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Here is one way to look at it: if V is a monoidal category and $\mathbf{B} V$ is the corresponding one-object bicategory, then a V-category in the usual sense is the same thing as a lax functor $\math …

Yes -- filtered/directed colimits commute with finite limits. See Mac Lane, Categories for the Working Mathematician, theorem IX.2.1. Edit: Oh, I thought you meant colimits, but it seems you meant …

Edit: the question was unclear, so prompted by the comments and answers I've tried to clarify things. A colleague has asked me whether, or under what conditions, it is possible, given a sufficiently …

I haven't seen this written down anywhere, but I've worked it out myself in a paper I'm working on. The paper isn't quite ready, so I'll just sketch the idea here. As Zhen says, you need to generali …

Yes, here: R. Garner, M. Shulman, Enriched categories as a free cocompletion. arxiv.

I don't think I have seen anything like this published before, but I have written up a similar definition here (see here too). One thing in your definition I would take issue with is that your 2-coen …

This page says that you may be able to get a copy by emailing Andrée Ehresmann. I don't know the exact answer to your question, but if you can't find a reference then it may be worth recalling that: …

I don't have it with me, and I can't recall the exact details, but I'm pretty sure Lambek & Scott's Introduction to Higher-Order Categorical Logic (link) is what you're looking for. In particular, th …

The following result seems to be well known: If T is a (V-)monad on a (V-)category C, then the forgetful functor $U^T \colon C^T \to C$ creates any limits that exist in C, and any colimits that exis …

Apart from Todd's recommendations, which I'd second, for monads and Lawvere theories there is Nishizawa and Power, Lawvere theories enriched over a general base, JPAA 213, 2009, and the references the …