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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)?


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!)

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According to B. Chorny arxiv.org/abs/math.AT/0401424 it is not known whether Isaksen's strict model structure on the pro-objects in a proper model category (and some generalizations) admit functorial factorizations. In the cited article [23] Isaksen makes a statement after theorem 1.1 that could be construed as stronger, but he probably only means to say that his construction of factorizations isn't functorial. Google doesn't let me read the relevant pages, so I can't verify this at the moment. –  Theo Buehler Feb 6 '11 at 8:19
    
@Theo: Thanks for the link, it definitely seems to point out that we don't know the answer for these pro-objects. –  Tyler Lawson Feb 6 '11 at 20:56
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Here is a recent attempt to construct functorial factorizations in a pro-category arxiv.org/abs/1305.4607 by Ilan Barnea and Tomer Schlank –  Boris Chorny Jun 20 '13 at 14:07

4 Answers 4

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.

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I should have said, you technically only need $R'$ to lift through the category of algebras for the pointed endofunctor part of the monad $\mathbb{R}$ (meaning, you can drop the coherence condition for the multiplication). This functor exists if and only if the arrows in the image of $R'$ lift naturally against their left factors with respect to the functorial factorization of $(\mathbb{L},\mathbb{R})$. And dually of course. –  Emily Riehl Feb 6 '11 at 20:04
    
Thanks a lot for your comments. Is the "usual" model category structure on Cat that you describe the one due to Thomason? –  Tyler Lawson Feb 6 '11 at 20:55
    
No, I meant the one with weak equivalences the categorical equivalences, fibrations the isofibrations, and cofibrations functors injective on objects. I often call this the ``folk'' model structure, but I seem to remember that Steve Lack, who has written on this sort of thing, doesn't like that name for some reason. –  Emily Riehl Feb 6 '11 at 22:08
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Dear Emily, Joyal feels the same way. The term that he prefers is the "canonical" model structure. Here's a link to his explanation: math.ntnu.no/~stacey/Mathforge/nForum/… –  Harry Gindi Feb 7 '11 at 5:36

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.

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In the following preprint: arxiv.org/abs/1305.4607, Tomer Schlank and myself show that there actually do exist functorial factorizations in Isaksen's strict model structure on pro simplicial sets (or more generally on $Pro(C)$, where $C$ is any proper model category with functorial factorizations). –  Ilan Barnea Dec 31 '13 at 10:35

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.

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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

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Dear Richard, are these algebraic notions compatible with exponentiation by small categories (That is, is this natural/algebraic WFS theory is an alternative to the theory of combinatoriality introduced by Jeff Smith)? –  Harry Gindi Feb 7 '11 at 5:31
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Sorry, what I wrote above seems to be complete rubbish. There is a perfectly good natural transformation G o F --> G under no assumptions at all, using only the functoriality of G. –  Richard Garner Feb 7 '11 at 5:56
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@Harry - one can always exponentiate a category equipped with an algebraic WFS by a small category, and again obtain a category equipped with an algebraic WFS. Whether or not this is a useful thing to do is another question. The WFS one obtains in this way is neither the injective nor the projective one; and even if the original one was cofibrantly generated, the induced WFS will typically not be, at least not in the classical sense. –  Richard Garner Feb 7 '11 at 6:00
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The point I really wanted to make, had my head been screwed on correctly, was that for an algebraic w.f.s., one obtains a natural transformation G --> G o G (by lifting) which is a section of the two projections G o G --> G, and moreover coassociative in the obvious sense. It is this transformation, which, for an ordinary w.f.s., would fail to be natural (or coassociative). –  Richard Garner Feb 7 '11 at 21:43
    
Actually, your side commentary is quite interesting. There are some monads in topology that require cofibrant input to be homotopically sensible, but typically will not produce cofibrant output. Having a more canonical version of cofibrant or fibrant replacement would be quite helpful. –  Tyler Lawson Feb 8 '11 at 2:30

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