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Stream: learning: questions

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

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 $X : \Delta^{\mathrm{op}} \to \mathbf{Set}$ one may think of a functor $(\operatorname{el} X)^{\mathrm{op}} \to \mathcal{M}$ from the opposite of the category of elements of $X$ to some category $\mathcal{M}$ as a generalized simplicial object. As a special case, when $X = \ast$ the category of elements is equal to $\Delta$ 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:

• It is well-known that $\mathbf{sSet}/X \simeq [(\operatorname{el} X)^{\mathrm{op}}, \mathbf{Set}]$ (and, of course, for any base category other than $\Delta$), therefore a functor $(\operatorname{el} X)^{\mathrm{op}} \to \mathcal{M}$ could be viewed as an "$\mathcal{M}$-valued fibration over the base space $X$" perhaps?
• The ideas of colimit and/or cover seem to be "better" formulated in terms of generalized simplicial objects. Given a diagram $F : \mathcal{I} \to \mathcal{C}$ its colimit may be computed as the coequalizer of a coproduct: I claim that this should be seen as the very particular, 1-category theoretical case of taking the nerve of $\mathcal{I}$ and then looking at the functor $(\operatorname{el} N(\mathcal{I}))^{\mathrm{op}} \to \mathcal{C}$ sending a sequence of composable arrows to the evaluation of the first object at $F$. If we take the left Kan extension along the projection to $\Delta^{\mathrm{op}}$ and truncate we exactly get the coequalizer/coproduct diagram. Truncation amounts to restricting along a final functor, which I feel is a distinctive feature of working "in low dimension"... ?

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 $H : \mathcal{I}^{\mathrm{op}} \times \mathcal{I} \to \mathcal{C}$ I define $(\operatorname{el} N(\mathcal{I}))^{\mathrm{op}} \to \mathcal{C}$ to send any sequence $x_0 \to \cdots \to x_n$ of composable arrows of $\mathcal{I}$ to $H(x_n, x_0)$, with the convention that the sequence $x_0$ of zero arrows is sent to $H(x_0, x_0)$. Then the colimit of this functor is equal to the coend of $H$.