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That their Cauchy completions are equivalent.

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 …

What you're asking is whether every limit in a functor category $[B,C]$ is a pointwise limit. The answer is yes if C is complete, but not always otherwise. Kelly gives an example in Basic Concepts o …

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 …

The second definition looks like the 'lax comma category' $C // T$, where a morphism $f \to f'$ is given by a 2-cell $f \to f'\phi$. The defining universal property is the same as for comma objects, …

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 …

As Fernando points out, you can't talk about isomorphisms in a semicategory, which means that they won't be as much use as categories in describing universes of mathematical objects. But the category …

This is exactly the notion of a closed category. See Eilenberg and Kelly's article in the 1965 La Jolla proceedings (Springer 1966). I think they also describe categories enriched in a closed catego …

One nice result is Street's theorem 14 in The formal theory of monads, generalized in Elementary cosmoi, which says that $C^T$ is isomorphic to the full subcategory of $[(C_T)^{\mathrm{op}}, \mathrm{S …

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 …

Firstly, the $W$-weighted limit $\lim^W F$ is defined to be a representation of $\operatorname{Psh}_D(W,C(-,F-))$; the definition you've given isn't even well-typed. There is no difference at all in …

Apparently it's 'well known' that if $P$ is a presheaf on $C$ then there is an equivalence $\widehat{C}/P \simeq \widehat{\int P}$, where $\int P$ is the usual category of elements and $\widehat{C} = …

According to the Elephant, and these notes, an object X in a category C is indecomposable if given an isomorphism $X \cong \coprod_i U_i$ there is a unique $i$ such that $X \cong U_i$ and $U_j \cong 0 …

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 …

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 …