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Stream: learning: questions

Topic: higher split coequalizers

Jonas Frey (Aug 12 2025 at 00:57):

Let $\Delta_m$ be the subcategory of the simplex category whose morphisms from $[n]$ to $[k]$ are monotone maps $f:[n]\to [k]$ preserving largest elements, i.e. $f(n)=k$.

This category appears in the walking/generic/free-standing adjunction as a hom-category, and I think it gives the shape of a kind of $\infty$-categorical split coequalizer, i.e. given a functor $F:\Delta_m^{op}\to \mathcal{C}$ into an arbitrary $\infty$-category $\mathcal{C}$, one can show that $F([0])$ is the colimit of the simplicial diagram given by precomposing $F$ with $(-+1):\Delta\to\Delta_m$.

Is something like this true? And if yes, what's a good reference?

Mike Shulman (Aug 12 2025 at 06:28):

That's one of my favorite facts! I know lots of references for it:

• Proposition 9.8 of J.P. May, The geometry of iterated loop spaces, 1972
• Sections 6-7 of Jean-Pierre Meyer. Bar and cobar constructions, I, 1984
• Lemma 6.1.3.16 of Jacob Lurie, Higher topos theory, 2009
• Theorem 5.3.1 of Riehl-Verity, The 2-category theory of quasi-categories, 2015
• Lemma 7.5 of Shulman, All (∞,1)-toposes have strict univalent universes, 2019

Jonas Frey (Aug 12 2025 at 10:53):

Great, thanks so much! I see you paper even has the part about the free adjunction!