Gabriel Saadia: Categorified convergence for categorified spaces

Abstract

There is a wild analogue of the notion of sequential convergence in a topological space; for a family of points and an ultrafilter on the indexing set one can define what it means for a point to be a limit of this sequence. It has been noticed by Barr [1] that this notion of convergence is enough to recover all the topological structure of a space. From this point of view the topological structure is now a generalized poset on the set of point extending the usual specialization order.

We categorify this story by replacing topological spaces with Grothendieck topoi. We can define a convergence notion indexed by ultrafilters that extends the categorical structure on the points of the topos. In the coherent case this convergence notion relates with the usual ultraproduct of models of a first-order theory and connects with Makkai's ultracategories [2].

We will prove that a Grothendieck topos with enough points can be recovered from this generalized convergence structure, thus extending the Makkai–Lurie duality between coherent topoi and ultracategories to all topoi with enough points.

[1] M. Barr, (1970), Relational algebras, 10.1007/BFb0060439
[2] M. Makkai, (1987), Stone duality for first order logic, 10.1016/0001-8708(87)90020-X.