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: theory: category theory

Topic: infinity-categorical generalisations of topological spaces

Madeleine Birchfield (Sep 05 2024 at 13:47):

A topological space consists of a set $X$ and a subframe $O(X)$ of the powerset frame $\Omega^X$, where $\Omega$ is the frame of truth values.

This should have an $\infty$-categorical analogue modulo size issues: a structure which consists of an $\infty$-groupoid $X$ and a sub-$(\infty, 1)$-topos $O(X)$ of the $(\infty, 1)$-functor $(\infty, 1)$-topos $\infty\mathrm{Grpd}^X$, where $\infty\mathrm{Grpd}$ is the $(\infty, 1)$-topos of $\infty$-groupoids.

Oscar Cunningham (Sep 05 2024 at 14:38):

The 1-categorical version is called an [[ionad]].

Madeleine Birchfield (Sep 05 2024 at 14:46):

So then I guess we can call these structures $\infty$-ionads.

Madeleine Birchfield (Sep 05 2024 at 14:48):

Now, every topological space is a preorder by way of the specialisation order

$$x \leq y \coloneqq \forall U \in O(X) .U(x) \Rightarrow U(y)$$

I wonder if there is a similar structure on $\infty$-ionads which turns an $\infty$-ionad $X$ into a $(\infty, 1)$-precategory.

If so, then the analogue of the $T_0$ separation axiom for topological spaces would be the complete Segal condition for the $(\infty, 1)$-precategory structure on an $\infty$-ionad turning it into a $(\infty, 1)$-category. And the analogue of the $T_1$ separation axiom would be the condition that the every morphism of the $(\infty, 1)$-category has a left inverse and a right inverse.

Kevin Carlson (Sep 05 2024 at 18:42):

There is indeed a right adjoint to the inclusion of categories in ionads that generalizes the poset of points of a space, and it seems plausible to hope that extends to the $\infty$-case.