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It is well known that (small) coproducts commute with connected limits in $\mathbf{Set}$. With which class of limits do finite coproducts commute? Ideally, we should furthermore like to know whether t …

My vague question is: given two diagrams in $D$, what comparisons between them induce a morphism of their limits? This is exactly the topic of Paré's paper Morphisms of Colimits: from Paths to Profu …

In Theorem 5.19 of Kelly's Basic Concepts of Enriched Category Theory, it is proven that a fully faithful functor $K \colon \mathcal A \to \mathcal C$ is dense if and only if $\mathcal C$ is the closu …

Assuming $\mathcal C$ and $\mathcal D$ are locally small and $\mathcal C$ is small-cocomplete, a functor $F \colon \mathcal C \to \mathcal D$ is a left adjoint if and only if it is small-cocontinuous …

For those coming across this question more recently, there is now an answer to the original question. In fact, the tricategory of pseudoprofunctors has been defined twice, independently, via different …

This construction is well known in enriched category theory, and has been studied in the increased generality of a class of weights (which recovers the finite discrete weights, describing finite (co)p …

I'm not sure I've followed your question exactly, so let me rephrase it to how I've understood it, and you can tell me if I've misunderstood. I shall use different letters to make sure I'm not acciden …

A locally small category $\mathscr C$ is called petite if, for every functor $F : \mathscr C \to \mathscr D$ with locally small codomain, and for every object $D \in \mathscr D$, the presheaf $\mathsc …

Let $S$ be a monad on a category $\mathbf C$. If $\mathbf C$ is cocomplete, and the category of algebras $\mathbf C^S$ admits reflexive coequalisers, then $\mathbf C^S$ is cocomplete. Thus, it is exis …

Two relevant papers are: Dupont's Interchange of filtered 2-colimits and finite 2-limits. Canevali's 2-filtered bicolimits and finite weighted bilimits commute in Cat. The former proves that finite …

A lax idempotent pseudomonad, also called a KZ doctrine or KZ monad, is a pseudomonad $(T, \mu, \eta)$ with the property that $T \eta \dashv \mu \dashv \eta T$. Lax idempotent pseudomonads were intro …

This is exactly the subject of the paper Coends of higher arity by Loregian and de Oliveira Santos.

We can view a category $C_0 \hookrightarrow C$ equipped with a specified subcategory as an $\mathscr M$-category, which is a category enriched in the category of injections and commutative squares. As …

A useful result due to Linton is that for a cocomplete category $C$ and monad $T$ on $C$, if the category of algebras $C^T$ admits reflexive coequalisers, then it is cocomplete (see here for a sketch …

For a category $\mathbf C$, the category of families of $\mathbf C$, denoted $\mathrm{Fam}(\mathbf C)$ is the free coproduct completion of $\mathbf C$. Explicitly, the objects of $\mathbf C$ are given …