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A recent question Why do we need model categories? reminded me of this long-standing confusion of mine -- I mentioned it in an answer there, and then decided to ask a separate question about it. I even dare not to use the tag.

What confuses me is this. There is an obvious resemblance between model categories and factorization systems. Yet, one thing goes totally wrong: in factorization systems, "left halves" tend to be "like epimorphisms" and "right halves" - "like monomorphisms". At the same time, in model categories cofibrations have distinct flavor of monos to them, and fibrations - of epis.

You see, I cannot even formulate this rigorously, yet I hope you agree that what I now described contains undeniable truth.

So what is this truth? Does this phenomenon have any explanation? Living with this puzzle for many years, the only consideration that I've been able to come up with is this: take one step from sets to categories. In sets, monos are just inclusions. In categories, the "correct" notion of mono starts to involve some amount of "epiness", since "good" monos in categories are full and faithful functors, and fullness involves some surjectivity condition. I don't even know whether there is some sort of such "first-step-consideration" from the opposite end, relating "betterness" of epis with some sort of additional requirements which have to do with monomorphy.

I repeat - this is a strange question: although it is full of most vague handwaving, I hope you agree that it touches on something very rigorous, which I just fail to capture.

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    $\begingroup$ I would say that there is an "obvious resemblance" between model categories and weak factorization systems, while what you describe, the (epi,mono) system, is an orthogonal factorization system. Orthogonal factorization systems are very special. For weak factorization systems (mono, epi) on set is a perfectly good example with generic features. See ncatlab.org/nlab/show/weak+factorization+system $\endgroup$
    – Chris Schommer-Pries
    Commented Nov 30, 2017 at 14:36
  • $\begingroup$ @ChrisSchommer-Pries It would be great if you could elaborate in an answer. Are there such (mono,epi) examples with some amount of uniqueness? I mean, in model categories the two classes do in a sense (although not without involvement of weak equivalences) determine each other, and moreover there is even some amount of uniqueness of liftings involved (again up to weak equivalences), just as in the orthogonal case. Is something similar observed in the (mono,epi) context? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Nov 30, 2017 at 14:39

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The (epi,mono) factorization system in Sets is part of a model structure on Sets whose weak equivalences are the epis, fibrations are monos and cofibrations are everything. This is a model for the homotopy theory of (-1)-truncated sets. One can also define a similar model structure on simplicial sets, where the weak equivalences are the maps which are surjective on pi_0 (i.e, homotopy epis) and whose fibrations are the Kan fibrations whose homotopy fibers are all (-1)-truncated (i.e., homotopy monos). This is a also a model for (-1)-truncated spaces/sets. A similar thing can be done in others contexts as well, for example, if you can replace simplicial sets by a model category of simplicial presheaves which presents a certain $\infty$-topos, then you can often left Bousfield localize $M$ such that the new weak equivalences are homotopy epis and the fibrations (at least between old fibrant objects) are the old fibrations which are also homotopy monos. The new model category will present the homotopy theory of (-1)-truncated objects in the original $\infty$-topos. I will go on a limb here and suggest that the (epi,mono) type factorization systems, which appear very typical from a 1-categorical point of view, are, from a model categorical point of view, essentially the particular case of (certain kinds of) (-1)-truncated homotopy theories.

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  • $\begingroup$ I like this very much, thanks! Can you provide any references? Do you know if this is functorial, providing an adjunction(reflection?)? Is there, driving analogy with skeleta and coskeleta of simplicial sets, also a co-truncation, giving another adjoint? Are there also (co)truncations at other levels? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Dec 1, 2017 at 5:20
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    $\begingroup$ For the construction of truncated model structures in categories of simplicial presheaves see Biedermann, On the homotopy theory of $n$-types. You can also look in the section on truncation in Rezk's toposes and homotopy toposes. The truncated model structure is obtained as a left Bousfield localization, so it's always related to the original one by a Quillen adjunction. As for cotruncation, cotruncated objects in $\infty$-topoi are very rare (in spaces it's only the empty set), but they do appear in other cases. For example, in rings the (-1)-cotruncated objects are the solid rings. $\endgroup$
    – Yonatan Harpaz
    Commented Dec 1, 2017 at 9:02
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This is probably not a full answer to your question, but I think it is a remark worth to make:

It is actually a couple of remarks:

1)If you have a weak factorization system where either the left class is made of epis or the right class is made of monos then you immediately get a unique (orthogonal) factorization system: any pair of diagonal filler in a lifting square will be either equalized by the mono on the right or co-equalized by the epi on the left.

2) model structure where one of the weak factorizations are unique/orthogonal factorization system are not the interesting ones. They corresponds to trivial model structures on reflective/co-reflective subcategories.

So It makes sense that the model structure we study in practice are "as far as possible" of satisfying the first condition. and this tends to means that cofibration will be monos and fibration will be epis (but it is not always the case).

Also notes that it follows from the axiom of model categories that trivial fibrations with cofibrant target are split epis and trivial cofibrations with fibrant domain are split monos, so unless the only weak equivalences are the isos you always gets that some cofibrations are non trivial monos and some trivial fibrations are non trivial epis.

Edit: so to sum up, this is closely related with Chris comment: it boils down to the difference between unique factorization system and weak factorization systems.

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  • $\begingroup$ Thank you, this is very interesting. And makes the question even more mysterious for me. Which is good. :) $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Nov 30, 2017 at 14:51

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