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: deprecated: id my structure

Topic: yoneda o over-yoneda

Noah Chrein (Mar 17 2023 at 20:59):

I recently found an easier way of describing hierarchical relations for higher structures and part of this description is a composition of two pretty basic concepts:

the over-yoneda embedding:
$\mathbb C \to \mathbb Cat _{/\mathbb C}$ which maps $c \mapsto (\mathbb {C}_{/c} \to \mathbb C)$

and the presheaf yoneda embedding:
$\mathbb C \to [\mathbb C ^ \text{op}, \textbf{Set}]$

I am asking about the composition $y^{\mathbb C_{/c}} := \mathbb C_{/c} \to \mathbb C \to [\mathbb C ^ \text{op}, \textbf{Set}]$

This "elaborates" all the representable subobjects of the representable $y^c$. For example if $[n]:\Delta$ you can imagine the image of $y^{\Delta_{/n}}$ with $\Delta^n$ at the top, and each of its $\Delta^{n-1}$ faces below it, each face seen as a different object (because in $\Delta_{/n}$, $f_0 \neq f_1$), with the $\Delta^{n-2}$ faces below it and so on. The degeneracies are also present in the image but its simpler to ignore them for exposition.

just wondering if $y^{\mathbb C_{/c}}$ already has a name and if its been studied. Everything above generalizes to the oo-cosmic setting, but that's irrelevant to this Q (I think).

Morgan Rogers (he/him) (Mar 18 2023 at 12:35):

It's probably good to know that $[(\mathbb{C}/c)^{\mathrm{op}},\mathbf{Set}] \simeq [\mathbb{C}^{\mathrm{op}},\mathbf{Set}]/y(c)$, and that the square you can construct out of the fibrations commutes.