherence condition, but I don’t think it’s written up anywhere in that form. The basic argument can be found (in a somewhat different setting) in the paper: “Topological and simplicial models of identity types” by Garner and van den Berg on the HoTT site: http://homotopytypetheory.org/references/ The main point, as you suspected, is that coherence follows from the construction of the required diagonal fillers out of the algebraic structure in a uniform way.
Posted by: Steve Awodey on December 13, 2011 2:10 PM | Permalink | Reply to this

### Re: Cellularity in Algebraic Model Structures

Hmm… more precisely, I think whan Richard and Benno prove is that if you have a “cloven wfs”, which is a “partial algebraicization” of a wfs (weaker than an awfs), and a pullback-stable choice of path objects, then the cloven-ness gives you coherence for the eliminator. So algebraicity helps with coherence, but only after you’ve ensured stability in some other way. (They go on to show one way to get stability. Steve and Michael did it in a different way.)

Recently I’ve become fonder of Voevodsky’s approach to coherence. There we first construct a “universal” fibration $\widetilde{U}\to U$, of which every “small” fibration is a pullback in some way. We interprete types by “named types”, meaning maps into $U$. We can then ensure strict coherence in the type theory for any categorical structure that is stable under pullback in the up-to-isomorphism sense, by performing the construction once in the universal case over $U$, classifying that by some map into $U$, and then just composing named types with that map. I believe this deals in one stroke with coherence for the eliminator (once you have stable path objects) as well as strict substitution for dependent types, and harmonizes well with the type-theoretic view that types are just terms of type $Type$.

Of course, one then has to figure out how to construct a universe. (-: But if you want to model univalence, then you have to do that eventually anyway.

Posted by: Mike Shulman on December 14, 2011 5:49 PM | Permalink | Reply to this

### Re: Cellularity in Algebraic Model Structures

I agree that Voevodsky’s approach is a neat way to solve these problems in one stroke. It not only deals with the coherence problem for J-terms and the like, but also the older problem of coherence of pullbacks for interpreting dependent types and substitution.

Posted by: Steve Awodey on December 15, 2011 2:27 PM | Permalink | Reply to this

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