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I am just starting to learn about enriched categories, so excuse me if I am asking something trivial.

Suppose $\mathcal{C}$ is a $\mathcal{V}$-enriched category $\mathcal{C}$, with $\mathcal{V}$ very rich in structure, for example $\mathcal{V}$ could be the category of $R$-modules for a commutative ring $R$. If the underlying category of $\mathcal{C}$ is cocomplete, how close is $\mathcal{C}$ to being cocomplete as an enriched category? Is there a general theory that answers this question?

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Yes. There's no guarantee of cocompleteness just from that for the underlying ordinary category. But if you assume that $\mathcal{C}$ is cotensored over $\mathcal{V}$, then the existence of colimits in the underlying category implies the existence of conical colimits in $\mathcal{C}$, and if $\mathcal{C}$ is tensored, every colimit can be written in terms of conical colimits and tensors, so that $\mathcal C$ is cocomplete. This result appears near the end of Chapter 3 of Kelly's enriched category theory monograph.

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  • $\begingroup$ There seems to be some confusion here. If you have tensors (resp. cotensors) then your ordinary limits (resp. colimits) are enriched conical limits (resp. colimits); but you still need to have cotensors (resp. tensors) in order to get all weighted limits (resp. colimits). $\endgroup$ – Zhen Lin Apr 10 '16 at 10:04
  • $\begingroup$ Oh, you're right, thank you. $\endgroup$ – Kevin Carlson Apr 10 '16 at 15:07
  • $\begingroup$ Thank you both...but...what categories have tensors/cotensors? There are absolutely no examples in the nLab and also I could not find any by googling...Is it in general too much to ask for a category to be both tensored and cotensored or are there many examples? $\endgroup$ – MikeNerent93 Apr 10 '16 at 16:21
  • $\begingroup$ Well, there certainly are many, given that plenty of enriched categories are complete and cocomplete so in particular tensored and cotensored-presheaves, their localizations, algebras for sufficiently nice monads... $\endgroup$ – Kevin Carlson Apr 10 '16 at 17:44

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