Questions tagged [2-categories]

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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 ...
3
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0answers
26 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 $\mathcal K$...
3
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60 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 $2$ ...
3
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1answer
104 views

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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67 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 ...
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57 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\...
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63 views

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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67 views

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\...
11
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1answer
153 views

“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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67 views

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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1answer
288 views

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 ...
4
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70 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-...
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140 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 ...
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1answer
71 views

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 ...
3
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89 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 ...
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26 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 ...
8
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1answer
207 views

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 ...
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1answer
81 views

Unitor identities for bicategories

In the standard definition of a bicategory, the unitors are required to satisfy the 'triangle identity' for any composable $1$-cells $f:Y\to Z,gX\to Y$. But it seems like we also want to commute for ...
6
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73 views

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)? ...
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113 views

How can we view monoids as lax functors?

Let $*_2$ be the terminal $2-$category, $V$ be a monoidal category with unit object $I$ and $\mathbf BV$ be its delooping, i.e the $2-$category with $ob(\mathbf BV) = \{*\}$ and $\text{Hom}_{\mathbf ...
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92 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(\...
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118 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 $...
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1answer
164 views

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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30 views

Morphism of pseudomonads induces pullback functors between pseudoalgebras

If $S$ and $T$ are monads on a category $C$, and $\lambda:S\to T$ is a morphism of monads, it is well-known that there is a functor $\lambda^*:C^T\to C^S$ which assigns to the $T$-algebra $(A,a:TA\to ...
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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 $...
8
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100 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 ...
5
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152 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 ...
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1answer
66 views

Has the covariant Hom-functor of the category of additive categories a left adjoint?

Let $\mathsf{Add}$ denote the (strict) 2-category of small additive categories and additive functors. Because categories of additive functors are itself additive, we have for each additive category $\...
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55 views

Examples of strictification of a weak category obtained from a generalisation of a strict category

I have made the following observation (hopefully a correct one) when reading the paper Orbifolds as stacks: They start with the strict $2$-category category of Lie groupoids, functors, natural ...
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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 ...
5
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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 ...
6
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1answer
146 views

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 ...
4
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1answer
371 views

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: $...
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113 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:\...
8
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2answers
578 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, ...
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102 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 ...
2
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0answers
56 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) \...
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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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1answer
133 views

“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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1answer
260 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 ...
6
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227 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 ...
4
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128 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 ...
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1answer
268 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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1answer
303 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
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1answer
159 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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1answer
183 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 ...
3
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0answers
65 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 \...
6
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1answer
265 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
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1answer
107 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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1answer
237 views

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 ...