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Tim Hosgood (Apr 03 2023 at 23:41):a while ago I asked on math.stackexchange if there was something like pi-base but for categories, where you could search for categories by property, by name, or as counterexamples to theorems (check out how cool pi-base is for this if you haven't seen it already! you can search for things like "T5=>T4?" and see some spaces that are witness to the fact that this isn't true).
recently, one of the maintainers for pi-base said that if any category theorists are interested in building a version of this for categories, then to reach out. i would love to do this but don't have the time or energy right now, but figured that there would probably be a bunch of people on this forum that would!
Morgan Rogers (he/him) (Apr 08 2023 at 21:47):I have been developing a network of relationships for geometric morphisms (and hence properties of toposes); I would love to develop that into a pi-base database at some point, but I guess a category pi-base is a prerequisite to that existing :rolling_on_the_floor_laughing:
Reid Barton (Apr 09 2023 at 05:56):That sounds really useful. I'm not sure if a general category theory pi-base would be a prerequisite. I would say "properties of geometric morphisms of topoi" is a partially independent subject, closer to geometry.
Ralph Sarkis (Aug 06 2023 at 19:57):There is now a database of small categories : https://smallcats.info