Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

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"... ?