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James Deikun (Jan 08 2023 at 13:24):Maps in the simplex category factor into "faces" and "degeneracies". Maps of simplicial sets, however, are not quite that simple. Nonetheless, there seems to be an intuitive notion of when one simplicial set is a "degenerate version" of another. It's surprisingly hard to pin down formally, though, even for 1-skeletal simplicial sets. Is there some established way to know when a map of simplicial sets is "pure degeneracy"? (Or "pure face" for that matter?)
Amar Hadzihasanovic (Jan 08 2023 at 13:39):Is there a reason you would like to reject the “obvious” answer that epimorphisms of simplicial sets generalise degeneracies, and monomorphisms generalise faces?
Amar Hadzihasanovic (Jan 08 2023 at 13:41):(Since restricted to the simplex category as a full subcategory of simplicial sets, monos are precisely faces and epis are precisely degeneracies)
Amar Hadzihasanovic (Jan 08 2023 at 13:42):(I'm counting identities as both faces & degeneracy, you may want to throw in “non-iso” if you'd rather not)
James Deikun (Jan 08 2023 at 13:52):Epimorphisms of simplicial sets don't really match the intuitive notion, since seemingly they contain "gluings" as well, like the map that glues all the vertices of a given simplex together. Monos might match the notion of "pure face" a little better since mapping a ("nondegenerate" or "trivially degenerate") simplex to a nontrivially degenerate simplex seems to require identifying some of the higher degenerate simplices on it.
James Deikun (Jan 08 2023 at 14:13):The category of simplicial sets is the Kleisli (and EM) category of a [[parametric right adjoint]] monad on the category of semisimplicial sets. I'm now wondering if "-generic" is actually the answer I'm looking for.
Amar Hadzihasanovic (Jan 08 2023 at 19:56):Oh, I see. I think I understand. The idea is that you should be able to factorise a map as
(collapse all simplices that are mapped to lower-dimensional simplices)
followed by
(do all identifications/gluings of simplices of the same dimension & include the image in the codomain)
and the first part is the “-generic” part, and the notion that you are looking for?
James Deikun (Jan 08 2023 at 20:04):Yeah, that's the idea.