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In Bénabou's 1967 Introduction to Bicategories, he mentions that, in a forthcoming part II, he would study bicategories $\mathcal K$ in which each composition functor $$\circ_{A, B, C} \colon \mathcal …

It is well known that a distributor/profunctor $A \not\rightarrow B$, i.e. a functor $B^{\text{op}} \times A \to \mathrm{Set}$, is equivalent to a two-sided discrete fibration from $A$ to $B$. Further …

Let $T$ be a 2-monad on a nice 2-category $\mathcal K$, so that the inclusion $T\text{-}\mathbf{Alg}_s \to T\text{-}\mathbf{Alg}_l$ of the 2-category of (strict) $T$-algebras and strict $T$-algebra mo …

The answer is no: it is possible to have a terminal resolution without having an Eilenberg–Moore object. Consider the 2-category $\mathbf{DagCat}$ of dagger categories, dagger functors, and natural tr …

For each small 2-category $\mathscr K$, the 2-category $[\mathscr K^\circ, \mathrm{Cat}]$ of 2-functors and 2-natural transformations has a universal property: it is the free cocompletion of $\mathscr …

Let $\mathscr K$ be a small 2-category. It follows from $\mathrm{Cat}$-enriched category theory that the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors is given by the 2-categ …

For every pseudomonad $T$ on the 2-category of (locally small) categories $\mathbf{Cat}$, we can consider the 2-category of $T$-pseudoalgebras and pseudomorphisms $T\text{-PsAlg}_p$, which is equipped …

A partial answer is contained in Betti's Formal theory of internal categories (page 49), where he states that the bicategory $\mathbf{Dist}(\mathcal E)$ of $\mathcal E$-internal distributors is the fr …

In Formal category theory: adjointness for 2-categories, Gray defines a tensor product of 2-categories, now more commonly known as the lax Gray tensor product, which I will denote by $\otimes_l$. For …

Let $\mathcal E$ be a category with pullbacks. A span $A \xleftarrow a X \xrightarrow b B$ has a right adjoint in the bicategory $\mathbf{Span}(\mathcal E)$ if and only if $a$ is invertible in $\mathc …

Let $\mathscr C$ be a finitely complete category. Let $x, y$ be objects of $\mathscr C$. We can describe the universal property of freely adjoining a morphism $x \to y$ to $\mathscr C$: it comprises a …

Since this question was written, there is a paper 2-dimensional bifunctor theorems and distributive laws by Faul–Manuell–Siqueira based on this idea. In particular, they prove a "bifunctor theorem" fo …

This follows from Lemma 2.3 and Proposition 3.2 of Creurer–Marmolejo–Vitale's Beck's theorem for pseudo-monads, together with fact that the bicategorical Yoneda embedding preserves bilimits. Presumab …

Yes, Theorem 3.4 of Gambino–Lobbia's On the formal theory of pseudomonads and pseudodistributive laws establishes that pseudomonad morphisms are in correspondence with liftings to pseudoalgebras. They …

A locally small category $A$ is called total if its Yoneda embedding $A \to [A^\circ, \mathbf{Set}]$ has a left adjoint. Such categories are necessarily small-cocomplete (since the presheaf category …