# Questions tagged [2-categories]

The 2-categories tag has no usage guidance.

134
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### Example of a pseudomonad on Cat whose pseudoalgebras are not the pseudoalgebras for a 2-monad

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 ...

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### Relations with "for each" composition and its properties (coming from profunctors with end composition)

$\newcommand{\sq}{\mathbin{\square}}$The usual composition $S\diamond R$ of two relations $R\colon A⇸B$ and $S\colon B⇸C$ is defined as follows:
For each $(a,c)\in A\times C$, we declare $a\sim_{S\...

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### Looking for Lawvere's "Closed categories and biclosed bicategories" lecture notes

The nLab page on closed bicategories reads
Closed bicategories were introduced by Lawvere in unpublished lecture notes Closed categories and biclosed bicategories (1971).
This work has also been ...

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### Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II

This is the second part to a previous question regarding left Kan extensions/lifts in the bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations, which has now been split into two ...

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### Existence and characterisations of left Kan extensions and liftings in the bicategory of relations I

The bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations has right Kan extensions and right Kan lifts¹, however I believe it does not have all left Kan extensions/lifts.
Is it ...

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### Existence and explicit descriptions for left and right Kan extensions and lifts in bicategories of spans

Given a category $\mathcal{C}$, it follows from Proposition 4.1 of Day's
Brian Day, Limit spaces and closed span categories, Lecture Notes in Mathematics, 420, 1974 (doi:10.1007/BFb0063100).
That ...

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### Universal property of Isbell duality

Let's take $\mathrm{C}$ be a category, let's have an adjunction $(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves(C)} \leftrightarrows \mathrm{Presheaves(C)}$. One such adjunction is ...

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### Comonoid homomorphisms in the bicategory of profunctors

Cartesian bicategories axiomatize the intuitively evident but mathematically elusive "cartesian" product on bicategories such as Rel, Span, and Prof. An important concept for cartesian ...

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### 3-functoriality of the lax Gray tensor product

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 ...

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### What are the internal adjunctions in the bicategory $\mathsf{Span}$?

Recently I've been trying to understand spans better, in particular how they relate to relations, as both may be thought of as "multivalued functions between sets" (see Bruni and Gadducci - ...

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### $\mathbf{E}_n$-algebras in nerves of 2-categories

In Example 5.1.2.4 of Higher Algebra, Lurie explains how there is a bijection between equivalence classes braided monoidal structures on one-category $\mathcal{C}$ and $\mathbf{E}_2$-algebra ...

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### Examples of bilimits that aren't 2-limits, and some related questions

Recently I've received an email from Sori Lee about an earlier question I had asked, and we ended up with a number of questions about 2-limits and 2-bilimits which I couldn't quite answer, and decided ...

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### Does the bicategory of additive categories admit bicolimits?

By bicolimit I mean what Kelly means in its "Elementary observations on 2-categorical limits". If we have a diagram (pseudofunctor) $G\colon\mathcal P\to\mathcal K$, the bicolimit of $G$ is ...

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### Classifying spaces of crossed modules

Let $\mathcal{G}$ be a strict topological $2$-group, i.e. a strict $2$-category with a single object, a space of invertible $1$-morphisms, a space of invertible $2$-morphisms and continuous structure ...

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### Adjoining a morphism to a finitely complete category

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 ...

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### Preservation of lax limits in categories of functors and lax natural transformations

Let $f:\mathbb{C} \to \mathbb{D}$ be a functor of 2-categories and let $\operatorname{Fun}(\mathbb{C},\operatorname{Cat})^{\operatorname{lax}}$ denote the 2-category of functors and lax natural ...

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### Morphisms of fibered categories which are compatible with the chosen cleavages

Let $\pi \colon F \rightarrow C$ and $\pi' \colon F' \rightarrow C$ be two fibered categories over the category $C$. A morphism from $\pi \colon F \rightarrow C$ to $\pi' \colon F' \rightarrow C$ is a ...

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### Reference for $2$-adjoint pairs and preservation of $2$-colimits

I know that similar questions have been asked in the past and, even if some useful explanations/clarifications have been given (so now I know or, at least, I believe I know what results I should ...

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### Conditions for natural transformations of weights to induce adjunctions of weighted limits

Suppose we have:
-) A $2$-category $\mathsf{J}$
-) A natural transformation of $\alpha : M \Longrightarrow W : \mathsf{J} \longrightarrow \mathsf{Cat}$
-) A functor $X:\mathsf{J} \longrightarrow \...

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### 2-completeness of stacks

I am looking for a reference which discusses the 2-categorical properties of the 2-category $St(C,J)$ of stacks and the stackification $\dashv$ inclusion of presheaves 2-adjunction.
My stacks are ...

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### What is a correct notion of an internal pseudofunctor?

Let $C$ be a category internal to a category $K$. It is well known (for example see Proposition 2.4 in the paper Higher Dimensional Algebra VI: Lie 2-Algebra by Baez and Crans https://digitalcommons....

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### Is totality a (large) cocompleteness condition?

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 ...

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### Lifting adjunctions along a localisation of 2-categories

Let $\mathcal S$ be a base category and let $Fib_\mathcal S^{split}$ be the 2-category consisting of split fibrations, fibered functors which strictly preserve the splitting and vertical ...

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### Overloading of the word "local" in category theory

The word "local" in category theory does not seem to have a precise definition in itself but it often appears as part of other terminology. To my understanding, it is then used in the ...

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### Which direction does a lax dinatural transformation go?

In 2-category theory we often encounter notions with noninvertible coherence 2-cells, in which case there is a choice about which way the 2-cell should point. Generally, one of these directions is ...

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### Do the representations of a 2-functor naturally form a contractible 2-category?

In 1-category theory a representation of a 1-functor $F:C\to Set$ is a 0-cell $X$ in $C$ together with a universal element $u\in FX$ such that the transformation $C(X,-)\to F$ is an isomorphism (=a 1-...

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### Unitors in Morton's definition of a double bicategory

I am confused by the definition of a double bicategory by Morton in (Definition 3.1.1. in https://arxiv.org/abs/math/0611930), but I need it, so I want to make sure I understand it correctly before I ...

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### Can a 2-category be defined by an endofunctor?

Take a category $C$. Define an endofunctor $F:C \rightarrow C$ that is identity on objects. This maps morphisms to morphisms, preserving source and target. This suggests that the endofunctor endows ...

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### If a monad in a 2-category admits a terminal resolution, does it admit an Eilenberg–Moore object?

Let $T = (t, \mu, \eta)$ be a monad on an object $A$ of a 2-category $\mathcal K$. In The formal theory of monads, Street proves (Theorem 3) that if $l \dashv r$ is the canonical adjunction associated ...

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### Do pseudo 2-limits commute?

It is a well-known fact that if $F:\mathcal{C}_1\times\mathcal{C}_2\rightarrow \mathcal{D}$ is a functor (between 1-categories), then $F$ has a limit if and only if $F:\mathcal{C}_1\rightarrow Fun(\...

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### Is the category of cochain complexes with terms in an additive category a 2-category?

$\def\hom{\operatorname{Hom}}
\def\bbZ{\mathbb{Z}}$This question is a follow-up to this other one. There the OP asks whether "the category of chain complexes" (can be interpreted in several ...

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### Examples and counterexamples to Lack's coherence observation

In Lack's A 2-categories companion, he states
There are general results asserting that any bicategory is biequivalent to
a 2-category, but in fact naturally occurring bicategories tend to be ...

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### What are the algebras for the laxification 2-monad?

Let $C$ be a small 2-category. Let $[C , Cat]$ denote the 2-category of strict functors to $Cat$, 2-natural transformations, and modifications. Let $[[ C, Cat ]]$ denote the 2-category with the same ...

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### 2-categories for the working algebraic geometer

I study algebraic geometry / number theory and from time to time I stumble upon 2-categorical (co)limits. I have two main examples in mind:
Example 1) In étale cohomology, the (triangulated) derived ...

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### Reference request for facts about bi(co)descent objects

I know the following facts are true, but I struggle to find adequate references for them:
Let $T$ be a pseudo-monad on a bicategory $\mathcal{C}$, and let $A$, $B$ be pseudo-algebras for $T$. Then, ...

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### Free $2$-category on a $2$-quiver

The construction of the free category on a quiver is standard in category theory.
Is there some $2$-dimensional analog of a quiver such that each $2$-quiver freely gives rise to a $2$-category? How ...

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### A 2-category of abelian categories?

Is there a reasonably well-behaved $2$-category of abelian categories, and if so, what are its objects (perhaps one needs to restrict one's attention to finite abelian categories ?), $1$-morphisms (...

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### Horizontal closure of 'almost $2$-categories'

Is there a reference discussing the notion of 'free horizontal closure' for an 'almost $2$-category', where all that's missing are some horizontal composites of $2$-cells?
The motivation for this ...

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### Adjoints to the forgetful functor from the $2$-category of monads

For the purpose of this post, we will identify a monad $T$ on a category $\mathcal{C}$ with a lax $2$-functor $T:{\bf 1}\to\mathfrak{Cat}$ such that $T(*)=\mathcal{C}$.
There is an obvious forgetful ...

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### How do the various homotopy 2-categories compare?

There are various models of $\infty$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in ...

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### (Co)limits in lax functor categories

Let $\mathcal I$, $\mathcal C$ be $2$-categories (or $(\infty, 2)$-categories, I'm interested in both cases) and assume that $\mathcal I$ is small, $\mathcal C$ has enough weighted (co)limits as you ...

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### Examples of 2-categories with multiple interesting proarrow equipment structures

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 ...

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### Universal property of the V-Mat construction

Internal categories and enriched categories can both be realised as monads in certain bicategories. If $\mathcal E$ is a category with pullbacks, then a monad in $\mathbf{Span}(\mathcal E)$ is a ...

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### Are locally fully faithful 2-functors closed under 2-pushout in 2-Cat?

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 ...

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### Adjunctions with respect to profunctors

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 $...

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### 2-morphism between circuits in a monoidal category

We are used to seeing equations between circuits in monoidal categories like this
I am wondering about morphisms between string diagrams. I think they are 2-cells. I found an example of a 2-cell ...

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### Adjoining extensions in bicategories

Given a bicategory $\mathcal K$, is there a universal construction of a bicategory $\mathcal K'$ and faithful locally fully faithful pseudofunctor $\mathcal K \hookrightarrow \mathcal K'$ such that ...

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### Long exact sequence of cohomology from 2-groups

I am trying to understand the following from Principal Infinity Bundles - General by Nikolaus, Schreiber and Stevenson.
So following the reference there to Nikolaus-Waldorf tells us that given any (...

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### Coherence for biadjunctions

There are many ways to give a definition of a biadjunction. For instance, one may say that a pseudofunctor $F:\mathcal{C}\rightarrow \mathcal{D}$ is left biadjoint to $G:\mathcal{D}\rightarrow \...

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### Examples of (co)lax idempotent pseudocomonads on Cat

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 ...