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stub for model structure on algebraic fibrant objects
just the bare minimum for the moment, no time...
expanded the entry model structure on algebraic fibrant objects: added an Idea-section and an Applications-section.
Also created stub entries
and tried to add all the relevnt cross-links to and from other relevant entries.
and interlinked with simplicial T-complex
I wrote out at model structure on algebraic fibrant objects all the details for the proof of the monadicity and the model structure, or rather all the details establishing monadicity and solidity of the forgetful functor, then pointing over to solid functor for the conclusion.
I added further details. Effectively, I have now spelled out the full proof and all the constructions that go into it.
added statement and proof that $Alg C$ is locally presentable/combinatorial if $C$ is – here
I think there is a terminological problem here, which I’m sorry I didn’t catch back when it was getting written on the page. A solid functor has semi-final lifts for all structured sinks, not just small ones; but as far as I can see, in general algebraic fibrant objects only have semi-final lifts for small structured sinks. (Though it seems possible that some kind of well-copoweredness condition might make them actually solid.) I don’t think I’ve heard of a name for functors with this property, but we could invent one since it seems a natural condition. Quasi-solid? Small-solid? Plastic? Supersolid?
By the way, I just learned that this paper of John Bourke proves Nikolaus’s result for any combinatorial model category without the assumption that acyclic cofibrations are monic. Instead of the semi-final lifts he uses the path object argument to prove acyclicity. We should add this to the page too, although certainly we should keep the explicit description of Nikolaus’s method too since it has other applications.
There’s also some abstract discussion of such semi-final lifts for categories of “algebraic injectives” in section 4 of this paper (also by Bourke). Maybe it would be better to move some of the results that work abstractly to a new page like algebraic injective.
Re #7: I like small-solid best, as the most descriptive.
Hmm… actually, now that I was reminded that all locally presentable categories are well-copowered, perhaps we do get full solidity in that case. Suppose we have a (possibly large) $U$-structured sink $\{U Y_i \to X\}$. Each of those maps induces its own map $X \to X_{H_i}$, which is an epimorphism. In general there might be a large number of these, but in a well-copowered category we can take cointersections of even large families of epimorphisms to obtain a single epimorphism $X\to X_{H}$, which ought to be what we need to make the rest of the proof go through.
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