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This is in response to a comment at the bottom of enriched factorization system.
I want to propose a notion of enriched weak factorization system. On account of the enriched version of the algebraic small object argument there will be many examples. I don’t know whether this appears in the literature. (If so, please tell me! I’d love to have a reference.)
Suppose is a tensored and cotensored -category. Say maps and in the underlying category of satisfy the enriched lifting property with respect to each other if the map from the hom-object admits a section in . The latter object is defined by pullback in the usual way.
An enriched weak factorization system consists of left and right classes of maps and a -functorial factorization such that the left class is precisely those things that lift on the left in the enriched sense against the right class and dually. Note that an enriched weak factorization system is an ordinary weak factorization system: both classes are closed under retracts.
Cofibrantly generated examples are produced by the enriched version of the algebraic small object argument. Note that this use of cofibrantly generated does not coincide with the usual one; more on this below.
To see this, note that and satisfy the enriched lifting property if and only if there is a solution to the canonical lifting problem in of tensored with against . The enriched version of the algebraic small object argument starts by forming the coproduct of such lifting problems over (but not over squares; that’s subsumed by the tensor) and then pushing out. This produces the “step-one” functorial factorization. Algebras for the functor that sends a map to its right factor are precisely things that satisfy the enriched lifting property against . The algebraic small object argument produces an algebraic weak factorization system whose right class is the free monad on this; hence its algebras again satisfy the enriched lifting property. Because the enriched algebraic small object produces a -functorial factorization, it is easy to see that coalgebras for the comonad satisfy the enriched lifting property with respect to algebras for the monad. The retract closures of maps admitting (co)algebra structures define the classes of the enriched (algebraic) weak factorization system.
Here’s an easy example to keep in mind: in the category of modules of a commutative ring the ordinary wfs generated by has as right class the epimorphisms. The enriched wfs generated by this has as right class the epimorphisms admitting an -module section. A souped up version of this example shows that the (trivial cofibration, fibration) wfs in the h-model structure on unbounded chain complexes of -modules is cofibrantly generated in the -module enriched sense (though not in the usual one).
More details (rather hastily written, alas) can be found here in notes I’ve writing for a course I’m teaching this term. The relevant lecture is today’s (April 3).
What do you think of this?
Very nice! I think I remember talking about this with you at least once before, but IIRC I kept getting sidetracked and disappointed by the fact that the “enriched lifting property” you get is different from the one in SM7. But clearly this one is also useful! Can you say anything about the h-model structure on topological spaces?
Yes, I suppose it’s clear the Hurewicz cofibrations are closed under tensors and Hurewicz fibrations are closed under cotensors. Assuming closed cofibrations are also closed on tensors, it would follow that both weak factorization systems satisfy the enriched lifting property.
Similar remarks (that are easier to verify) hold for the h-model structure on unbounded chain complexes over a ring, where I’m think of enrichments over abelian groups or -modules (if the ring is commutative). Indeed, in the latter case, the (trivial cofibration, fibration) weak factorization system is generated by the set interpretted in the usual way. Furthermore, the functorial factorization produced by the enriched algebraic small object argument exactly factors a map through the mapping path space. Richard Garner first observed this back when you were still at Chicago.
…and clearly your memory is better than mine :) Now that you mention telling me about this, I’m starting to get a vague recollection. In any case, the mathematical content seems to have transferred to my subconscious. No wonder everything was so easy to prove!
Is knowing that the h-model structures are “cofibrantly generated” in the enriched sense useful? Does it let us do anything with them that we can usually only do with cofibrantly generated model structures?
I have added to the entry enriched factorization system a few lines to indicate how the non-orthogonal case works.
I have also added a References-section with a pointer to your (Emily’s) notes.
I was going to add an Examples-section remark that – if maybe in slight disguise – enriched lifting is present all over the place in Reedy model category theory: whenever the matching objects are co-represented by powering out of cell boundaries (as happens for all “EZ categories”). But now I am running out of steam for that.
To clarify, I don’t know of any generators for the (cofibrations, trivial fibrations) side; just the other one.
Actually I’d love to have a list of “things we can only do with cofibrantly generated model structures.” I do know that this observation allows you to construct projective model structures. What other constructions should I think about?
Bousfield localization?
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