Post #1639319
2026-04-23 20:31 UTC
Replies (3)
-
@cbaberle@mathstodon.xyz 2026-04-23 20:59
@maxsnew I tend to think of sheaves in terms of converging processes. To back up, think of a category as a collection of "interfaces" and ways of moving/translating information from one interface to another. Then a presheaf X can by thought of as assigning a set X(I) of "systems with interface I" to each object I of the category, which is compatible with the structure of translations in that you if you have a system with interface J and a translation f : I -> J, you can construct a system with interface I. Now suppose you additionally have a coverage on your category. Informally, what a covering family f_i : U_i -> U tells you is the U_i "converge to" U via the f_i, in the sense that they jointly account for all the "information" in U. Then the sheaf condition tells you that, if you have a collection of systems s_i : X(U_i) that are suitably compatible with such a covering family, then there is a unique system in X(U) which they "converge to" under that coverage.
-
@trebor@types.pl 2026-04-23 21:11
@maxsnew All propositions are "open" in a locale, because we can't even talk about non-open things. We can think about sublocales (which are not given by propositions but some kind of modality), but those are cursed. Maybe you can talk about openness in ionads though.
-
@jonmsterling@mathstodon.xyz 2026-04-23 21:13
@maxsnew Yes — and Uemura has already begun to develop synthetic topos theory using this analogy. There are other recent perspectives that may also be of interest, which perhaps @olynch can comment on…