# Questions tagged [2-categories]

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116
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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 in 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 ...

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### Strict 2-Category with Lax Tensor?

I am faced with an operation $\otimes$ on a strict 2-category $C$ which walks and talks like a tensor, except that it only satisfies a "lax" interchange law. To be precise: for any $f,g,h,k\...

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### Factoring a natural transformation through a functor

This seems fairly similar to density. Suppose I have three categories $A,B,C$, and a functor $L: B \to C$ so that every natural transformation $f: L.F \Rightarrow L.G$, for a parallel pair $F,G: A \to ...

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### Lax algebras for pseudomonads and monads in Kleisli bicategories for the induced pseudocomonad

In Day–Street's Lax monoids, pseudo-operads, and convolution, they remark without proof:
There are general principles involved here. Suppose $(T, m, j)$ is a pseudomonad on any bicategory $\mathcal K$...

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### "Character" theory via dualisable $2$ categories

One interesting way to describe the ordinary (over $\mathbb{C}$) character theory of finite groups is to view the categories $Rep(G)$ together in a $2$ category with bimodules as morphisms. This $2$ ...

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### Coherence $2$-cells in braided monoidal bicategories

In a braided monoidal category $(\mathcal{C},\otimes_{\mathcal{C}},\mathbf{1}_{\mathcal{C}},\alpha,\lambda,\rho,\beta)$, we have $\beta_{\mathbf{1}_{\mathcal{C}},\mathbf{1}_{\mathcal{C}}}=\mathrm{id}_{...

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### Strictifying monoidal 2-functors

Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a (weak) monoidal 2-functor between two strict monoidal 2-categories. Up to replacing $\mathcal{C}$ by an equivalent strict monoidal 2-category, can I ...

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### $2$-dimensional adjunctions via co/Cartesian fibrations $\mathcal{M}\longrightarrow[1]$

Recall [HTT, Definition 5.2.2.1]:
Definition 5.2.2.1. Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$-categories. An adjunction between $\mathcal{C}$ and $\mathcal{D}$ is a map $q\colon\mathcal{M}\to\...

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### Is there a bicategorical Yoneda lemma for marked lax transformations?

The bicategorical Yoneda lemma (see [Johnson–Yau, Chapter 8]) states that, given a bicategory $\mathcal{C}$ and a pseudopresheaf $\mathcal{F}\colon\mathcal{C}^{\mathsf{op}}\to\mathsf{Cats}_{\mathsf{2}}...

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### Can we simplify the definition of a biadjunction using coherence for bicategories?

In Biequivalences in Tricategories,
Gurski defines a biadjunction of bicategories as a sextuple $(F,G,\eta,\epsilon,\Gamma,\Sigma)$ with
$F\colon\mathcal{C}\longrightarrow\mathcal{D}$ and $G\colon\...

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### $2$-dimensional adjunctions via pre/post-composition

Recall that in the setting of $1$-categories, given functors $L\colon\mathcal{C}\longrightarrow\mathcal{D}$ and $R\colon\mathcal{D}\longrightarrow\mathcal{C}$, the following conditions are equivalent:
...

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### "Very lax" $2$-dimensional co/limits

In the setting of $1$-categories, there are two (unweighted) variant notions of limits, namely limits and colimits. For bicategories, there are fourteen of them:
Here $\mathsf{LaxCones}(\Delta_{X},D)\...

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### Monads and modules, and the bicompletion under Kleisli and Eilenberg–Moore objects

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 ...

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### Characterization of functors whose right adjoint is monadic?

Let $F: \mathcal A^\to_\leftarrow \mathcal B: U$ be an adjunction, and suppose we want to know whether the comparision functor $\mathcal B \to Alg^{UF}$ is an equivalence, where $Alg^{UF}$ is the ...

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### Coherence for closed bicategories

A right-closed bicategory [1] is a bicategory that has all right extensions (i.e. right adjoints to precomposition with a fixed 1-cell). A one-object right-closed bicategory is therefore a right-...

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### A bicategorical representation theorem

The representation theorem in $1$-category theory states that a presheaf $\mathcal{F}$ is representable iff $\int_\mathcal{C}\mathcal{F}$ has a terminal object. When one tries to formulate a ...

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### Reference for proof that vertical composition of pseudonatural transformations is a pseudonatural transformation

Is there a standard reference for the fact that the vertical composition of two pseudonatural transformations between pseudofunctors between bicategories is a pseudonatural transformation? Recall that ...

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### Reference for "taking adjuncts preserves Kan extensions"

I'm using a result similar to the one below, and I would like to know if there is a reference that I can cite. It's easily proved, by "following your nose". The cell $G\phi.\eta_A$ is often ...

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### Coherence for monoidal 2-categories vs coherence for braided monoidal categories

Gordon, Power and Street have proven that every monoidal 2-category is equivalent to a Gray monoid. This means that the only coherence 2-isomorphisms we have to be concerned about are the ...