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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 …

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 …

In The Formal Theory of Monads, Street proves that a 2-category $\mathscr C$ admits the construction of algebras when the inclusion $\mathscr C \to \mathbf{Mnd}(\mathscr C)$ has a right adjoint. In Th …

An answer to the second question has been provided in the recent preprint What is the universal property of the 2-category of monads? by Lack–Miranda. The 2-monad morphism $\mathbf{Mnd}(\mathscr C) \t …

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 …

Proarrow equipments (also known as framed bicategories) are identity-on-objects locally fully faithful pseudofunctors $({-})_* \colon \mathcal K \to \mathcal M$ for which every 1-cell $f_*$ in the ima …

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 …

I asked Peter Johnstone about this citation, and he told me that he included the reference, despite not having a copy of the manuscript, since John Gray had been promising a book on the subject for ma …

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 …

It is known that fully faithful functors are closed under pushouts in Cat (e.g. Lemma 4.9 of this paper). Are locally fully faithful 2-functors closed under (strict) 2-pushouts in the 2-category 2-Cat …

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 $P : W° \times Y \to \mathbf{Set}$ and $Q : X° \times V \to \mathbf{Set}$ be profunctors, and let $L : X \to W$ and $R : Y \to V$ be functors. Suppose that $$P(Lx, y) \cong Q(x, Ry)$$ natural in $ …

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 $\mathcal K$ be a bicategory. For every pseudomonad $T : \mathcal K \to \mathcal K$, does there exist a 2-monad $S : \mathcal C \to \mathcal C$, where $\mathcal C$ is a 2-category biequivalent to …