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Reid Barton (Oct 06 2020 at 18:29):Is there an easy way to tell when the topos of presheaves on (a small category) is a coherent topos?
Reid Barton (Oct 06 2020 at 18:31):If has finite limits then it's true (maybe more or less by definition, depending on the definition). In general it's not true. For example, if is a discrete category on an infinite set then the terminal object of the presheaf category is not even finitely generated.
Reid Barton (Oct 06 2020 at 18:33):For simplicial sets I think it's true and that the coherent objects are the finite simplicial sets, and basically the reasons are that the product of two simplices can be formed by taking finitely many simplices and quotienting out by an equivalence relation which is also generated by finitely many simplices. But it's not so clear to me how to reduce this to simple statements about the index category .
Fawzi Hreiki (Oct 06 2020 at 18:35): Fawzi Hreiki (Oct 06 2020 at 18:35):Thm 2.1 might answer your question
Reid Barton (Oct 06 2020 at 18:45):Yes, thanks! I think I came across this paper in searches but couldn't easily find a pdf copy.
Reid Barton (Oct 06 2020 at 18:50):I guess I already knew that simplicial sets are the classifying topos for a coherent theory, namely intervals. But this somehow seems quite indirect.
Fawzi Hreiki (Oct 06 2020 at 18:53):Check out chapter 8 of Mac Lane-Moerdijk
Fawzi Hreiki (Oct 06 2020 at 18:54):I've only briefly read through it but it explains this fact in more detail