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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 …

Here is a class of examples different to Tom's: if your underlying monoidal category C is closed, then a strong monad on C is the same as a C-enriched monad, i.e. one that respects the enrichment of C …

That their Cauchy completions are equivalent.

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

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 …

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 …

Our adjunction isomorphism $\phi$ does not necessarily involve the assignment above. Yes, it does, by Yoneda's lemma. The bijection $\phi$ is natural, hence its value $\eta_x$ at the identity de …

The bicategory of elements of a Cat-valued functor is defined in e.g. Street's Fibrations in bicategories; it's the same as the usual one, with 2-cells as described here. Its property of classifying …

I don't know of any definition involving Kan extensions, but (co)ends can be expressed as (co)limits weighted by a hom functor (see e.g. here), so that for $F \colon C^{op} \times C \to D$ the end $\i …

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 …

I assume that by 'cartesian' you mean 'having cartesian products'. In that case the answer is no. You've more or less said this yourself: for $C^{\mathbf{3}} \to C^{\mathbf{2}}$ or the composite $C^ …

The answer (to the first question) is yes: reflections are always monadic, and the associated monad is idempotent.

Monadicity theorems like Beck's give sufficient conditions on a functor $U \colon B \to A$ with a left adjoint F to be equivalent (in the slice category Cat/A) to the forgetful functor out of the cate …

There was some discussion here about special cases of Yoneda's lemma, including the usual examples of Cayley's theorem for groups and Dedekind's embedding theorem for posets. It also seems that a goo …

First, a disclaimer: I am not even close to being an analyst. Second, I don't know of any applications of category theory to the areas of analysis that you mention. I don't think we have got to that …