Questions tagged [2-categories]
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147 questions
17
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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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16
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1
answer
521
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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 sixteen of them:
Each of these notions has an ...
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14
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3
answers
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2-category theory
I know that we can do a lot of 2-category theory, seeing 2-categories as Cat-enriched categories. Yet, I know that there are some limitations of this approach.
I also know that there are many articles ...
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13
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1
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718
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On the coherence theorem for bicategories
The coherence theorem for bicategories, as usually stated, reads
Any bicategory $B$ is biequivalent to a (strict) 2-category.
It is possible to give an explicit construction of the strictification ...
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13
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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 ...
- 10.7k
10
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1
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466
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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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10
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1
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333
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2-monads for categories with a class of (co)limits
This question concerns the strictness of (co)completions, at various levels of generality.
In Blackwell–Kelly–Power's Two-dimensional monad theory, the authors state
For instance, the 2-category $\...
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10
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1
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405
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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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10
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0
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145
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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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9
votes
1
answer
727
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Yoneda Lemma for internal presheaves
I'm looking for a reference explaining under what conditions the internal Yoneda lemma holds; in particular, I am wondering if it is known what properties of the ($2$-)category of categories are ...
- 572
9
votes
2
answers
434
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Pushouts of commutative pseudomonoids
Let $(\mathcal{C},\otimes)$ be a symmetric monoidal bicategory. Assume that $\mathcal{C}$ has bicategorical coequalizers which are preserved by $\otimes$ in each variable. My question is if then the ...
- 63.1k
9
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1
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356
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What is a bipullback of lax functors?
$\require{AMScd}$The following question is somewhat technical, and since I firmly believe this has a small hope to be true only using all the assumptions, I am forced to introduce them all: I don't ...
- 13.6k
9
votes
0
answers
104
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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 ...
- 10.7k
9
votes
0
answers
129
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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 ...
- 10.7k
9
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0
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199
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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 ...
- 11.8k
9
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0
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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(\...
- 10.7k
9
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0
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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 ...
- 42.4k
9
votes
0
answers
358
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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}(...
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9
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0
answers
323
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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 ...
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8
votes
2
answers
659
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Is dgCat a category or a 2-category?
Let us consider dgCat, the "collection" of all small dg-categories. In On differential graded categories and Lectures on dg categories the authors state that they form a category, i.e. dgCat has ...
- 9,019
8
votes
2
answers
942
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Does the Grothendieck construction produce a 2-category or a category?
Let $F : \mathcal{C} \to \mathbf{Cat}$ be a lax 2-functor. Then we can form a category $\int F $ which is the Grothendieck construction on F.
There's a number of resources detailing this construction, ...
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8
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2
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472
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How to construct a free 2-group on a groupoid?
Let G
be a groupoid. I'm wondering how to construct the free 2-group on G.
By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$
equipped with a functor $i:G\longrightarrow\mathcal{F}\...
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8
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1
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954
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A question on the Grothendieck construction
The usual Grothendieck construction is for pseudofunctors $I^{op}\to Cat$, where $I$ is a 1-category and $Cat$ is the 2-category of 1-categories. The Grothendieck construction produces a category with ...
- 1,271
8
votes
1
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778
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J.W. Gray's monumental work notes on the formal theory of internal (2-)categories
In the book "Topos Theory" of Peter Johnstone (Topos Theory, LMS Monographs no. 10. Academic, 1977) one finds at page 41 in Chapter 2:
"For a detailed account of internal categories ...
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8
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1
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530
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Is a weak monoidal category a monoid object in some category?
A monoid in the Category Cat is a strict monoidal category according to Wikipedia. Is it possible to weaken the monoid so that its realisation in Cat is a weak monoidal category? Do we shift up a ...
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8
votes
1
answer
361
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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 $...
- 10.7k
8
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0
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264
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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'...
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7
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2
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593
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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 ...
- 511
7
votes
2
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584
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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 ...
- 11.8k
7
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1
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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 ...
- 225
7
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1
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159
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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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7
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1
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Notions of Lie 2-groupoids
The term Lie $2$-groupoid is used in the literature in more than one context. Some examples are given below:
Ginot and Stiénon's paper $G$-gerbes, principal $2$-group bundles and characteristic ...
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7
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2
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558
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Kan extensions in concrete 2-categories
Kan extensions make sense in any 2-category. I am interested in Kan extensions in "concrete" 2-categories consisting of actual categories with some sort of structure (e.g., finite products, finite ...
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7
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1
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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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7
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1
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366
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Are sifted (2,1)-colimits of fully faithful functors again fully faithful? (And a de-categorified variant)
1) Suppose that I have a sifted diagram of categories $\mathcal{C}_i$, another of the same shape $\mathcal{D}_i$, and that I have a system $F_i:\mathcal{C}_i\to\mathcal{D}_i$ commuting with the ...
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7
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1
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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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7
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2
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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 ...
- 10.7k
7
votes
1
answer
345
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Has anybody studied strict/pseudo morphisms of monads?
There is a notion of morphism from a monad $T:\mathscr C\to \mathscr C$ to another one $T':\mathscr C'\to \mathscr C'$. It arose here on MO e. g. in "Functors between monads": what are these ...
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109
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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 ...
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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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266
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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
$...
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161
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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 $...
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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 ...
6
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2
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220
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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\...
- 11.8k
6
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1
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255
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Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first?
Let $C$ be a site and $CAT$ the 2-category of categories. Given a contravariant 2-functor $A:C\rightarrow CAT$, we can of course consider the associated stack. This is done by first considering the ...
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6
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1
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450
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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 (...
6
votes
1
answer
355
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Is the 2-сategory of groupoids locally presentable?
I am wondering if the 2-сategory of groupoids is locally presentable. Locally presentable means the category is accessible and co-complete.
It has been pointed out that the category of groupoids is ...
- 1,313
6
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1
answer
309
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Does the Dwyer-Kan model structure make dgCat a model $2$-category?
Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category.
Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...
- 9,019
6
votes
2
answers
944
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Twisted-arrow construction for 2-categories
I've been looking over Lurie's DAG X, and he introduces a combinatorial construction called the twisted arrow construction for simplicial sets that generalizes the following ordinary categorical ...
- 19.6k
6
votes
1
answer
267
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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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