How canonical is cofibrant replacement?

Question

Quillen's original definition of a model category included noncanonical factorization axioms, one being that any map can be factored into a cofibration followed by an acyclic fibration. More recent references have strengthened this axiom by assuming that this is actually a functorial factorization.

If you take a "classical" model category (not exactly in Quillen's sense - I would still like to assume that it has all small limits and colimits), it becomes natural to ask whether it has such a functorial factorization. More, one might wonder how canonical this factorization is, even just on the level of objects.

So in this situation, let's say that we have either:

• a very large category whose objects are "factorization functors", and whose morphisms are natural transformations between them (of necessity natural weak equivalences), or
• a very large category whose objects are functorial cofibrant replacements for objects, and whose morphisms are again natural weak equivalences.

Are there easy examples where these categories are empty? Are there examples where they are nonempty, but the functorial factorization is "noncanonical" in the sense that the category of factorizations is noncontractible? Does this category of factorizations become contractible under stronger assumptions (such as cofibrant generation)?

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Added later: It turns out that the question of homotopy type has a boring answer (and, embarassingly, one answerable by the standard techniques). Let's suppose we have one factorization functor $F$, so that any arrow $g:x \to y$ factors canonically as $$x \stackrel{c(g)}{\to} F(g) \stackrel{f(g)}{\to} y.$$ Then, naturally associated to any other such functor $G$, we get a third replacement functor abusively written as $G \circ F$, obtained by applying $G$ to the map $c(g)$; this gives a factorization $$x \to (G\circ F)(g) \to F(g) \to y$$ and the first map, together with the composite of the latter two maps, gives a new factorization.

There are natural transformations of factorization functors $F \leftarrow G \circ F \rightarrow G$, and this is natural in $G$; this provides a two-step homotopy contracting the space of factorization functors down to the constant $F$.

(However, the responses to the first question have already informed me quite a bit!)

Answer 1

I don't know the answer to your questions off the top of my head, but I think algebraic weak factorization systems (the new consensus terminology for what were originally called natural weak factorization systems) are the right context to search for the answer. I'll try to briefly explain why.

Loosely, an algebraic weak factorization system on a category $\mathcal{M}$ consists of a comonad $\mathbb{L}$ and monad $\mathbb{R}$ on the arrow category $\mathcal{M}^\bf{2}$ such that these fit together to form a functorial factorization (ie, a section of the composition functor $\mathcal{M}^\bf{3} \to \mathcal{M}^{\bf 2}$, $\bf{2}$ and $\bf{3}$ being the poset categories for these ordinals). A main point is that the arrows which admit coalgebra structures for $\mathbb{L}$ lift canonically against those arrows which admit algebra structures for $\mathbb{R}$. So the comonad-monad functorial factorization also algebraicizes the construction of lifts.

Here's why I suspect this is relevant to your question. A priori, the existence of pointwise lifts does not give rise to a natural transformation between two functorial factorizations for the same weak factorization system. But suppose we had some other functorial factorization $(L',R')$ for the underlying weak factorization system of $(\mathbb{L},\mathbb{R})$. Then if the functor $R' \colon \mathcal{M}^{\bf 2} \to \mathcal{M}^{\bf 2}$ factored through the category of algebras for $\mathbb{R}$ (as, for instance, $R$ tautologously does), then there would be a morphism $(L,R) \to (L',R')$ in the category you describe. Or dually, if $L'$ factored through the category of $\mathbb{L}$-coalgebras, then there would be a morphism $(L',R') \to (L,R)$ given by the canonical solutions described above to the lifting problems.

Garner's small object argument produces algebraic weak factorization systems for any cofibrantly generated ordinary weak factorization system (or model category), so this algebraic setting is a lot more common than you'd think. (Incidentally, the best paper of his to read is Understanding the small object argument.) Interestingly, more things are cofibrantly generated than were before, because his small object argument works for generating categories (ie, one can have morphisms in the form of squares between the arrows in the generating set; in other words, one can ask that the trivial fibrations lift "coherently" against the generating cofibrations). For example, the usual model structure on ${\bf \mathrm{Cat}}$ induces one on the functor category ${\bf \mathrm{Cat}}^{\mathcal{A}}$ where the fibrations and weak equivalences are defined representably. This isn't cofibrantly generated in the usual sense, but it is in algebraic context. I describe the generating categories in my paper.

I'm quite interested in the sort of question you posed and would be happy to talk more offline, if you'd like to get in touch.

Answer 2

While Emily was too modest to say so, the history is that Garner developed a beautiful refined small object argument for the construction of algebraic weak factorization systems (his paper Understanding the small object argument), but it was Emily (her paper Natural weak factorization systems in model structures) who algebraicized model categories. Also, while I may be misremembering, I'm pretty sure that Isaksen, after his initial paper, proved that his model structure on pro-simplicial sets cannot admit functorial factorizations.

Answer 3

Dear Tyler,

The other replies here have summarised quite well what has been done by Emily and myself in this regard, but it might be useful to point something out concerning the later addition you have made to your question. You describe a construction that, from two functorial factorisations $F$ and $G$, produces a third, $G \circ F$. This, as you observe, comes equipped with a natural transformation $G \circ F \to F$. However, in trying to define similarly a natural transformation $G \circ F \to G$, one must use lifting properties; the component at $g$ being obtained by considering a square with $x \to G \circ F(g)$ on its left, and $G(g) \to y$ on its right. The problem is that these components are highly unlikely to constitute a natural transformation; the naturality squares most likely do not commute. Even if the functorial factorisations in question were derived from Quillen's small object argument, and the liftings chosen were the ones described in that argument, naturality is still unlikely to obtain: for the construction of those liftings requires the making of some non-canonical choices (more specifically, the choices made in Hovey's book, p.33, line 6, "there is a $\beta < \gamma$") which materially affect the resulting fillers, and are unlikely to cohere for different choices of $g$.

The notion of "natural" / "algebraic" weak factorisation system could be understood as an attempt to rectify precisely this problem. Here one has provided not only the functorial factorisation, but in addition, canonical choices of filler, which, in particular, will cohere sufficiently to allow the construction of the natural $G \circ F \to G$ sought above. That every (cofibrantly generated) weak factorisation system can be made into an instance of this notion relies on modifying the small object argument in such a way as the chosen fillers it provides no longer rely on the making of non-canonical choices (to be more precise: the non-canonical choices are still there, but the outcome is now independent of them).

Richard

Answer 4

Have you looked at the work of Richard Garner and of Emily Riehl?

(Edit: having read Peter's answer, I've decide that I'm out of my depth when it comes to knowing which of them did what. So I'll just say "they" in what follows.)

I'm not an expert on this, but here's what I think I know. They have a notion of "algebraic" model category, which I think is rather more than having functorial factorizations. The idea, I believe, is that you know not just whether something is a fibration or cofibration, but why it is. (This builds on work of Marco Grandis and Walter Tholen on "natural weak factorization systems".)

That sounds like it's asking a lot, but they have a small object argument implying that any cofibrantly generated model category can be algebraicized. So, for example, this gives you a fibrant replacement monad (I mean a genuine monad, not just up-to-something), a cofibrant replacement comonad, and a distributive law of one over the other.

All I can find about this on Garner's website is this; I suspect he's done more, though. (Edit: Emily points out in her answer that his paper Understanding the small object argument is a better source.) The previous paragraph came from my notes from this talk by Riehl, and there's an associated paper of hers.