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Bernd Losert (Jul 18 2023 at 17:06):I am reading Algebraic Theories by Ademak et al. But they only cover the case where the algebras are functors into Set. Is there a text that covers algebras whose codomain is not Set but something else like Top or Ord?
Kevin Arlin (Jul 18 2023 at 17:10):The topological/ordered part of the category theory is much harder in such cases than the algebraic part, so it doesn't make much sense to think of such a book as being on algebraic theories at all. Most of the time, topological gadgets have a topological functor to the category of set-based gadgets, so the categorical part of the theory becomes a matter of stacking the theory of topological functors on top of the book that you're reading.
Bernd Losert (Jul 18 2023 at 18:19):In other words, there is no such book. Maybe that's an opportunity for me to write one because I am compiling a list of results on this subject.
Nathanael Arkor (Jul 18 2023 at 18:31):What sort of results are you looking for?
Bernd Losert (Jul 18 2023 at 18:39):My research is mostly focused on continuous actions on convergence spaces. Right now I am doing research about continuous partial group actions on convergece spaces, and in particular whether this category has directed colimits.
I am basically looking for results about what kinds of limits/colimits are preserved/detected/reflected/lifted/created by the different forgetful functors from these kinds of categories.
Nathanael Arkor (Jul 18 2023 at 18:47):You may find Todd Trimble's page on multisorted Lawvere theories useful.
Leopold Schlicht (Jul 18 2023 at 22:43):Bernd Losert said:
I am reading Algebraic Theories by Ademak et al. But they only cover the case where the algebras are functors into Set. Is there a text that covers algebras whose codomain is not Set but something else like Top or Ord?
I asked a related question on MathOverflow and haven't received an answer, so I think it's safe to assume it's nothing people worked on a lot.
Jean-Baptiste Vienney (Jul 18 2023 at 23:47):Maybe the notion of enriched Lawvere theories would be useful?
Chris Grossack (they/them) (Jul 18 2023 at 23:52):You might also be interested in recent work of Yuto Kawase on "Relative Algebraic Theories", presented at CT2023. These are algebraic theories defined over a Locally Finitely Presentable category instead of (as is classical). In particular, this gives results for ordered algebras (taking to be the category of posets). It doesn't quite do what you want, since is not LFP, but it's still extremely interesting, and Kawase proves an analogue of Birkhoff's HSP theorem in this (very general!) setting.
You can find the slides here and the full paper here
Bernd Losert (Jul 20 2023 at 04:04):I will check these out. Thanks.