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James Deikun (Nov 20 2025 at 13:02):I know there's work about the tensor product and hom and (small) biproducts of presentable categories; is there something about the further linear algebra (modules and tensors over monoids, e.g.)? I recently worked out that a commutative monoid in presentable 1-categories seems to be a presentable [[Benabou cosmos]] and a module, a copowered and powered enrichment of a presentable category. It sounds interesting, and surely someone must have noticed before.
Chris Grossack (she/they) (Nov 20 2025 at 18:15):Yes, there's definitely work in this area since these kinds of categorified monoid actions are very important for topological quantum field theories. One good reference that you might already know is the "EGNO" book Tensor Categories, which is not in the presentable setting but still shows what kinds of things one can do with categorified modules/etc. It's my impression that in the presentable case a lot of this stuff can be found in at least one of Lurie's books (probably higher algebra, as a special case of something to do with operads?) but I'm not actually 100% sure on that
James Deikun (Dec 12 2025 at 02:28):The first talk of the Pacific Category Theory seminar by Richard Garner was quite heavily involved in this although it wasn't advertised that way. If you drop the idea that you're dealing with symmetric pseudomonoids, then you can look at what you get just as a pseudomonoid, which is an asymmetric presentable [[Benabou cosmos]] for a monoid and a module is, I think, an enriched category that has powers and copowers on one side? And then for each presentable category you can create a bicategory of module structures (enrichments) that is fibered over the bicategory of pseudomonoids.
And the internal hom of in presentable categories, which consists of its cocontinuous endofunctors, has a canonical pseudomonoid structure (it's actually a straight-up monoid in its standard presentation, but equivalent presentations that are easier to understand and work with might lose the strictness). This comes along with a canonical enrichment of , which is determined by the associated copower, functor application. This enrichment is a terminal object in the bicategory above, making it a classifying object for all enrichments of .
This means you can compute one "most informative" enrichment for any presentable category and if you know some way of classifying the cocontinuous endofunctors on this information translates to less-tautological information about its possible enrichments. Garner gave lots of interesting and surprising examples, so when the recording of the talk comes up I highly recommend it.