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Matteo Capucci (he/him) (Sep 23 2020 at 16:22):Does the definition of limits/colimits in an enriched category differ from the usual one? Universal properties doesn't seem to really work since you're asking about 'existence and uniqueness' in an hom-object which could be something else than a set (e.g.a number in [0,infty])
Dan Doel (Sep 23 2020 at 16:30):Maybe that's another good reason to not define (co)limits with existence/uniqueness. :)
Matteo Capucci (he/him) (Sep 23 2020 at 16:35):I really like it usually :joy: but yeah, I guess
Dan Doel (Sep 23 2020 at 16:35):I imagine you can still do it, though, with generalized elements of the hom based in the monoidal unit or something.
Matteo Capucci (he/him) (Sep 23 2020 at 16:36):I think it's easier to just pick another equivalent definition, e.g. As adjoints to the constant functor
Dan Doel (Sep 23 2020 at 16:38):I think enriched (co)limits are often generalized to "weighted" (co)limits, though, which might even further generalize that choice of 'base'.
fosco (Sep 23 2020 at 17:08):Yes, co/limits in an enriched category differ substantially because the hom-objects may lack "elements". Yet, in many concrete cases (when, e.g. your base of enrichment is concrete over ) you can still find an element in the carrier of . The important point is that a colimit isn't always conical, in full generality, but just weighted (see chapter 4 here, but just because it's the most handy reference I can provide atm).
The special case of -weighted co/limits, and a lot of examples, are studied here: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0004972700002781 The definition for general bases of enrichment does not differ very much, and that's already a wide enough ground. Notably, laxified versions of co/limit can be reduced to weighted ones; it is a result of Street, contained in "Limits indexed by category-valued 2-functors".