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One paper about this is Elliott's Towards a theory of classification, where such functors are called classification functors, and there is a nice collection of examples. (Though note that a category …

This is not really an answer to your question, but I'd like to point out that your "extension" definition probably isn't quite what you want yet. Of course this may well depend on your intended applic …

Here are some additional thoughts and examples. 1) The argument presented in the OP applies whenever one has a monoidal category $(\mathsf{C},\otimes,1)$ in which the unit object $1$ is a separator a …

Let us say that a monomorphism $U\subseteq A$ is nontrivial if $U$ is nonempty. Since every nontrivial monomorphism in $Set$ is split, every $F : Set\to Set$ takes nontrivial monomorphisms to monomorp …

Here are some remarks and pointers to the literature which are too long for a comment. The map $$ \mathsf{Dist}(X_1) \times_{\mathsf{Dist}(X)} \mathsf{Dist}(X_2) \longrightarrow \mathsf{Dist}(X_1 \t …

The existence of such an adjoint triple in particular requires the strongly connected components functor to be a left adjoint. But in fact this fact is not a left adjoint, since it doesn't preserve co …

A very natural enlargement of the category of measurable spaces is $\mathsf{Stoch}$, the category of measurable spaces and Markov kernels, with composition given by the Chapman-Kolmogorov equation. Th …

What can you do with the Kleisli category (of a probability monad) that you can't do with the Eilenberg-Moore category? The first paragraph of kirk sturtz's answer provides a good high-level summary …

Note: As pointed out by Peter LeFanu Lumsdaine in the comments, the example constructed in this answer does not work in its current form, since naturality of the associator fails. (This only affects t …

The algebras of this monad and closely related ones have been introduced by Pumplün and Röhrl. For example the introduction of a paper by Börger and Kemper provides a good summary, Pumplün and Röhrl …

You haven't specified what the morphisms in your "symmetric simplex category" are supposed to be, and there are two natural choices: Functors, or Natural isomorphism classes of functors. In the fi …

Tannaka-Krein duality shows how to recover a group $G$ from its category $\mathbf{Rep}(G)$ of finite-dimensional complex representations and the forgetful functor $F:\mathbf{Rep}(G)\to \mathbf{Vect}_{ …

Yes, it is possible to recover the manifold through the following steps: Smooth Serre-Swan theorem: $\mathbf{Vect}(X)$ is equivalent to the category of finitely generated projective modules over $C^\ …

The answer is yes: $1 \ast c = c \ast 1$ implies $c \in B$. To see this, let me first translate the statement into fully categorical language. If we restrict to $\|c\| \le 1$ wlog, then the elements …

I'd like to argue that the current definition is too minimal to allow for much theory development, since the only substantial axiom is the pasting condition. In particular, it would be possible to tak …