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Morgan Rogers (he/him) (Apr 05 2020 at 11:45):A problem that I'm gradually attacking can be stated as follows:
How can I tell that a monoid is commutative without looking at it?
Morgan Rogers (he/him) (Apr 05 2020 at 11:48):In the representation theory of groups, I can tell a group is commutative if all of its -linear representations are direct sums of one-dimensional ones. But this seems like overkill: is huge, whereas the equational condition of commutativity is very simple. So can I tell that a monoid is commutative by only looking at its actions on sets, say?
Morgan Rogers (he/him) (Apr 05 2020 at 11:53):More precisely: given a category of (right) actions of a monoid (which is always a Grothendieck topos), there is a canonical way to recover a presenting monoid, which is necessarily unique when the monoid is commutative. But I really want to know what characterises the class of such categories, rather than having to reconstruct the monoid in order to check it. I'll post partial answers here in the near future.
Morgan Rogers (he/him) (Apr 07 2020 at 15:54):(This is one of the things I'm working on with @Jens Hemelaer)
sarahzrf (Apr 07 2020 at 20:49):interesting
sarahzrf (Apr 07 2020 at 20:50):when you say "a category of (right) actions of a monoid", do you mean always the full category or maybe some subcategory of it?
sarahzrf (Apr 07 2020 at 20:50):also, this is the same as presheaves, right?
Morgan Rogers (he/him) (Apr 08 2020 at 08:39):I mean the full category in the first instance, which is just the topos of presheaves on the monoid qua category; imposing continuity with respect to some topology gives a full (coreflective) subcategory which is also a topos. One can also ask about subtoposes (full reflective subcategories) but we need to complete work on how properties of monoids lift to their presheaf toposes to make meaningful progress on that front.