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Stream: learning: questions

Topic: Examples of generalized simplicial objects (and beyond)?


view this post on Zulip Moana Jubert (Feb 05 2025 at 11:10):

Hello everyone,

While reading Hirschhorn's chapter on the Reedy model structure he mentions (15.1.14) that given a simplicial set X:ΔopSetX : \Delta^{\mathrm{op}} \to \mathbf{Set} one may think of a functor (elX)opM(\operatorname{el} X)^{\mathrm{op}} \to \mathcal{M} from the opposite of the category of elements of XX to some category M\mathcal{M} as a generalized simplicial object. As a special case, when X=X = \ast the category of elements is equal to Δ\Delta and we get the usual simplicial objects.


Do you have examples or uses of such concept elsewhere?

I have thought about it in two different ways:

view this post on Zulip Moana Jubert (Feb 13 2025 at 11:08):

Bumping. In the meantime, I figured that coends can probably be reformulated this way as well: given a functor H:Iop×ICH : \mathcal{I}^{\mathrm{op}} \times \mathcal{I} \to \mathcal{C} I define (elN(I))opC(\operatorname{el} N(\mathcal{I}))^{\mathrm{op}} \to \mathcal{C} to send any sequence x0xnx_0 \to \cdots \to x_n of composable arrows of I\mathcal{I} to H(xn,x0)H(x_n, x_0), with the convention that the sequence x0x_0 of zero arrows is sent to H(x0,x0)H(x_0, x_0). Then the colimit of this functor is equal to the coend of HH.