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John Onstead (May 02 2024 at 14:25):When you want to enrich in a category, a good base of enrichment usually consists of a symmetric closed monoidal category, perhaps also requiring limits or colimits. If the category you want to enrich in lacks this structure, it becomes difficult to describe a consistent Yoneda theory for categories enriched in it, and do all the things you ordinarily would want to do in category theory.
So here's my question: why can't you "enrich" in an arbitrary category C, perhaps a monoidal category without these key structures, simply by enriching instead in its presheaf category [C^op, Set]? After all, presheaf categories are toposes, so they have all the good properties you would want from a base of enrichment. The only requirement would then to ensure that all hom functors from categories enriched in [C^op, Set] into [C^op, Set] land in the image of the yoneda embedding C -> [C^op, Set] so that all hom objects can be thought of as being objects of C.
Nathanael Arkor (May 02 2024 at 15:35):So here's my question: why can't you "enrich" in an arbitrary category C, perhaps a monoidal category without these key structures, simply by enriching instead in its presheaf category [C^op, Set]?
You can, and this leads naturally to the notion of [[locally graded category]].
Nathanael Arkor (May 02 2024 at 15:38):If the category you want to enrich in lacks this structure, it becomes difficult to describe a consistent Yoneda theory for categories enriched in it, and do all the things you ordinarily would want to do in category theory.
Although I would also counter this: I think it is an easy impression to get, because most authors work with a lot of structure on their bases of enrichment for convenience, but category theory behaves well even without all of these assumptions. One of the main obstructions it that a lot of the basic theory has not been developed (e.g. to the level of Kelly's Basic concepts), which makes it inconvenient to do.
Nathanael Arkor (May 02 2024 at 15:39):(This is one of the motivations for developing formal category theory.)
John Onstead (May 02 2024 at 22:01):Thanks, this seems to answer the question. I have heard of a "locally graded category" before. but I suppose I hadn't connected these two ideas together! I'll certainly have to check out more about them and what they can be used for.
Morgan Rogers (he/him) (May 03 2024 at 06:12):It's not true that you can enrich in an interesting way over an arbitrary monoidal category, and I expect that may remain the case even if one extends to presheaves. See this MO answer I wrote about the fact that enrichment sometimes trivialises. Of course there will be at least some morphisms to consider once you take presheaves on , so I'll be interested to hear if that gives some non-trivial categories or if you just end up being essentially Set-enriched.
Sam Staton (May 15 2024 at 08:02):Sorry I'm late. Just to note following your original question @John Onstead that even Kelly does indeed suggest to use this Yoneda universe enlargement at the end of Ch 2 and spelt out in 3.11 of Basic concepts.
Sam Staton (May 15 2024 at 08:09):@Morgan Rogers (he/him) it's a nice illustration in your MO post. I agree that enrichment in is useless. However, enrichment in is really useful. For example, a PROP can be regarded as the same thing as an enriched monad on that is commutative and preserves colimits. Maybe this is the kind of non-trivial example you were looking for.
(I found this angle on PROPs useful because it suggested to weaken the colimit preservation requirement to sifted-colimit preservation. This gives an algebraic framework combining features of both Lawvere theories and PROPs, which I used for presenting quantum theory, here. That set-up is also -enriched.)