# Questions tagged [2-categories]

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51
questions

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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 ...

**3**

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**0**answers

53 views

### Applications of Day convolution for monoidal bicategories

Day convolution is a very powerful tool to build monoidal structures on categories of functors from a pro/monoidal $\mathcal{V}$-category. For instance, it is used in stable homotopy theory to ...

**4**

votes

**1**answer

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### What is the notion of a group object and its action in a 2-category?

It is well known that a group object in a category $C$ (with terminal object $1$ and such that any two objects of $C$ have a product) is defined as an object $G$ in $C$ with the following morphisms:
$...

**3**

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100 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:\...

**6**

votes

**2**answers

524 views

### 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, ...

**5**

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### 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 ...

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51 views

### A name in literature for a certain kind of 2-categories

Let $tr_2: \mathrm{sSet} \to \mathrm{sSet}_{\le 2} $ be the 2-truncation functor.
Let $C$ be a 2-truncated simplicial set such that every horn $tr_2( \Lambda^2_1) \to C$ extends to $tr_2(\Delta_2) \...

**4**

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54 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 ...

**3**

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**1**answer

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### “discrete” objects of a $2$-category

Let $\mathcal{K}$ be a $2$-category. Is there a special name of those objects $B \in \mathcal{K}$ which have the property that the category $\mathrm{Hom}_{\mathcal{K}}(B,C)$ is essentially discrete ...

**5**

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**1**answer

225 views

### Diagonal of a diagram of codescent objects

Given the following diagram in a $2$-category, in which squares of the same "type" commute, where each column and each row is a strong codescent diagram (Edit: it should be reflexive as well), is ...

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208 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 ...

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### 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 ...

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**1**answer

263 views

### Does the 1-category construction of a topos of presheaves extend to the 2-Category of Groupoids?

In the case of 1-categories, we know there is a functor category
$PSh(C):=[C^{op},Set]$, where $C$ is a small category,
and this functor category is a topos. I am hoping this will extend to the case ...

**6**

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**1**answer

282 views

### 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 ...

**3**

votes

**1**answer

147 views

### On the Group Structure of Morphism Set of a Strict 2-Group

The standard definition of a strict 2-group says that it is a strict monoidal category in which every morphism is invertible and each object has a strict inverse.
Also it is a well known fact that a ...

**3**

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**1**answer

144 views

### Weak enrichment and bicategories

I'm trying to find examples where the following perspective on bicategories is developed.
We can define a 2-category as being enriched in Cat, where Cat is treated as a monoidal category using the ...

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64 views

### On cofibrations of simplicially enriched categories

Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells.
We have a canonical inclusion functor ,
$$i: C \...

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253 views

### 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, $1$-...

**2**

votes

**1**answer

97 views

### When does a 2-functor or 2-monad of Cat lift to a psuedofunctor or pseudomonad on Prof?

I'm currently reading Richard Garner's paper Polycategories via pseudo-distributive laws, and a central construction is the lifting of the symmetric strict monoidal category 2-monad to a pseudomonad ...

**7**

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**1**answer

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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 ...

**6**

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166 views

### 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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votes

**3**answers

276 views

### On 2-actions of strict 2-groupoids?

I'm looking for an opinion if the following makes sense.
A linear representation of a groupoid $\mathcal{G}$ is a functor $$\nabla: \mathcal{G}\longrightarrow \mathsf{Vect}_{\mathbb K},$$ where $\...

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180 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}(...

**6**

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537 views

### 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 ...

**6**

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**1**answer

218 views

### 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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352 views

### 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 ...

**7**

votes

**1**answer

389 views

### 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 ...

**2**

votes

**1**answer

84 views

### Lax co/limit as evaluation on terminal/initial

A quick question about lax co/limits.
Strictly, when $F : J\to \bf A$ is a diagram and $J$ has an initial object $\varnothing$, then $\varprojlim F \cong F(\varnothing)$; dually, if $\cal J$ has a ...

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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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211 views

### 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 been ...

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**1**answer

268 views

### Comma category as weighted limit

Let $F : C \to E$ and $G : D \to E$ be functors. Consider the comma category $(F \downarrow G)$ with its projections $\pi_1 : (F \downarrow G) \to C$ and $\pi_2 : (F \downarrow G) \to D$. Using the ...

**0**

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**1**answer

94 views

### Equality of lax natural transformations in the constructive approach

In the constructive approach to category theory, a category comes equipped with an equality (an equivalence relation) between its morphisms but not between its objects.
Let C and D be such categories,...

**4**

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**1**answer

267 views

### Whiskering approach to strict 2-categories: literature reference needed

I am familiar with the nLab web page that nicely lays out the axioms needed to define strict 2-categories using whiskering as opposed to horizontal composition of 2-cells. However, I am old fashioned ...

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77 views

### Reference for the existence of bicolimits in groupoids and categories?

I am looking for a reference of these, I would say, very well known facts. (strangely though finding a reference was bit trick for me).
Let $C$ be a category and $F:C\rightarrow Cat$ a 2-functor in ...

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49 views

### Pseudopullback of dimension three

What is the name of the appropriate analogue of the pseudopullback for dimension three?
That is to say, a pseudonatural equivalence $fg\simeq hj $ which is universal in the obvious sense...
Thank ...

**2**

votes

**1**answer

81 views

### Why does vertical multiplication in 2-groups not follow the same order as horizontal one when constructed from crossed modules?

Consider a 2-group (seen as a 2-category with only one object $\star$) constructed from a crossed module $(G,H,t,\rhd)$ ($\circ$ will denote 1-morphisms composition, $\circ_{h}$ 2-morphisms horizontal ...

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493 views

### 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 ...

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71 views

### Cancellation property of groupoidal cartesian fibrations

I have an issue concerning a property of "left cancellation" for groupoidal cartesian fibrations of ∞-cosmoi (but everything works fine in a 2-category as well).
A 1-cell $p: E \to B$ is called ...

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**2**answers

194 views

### Completion under weighted limits/colimits

Is there any further reference besides "Basic Concepts of Enriched Categories" (Kelly) for completion under T-(weighted) limits/colimits?
(in which T is a set of weights)
Thank you in advance

**10**

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**1**answer

464 views

### 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 ...

**7**

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318 views

### 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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**2**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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votes

**1**answer

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### Kan extension pseudonatural transformations

Consider the 2-category $[S, H] $ of 2-functors $S\to H $ (in which, obviously, $S$ and $H$ are 2-categories). And consider a (possibly fully faithful) functor $T: S\to Z $
For simplicity, let's ...

**5**

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**1**answer

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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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380 views

### 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}\...

**2**

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**0**answers

96 views

### Pseudofunctors of 2-variables and Gray tensor product of bicategories

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be bicategories. We say a 'pseudofunctors of 2-variables' consists of two families of pseufunctors
Fix $A\in obj\mathcal{A}$, we have a pseudofunctor $f(A,-...

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### Bicategorical limits with parameters

(This question was asked in https://math.stackexchange.com/questions/741334/bicategorical-limits-with-parameters with no answer.)
Let $F(-,-)\colon \mathcal{A}\times \mathcal{B}\to \mathcal{C}$ be a ...

**6**

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**1**answer

773 views

### 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 ...

**4**

votes

**2**answers

355 views

### Transporting algebraic structure along adjoint equivalences

I have two questions, one general and the other particular to the case I am interested in.
The 'homotopically correct' notion of equivalence of categories is an adjoint equivalence (from one point of ...

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151 views

### Recognition principle for 2-categories (2-groupoids)

Given a 2-category (i.e. bicategory) $C$ there is a unitary geometrical nerve whose 0-simplices are objects of $C$, 1-simplices are 1-arrows of $C$, 2-simplices are 2-commutative triangles (in certain ...