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Questions tagged [2-categories]

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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 ...
varkor's user avatar
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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(\...
JeCl's user avatar
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9 votes
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104 views

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 ...
varkor's user avatar
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9 votes
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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 ...
varkor's user avatar
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9 votes
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199 views

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 ...
Emily's user avatar
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9 votes
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192 views

What is the relationship between free bicompletion and the Isbell envelope?

Given a small category $\mathbb C$, we can form the free cocompletion $\mathbf y : \mathbb C \to \mathcal P(\mathbb C)$ and the free completion $\mathbf y^\circ : \mathbb C \to \mathcal P^\circ(\...
varkor's user avatar
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9 votes
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166 views

Proper model category for "categories with finite limits"

I'm looking for a Quillen model category which model the $2$-category of 'small category with finite limits (and functors between them preserving finite limits)': Left proper, right proper, Enriched ...
Simon Henry's user avatar
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9 votes
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358 views

The category of elements corresponding to a coend as a higher colimit

Let $D: \mathcal{C} \to \mathbf{Set}$ be a diagram of sets, then we can obtain the colimit of $D$ as the set of connected components of the category of elements of $F$, which we denote by $\mathrm{el}(...
Adrian Clough's user avatar
9 votes
0 answers
323 views

To what kind of generalized Lawvere theory does the "free cartesian closed category" 2-monad on $\mbox{Cat}_g$ correspond?

Thinking of Cat as a mere 1-category, there is a 1-monad $\Lambda$ for the free cartesian closed category on a category. To every category X it assigns the category $\Lambda(X)$ whose objects are ...
Mike Stay's user avatar
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8 votes
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Double-categorical refinement of twisted arrow category: does it have a name?

Let $C$ be a category. The twisted arrow category $Tw(C)$ can be refined to a double category $TTw(C)$ by making morphisms on the left "vertical" and morphisms on the right "horizontal". Question: I'...
Tim Campion's user avatar
7 votes
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109 views

The Barr-Kock lemma for regular 2-categories

There is a nice result for regular 1-categories, which I quote from page 441 of Borceux & Bourn's textbook "Mal'cev, Protomodular, Homological and Semi-Abelian Categories". This is ...
Cory Gillette's user avatar
7 votes
0 answers
184 views

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 ...
JeCl's user avatar
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7 votes
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266 views

Relation between two limit presentations of Eilenberg--Moore objects

Let $\mathbb{T}=({\cal T}\colon C\to C,\mu,\eta)$ be a monad (in the $2$-category $\mathsf{Cat}$), which we view as a $2$-functor $\mathbb{T}\colon\mathsf{B}\Delta_{\mathrm{a}}\to\mathsf{Cat}$ (where $...
David Kern's user avatar
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161 views

Coherence for pseudomonads and their pseudoalgebras

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 $...
varkor's user avatar
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7 votes
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428 views

Left Kan extensions of "strong" monoidal functors

Consider the 2-category $\mathsf{MonCat}$ where objects are monoidal categories, 1-cells are strong monoidal functors, and 2-cells are monoidal natural transformations. Given arrows $f: \mathsf{C} \to ...
Eigil Fjeldgren Rischel's user avatar
6 votes
0 answers
75 views

What are the algebras of the powerset intersection (oplax) monad?

The assignment $X\mapsto\mathcal{P}(X)$ and $f\mapsto f_*$ (direct images) defines a functor $\mathcal{P}\colon\mathsf{Sets}\to\mathsf{Sets}$. This functor has a monad structure whose multiplication $\...
Emily's user avatar
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6 votes
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98 views

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 ...
varkor's user avatar
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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 ...
Emily's user avatar
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6 votes
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Codescent objects in Morita 2-category?

Let $\mathbf{Bimod}$ be the 2-category of algebras, bimodules and bimodules maps over a field $\mathbb{k}$. Does this 2-category have codescent objects (see my attempt at giving a definition below)? ...
JeCl's user avatar
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6 votes
0 answers
118 views

Retracts in the bicategory of spans

I would like to show that the category of sets and spans between them, seen as a $(2,1)$-category, is Cauchy complete, i.e. has splitting of (homotopicaly coherent) idempotent. Ideally I would also ...
Simon Henry's user avatar
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6 votes
0 answers
245 views

Uniqueness of the $(2,2)$-category theory of $(\infty,1)$-categories?

The question, as in the title, may be very simply stated as follows: Main Question: Can the homotopy $(2,2)$-category of $(\infty,1)$-categories be characterized as the unique $2$-category upto ...
Saal Hardali's user avatar
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On Isbell's "On coherent algebras and strict algebras"

In section 1.2 (p. 284) of his "Coherence theorems for lax algebras and for distributive laws," Kelly writes: It is a special case of the assertion of Isbell in [7], but this assertion has ...
Kate's user avatar
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0 answers
66 views

Bicategories in which the composition functors $\circ_{A, B, C}$ admit right adjoints

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 ...
varkor's user avatar
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5 votes
0 answers
144 views

2-category structure on Mod(R)

Apologies for the basic question but I'm curious to know if there is an ``interesting" $2$-category structure on the category of modules over a ring $R$. Essentially what is not clear to me if $M,...
curious math guy's user avatar
5 votes
0 answers
89 views

Free cocompletion of a 2-category under pseudo colimits, lax colimits, and colax colimits

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-...
varkor's user avatar
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5 votes
0 answers
107 views

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 ...
Ulrich Pennig's user avatar
5 votes
0 answers
349 views

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 (...
Dat Minh Ha's user avatar
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5 votes
0 answers
120 views

Simplicial matrices and the nerves of weak n-categories II, III, and IV

Duskin introduced his nerve functor (see the nLab or Kerodon) in the paper Duskin, John W. Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories. [Link]. While three ...
Emily's user avatar
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4 votes
0 answers
80 views

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 ...
Nico's user avatar
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4 votes
0 answers
207 views

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 ...
Elías Guisado Villalgordo's user avatar
4 votes
0 answers
108 views

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 \...
JeCl's user avatar
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4 votes
0 answers
111 views

$2$-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: ...
Emily's user avatar
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4 votes
0 answers
95 views

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-...
varkor's user avatar
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4 votes
0 answers
85 views

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 ...
JeCl's user avatar
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4 votes
0 answers
63 views

Describing modifications using limits

It is well know that in (1-)category theory, one can describe the set of natural transformations between two functors by an end formula. I would like to know whether some similar description is ...
JeCl's user avatar
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4 votes
0 answers
151 views

Adjunctions in a weak $2$-category

Is the notion of an adjunction well defined in an arbitrary weak $2$-category? In particular, it seems like we need at least invertible unitors in the ambient $2$-category to write down the triangle ...
Alec Rhea's user avatar
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3 votes
0 answers
49 views

Lax morphism classifiers via lax-idempotentification

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 ...
varkor's user avatar
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3 votes
0 answers
55 views

Universal property of 2-presheaves and pseudo/lax/colax natural transformations

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 ...
varkor's user avatar
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3 votes
0 answers
85 views

On the category Fib of 2-fibrations

I have recently been reading on 2-fibrations. It is well-known (from Hermida) that the codomain functor $cod \colon \textbf{Fib} \to \textbf{Cat}$ taking each fibration to its base category from the 2-...
Siya's user avatar
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3 votes
0 answers
124 views

$\mathbf{E}_n$-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 ...
Pulcinella's user avatar
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3 votes
0 answers
114 views

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 ...
Simone Virili's user avatar
3 votes
0 answers
75 views

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 \...
theAdmiral's user avatar
3 votes
0 answers
152 views

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 ...
Alec Rhea's user avatar
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3 votes
0 answers
71 views

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 ...
Alec Rhea's user avatar
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3 votes
0 answers
70 views

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 $\mathcal K$...
varkor's user avatar
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3 votes
0 answers
78 views

"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 $2$ ...
Chris H's user avatar
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3 votes
0 answers
85 views

$2$-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\...
Emily's user avatar
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3 votes
0 answers
136 views

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 ...
Roald Koudenburg's user avatar
3 votes
0 answers
125 views

Accessible 2-category of presheaves

Let $A$ be a locally small category and let $\mathbf{Cat}$ be the 2-category of small categories, functors and natural transformations and let $Ps(A)$ be the 2-category of presheaves (the objects are ...
Wilson Forero's user avatar
3 votes
0 answers
130 views

A Whitehead theorem for 3-categories

Let $F:\mathscr{C}\rightarrow \mathscr{D}$ be a 3-functor between 3-categories. Are the following two properties known to be equivalent? $F$ is a 3-equivalence, meaning that there is a 3-functor $G:\...
JeCl's user avatar
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