Questions tagged [2-categories]

The notion of a [$2$-category](https://en.wikipedia.org/wiki/Strict_2-category) generalizes that of category by considering "morphisms between morphisms". In other words, it is a category enriched over $\mathbf{Cat}.$ It generalizes further in higher category theory to $n$-categories, which have $k$-morphisms for all $k\le n.$ However this should not confused with the (related) notion of [double categories.](https://ncatlab.org/nlab/show/double+category)

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Is “semisimplification” a 2-functor?

Consider the following 2-category $\mathcal K$: objects are finite length abelian categories (i.e., abelian categories where every object has finite length) morphisms are exact functors (preserving ...
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Examples of adjoint 2-functors

I hope this is not too vague, but I'm learning about 2-dimensional categories and I would like more examples of adjoint 2-functors to study. Could somebody tell me some interesting 2-adjunctions ? ...
Richard Southwell's user avatar
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Endomorphisms of an equivalence of categories

For $F:C^{op}\rightarrow C$ an isomorphism of categories, it is easy to see that $End(F)\cong End(\operatorname{id}_C)$ and $Aut(F)\cong Aut(\operatorname{id}_C)$ as sets, where we consider the hom-...
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2-pullbacks in 2-functor 2-categories

Let $\mathcal{C}$ and $\mathcal{D}$ be 2-categories, and let $[\mathcal{C}, \mathcal{D}]$ be the 2-category of 2-functors from $\mathcal{C}$ to $\mathcal{D}$. If $\mathcal{D}$ has 2-pullbacks, does it ...
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The free 2-category on a 1-category with respect to pseudo-functors into 2-categories

Let $\mathbb{C}$ be an ordinary 1-category. I'm interested in the following potential construction. Is there a 2-category $\widetilde{\mathbb{C}}$ (equipped with a pseudo-functor $\eta : \mathbb{C} \...
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Obtaining a $1$-category from a $2$-category

Let's say we are given a strict $2$-category $C$. One can obtain a $1$-category from this $2$-category by deleting the $2$-cells (and forgetting horizontal composition). Similarly, one can consider ...
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Transporting pseudo-monad structure along a pseudo-natural equivalence

Suppose that we have a pseudo-monad $\mathbb{T}$ (with underlying pseudo-functor $T$) on a $2$-category $\mathcal{K}$. Suppose also that there is a pseudo-natural equivalence between $T$ and another ...
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Can categories with suspension be considered as Cat-enriched presheaves over certain strict 2-category?

I came across the concept of a category with suspension $(\mathcal{C},\Sigma)$ which is defined as a category $\mathcal{C}$ together with an endofunctor $\Sigma:\mathcal{C}\rightarrow \mathcal{C}$. We ...
Zhenhui Ding's user avatar
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Left adjoint to the inclusion of semicocartesian symmetric monoidal categories in symmetric monoidal categories

Let $\mathbf{SMC}$ be the $2$-category of symmetric monoidal categories and strong symmetric monoidal functors. Also, let $\mathbf{SMC}_{0}$ be the full sub-$2$-category of $\mathbf{SMC}$ on the ...
Geoffrey Trang's user avatar
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Characterisation of terminal category in the 2-category sense

On nlab https://ncatlab.org/nlab/show/terminal+category, it is stated that a category is terminal in the 2-category sense precisely when it is inhabited and indiscrete. I wanted to try to prove this ...
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Construction of 2-category of monoidal categories and (lax) monoidal functors as strict algebra category of a 2-monad

As motivation for 2-monads, I would like to understand an explicit construction of the 2-monad $T$ of which derived 2-category $T-\operatorname{Alg}_l$ of algebras as described in Lack's 2-categories ...
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Motivating the definition of adjoint equivalence

Recently I have been trying to convince myself that the most natural definition for equivalence between categories is the notion of adjoint equivalence rather than simply equivalence. Of course, every ...
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Equation in the paper "homotopy limits for 2-categories"

I do not understand a step in the paper "Homotopy Limits for 2-Categories" by Nicola Gambino. The two longer calculations on page 22 together imply that \begin{align} \textrm{Ps}(\mathscr J,\...
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Does pslim have a Cat-enriched left adjoint?

I use the notation and 2-categorical conventions from the papers by Lack, Kelly, Power, et. al. about 2-monad theory and 2-limits. When $K$ is a 2-category which is complete and has copower, then one ...
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are comma-objects semi-coflexible

I need someone to read through my proof, because I feel very uncertain about 2-categorical limits. A strict indexed category $C:\mathscr S^{op}\to Cat$ is semi coflexible when every pseudo-...
Nico's user avatar
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Meaning of "functorial" in Proposition 1.3.5 of Hovey's Model Categories

For any category $C$, with a terminal object $*$, denote by $C_*$ the coslice category $*/C$. There is an adjoint pair, denoted by $V_C\dashv U_C:C_*\to C$, where $U_C$ is the forgetful functor. Let $...
Jerry Scott's user avatar
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Natural transformations of functors $\rm Cat\to Cat$

For any two categories $C$, $D$ denote by $[C,D]$ the category of functors $C\to D$. Fixed two categories $C$, $C'$, there are two functors $\rm Cat\to Cat$, namely $[C,-]$ and $[C',-]$; suppose to ...
Jerry Scott's user avatar
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Cat-enriched right adjoint

In Streicher's text "Fibered categories" it is described how to construct a 2-functor $Sp:Fib_\mathscr S\to 2Cat_s(\mathscr S^{op},Cat)$ such that $R = Sp\circ \int$ is right 2-adjoint to ...
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A local approach to adjunctions in a 2-category

I have trouble understanding the following pragraph in Stephen Lack's A 2-Categories Companion. You can find it near the bottom of page 8, here is a link the article. The local approach to ...
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How does the 2-yoneda lemma exactly work

I understand the 1-Yoneda lemma well, and I know how to use it in proofs, but I have trouble understanding how the 2-Yoneda lemma exactly works. Say for simplicitly I have a strict 2-category $K$ and ...
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What categorical notion of "equivalence" does $B:\text{Grp}\to \text{ConnGrpd}$ give?

I've been taking a course on Category Theory recently and really enjoying it. The lecturer today was discussing the notion of equivalence in more detail and showed that every connected groupoid $\...
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Adjoint functors in 2-category theory

Let $\cal{C},\cal{D}$ be two categories. Let $F:\cal{C}\rightarrow\cal{D}$ and $G:\cal{D}\rightarrow\cal{C}$ be a pair of adjoint functors such that $F$ is left adjoint to $G$. Suppose that $\cal{D}$ ...
Toney Leung's user avatar
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Proof that the multiplication of a group is recovered by the repeated 2-pullback of the unique point of its delooping

Given a group $G$ with delooping $BG$, $BG$ has a unique point $\iota:\star\rightarrow BG$ and $G$ is equivalent to the 2-pullack of $\iota$ along itself. Furthermore, the repeated 2-pullback $\iota\...
Alexander Praehauser's user avatar
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Question about a forgetful 2-functor in Tom Leinster's *Higher Categories, Higher Operads*

In their book Higher Operads, Higher Categories Tom Leinster claims, that unbiased weak 2-categories may be defined as weak algebras for a strict 2-monad on the strict 2-category of Cat-enriched ...
Nico's user avatar
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Initial component lax terminal object

Given a bicategory $K$, Niles Johnson and Donald Yau define an initial component lax terminal object in $K$ to be an object $T$ together with a lax transformation $k: 1_K\to \Delta_T$ such that $k_T$ ...
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Examples of Pseudo-adjunctions

Let $K$ and $L$ be strict $2$-categories. A pseudo-adjunction consists of two pseudo-functors $F: K \to L$ and $U: L \to K$ together with a pseudo-natural equivalence $\phi_{X,Y}: L(FX,Y) \simeq K(X,...
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2-categorical universal property of the classifying category of a type theory

For example let us say we are in the setting of cartesian closed categories and the simply typed $\lambda$-calculus. Let $\mathtt{strCCCat}$ denote the $2$-category of strict cartesian closed ...
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Universal characterization of the arrow (interval, ordinal, 1-simplex) object? (Is $2$-category theory actually needed?)

In the "category of categories" $Cat$ (see e.g. here for a rigorous definition), there is a category $A$ with two objects and exactly one non-identity morphism. (Cf. "two-valued object&...
hasManyStupidQuestions's user avatar
2 votes
2 answers
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The functor that maps a (co-)monad to its (co-)Eilenberg-Moore category

I have noticed that the function that maps a monad $T : C \to C$ to the Eilenberg-Moore category $C^T$ can easily be extended into a functor $E_C$ from the category of monads $\textbf{Mnd}_C$ to $\...
Bob's user avatar
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From monads to comonads by the calculus of mates

If a comonad $D$ is left adjoint to an endofunctor $T$, then $T$ can be made into a monad: its unit and multiplication are given respectively by the mates of the counit and comultiplication of $D$. ...
Bob's user avatar
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1 vote
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Correct Definition of a Universal 2-Arrow In a 2-Category?

I having been thinking of how to formulate the idea of a universal 2-arrow (or 2-cell) in a 2-category. As some background, I am learning about Kan extensions. In $Cat$, I understand that a left Kan ...
IsAdisplayName's user avatar
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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 have an example of a 2-cell ...
mathlete42's user avatar
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Classification of biclosed monoidal structures on the $2$-category of $2$-categories

This paper proves that the category of small categories and functors between them admits exactly two monoidal biclosed categories: the cartesian tensor product and the funny tensor product. Is there a ...
Emily's user avatar
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Equivalence and isomorphism of $\hom$s in $2$-categories/bicategories

In a usual $(1)$-category $C$, if $A$ is an object and $B\cong B'$ are isomorphic objects, then the hom-sets $\hom(A,B)\cong \hom(A,B')$ are in bijective correspondence, because composition with the ...
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Why is a monad on self C equivalent to a strong monad on C

In Chapter 12 of Call-by-Push-Value, Levy states that a strong monad on a Cartesian category $\mathcal C$ is equivalent to a monad in the 2-category of $[\mathcal C^{\mathit{op}},Sets]$-enriched ...
Kevin Clancy's user avatar
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Right extensions along right adjoints

I'm reading The Formal Theory of Monads by Ross Street, and I am stumped at the fourth theorem. It seems that its proof has been elided, so I tried to come up with one myself. This theorem requires ...
Kevin Clancy's user avatar
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1 answer
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Definition of an algebra over a monad by using equalities between natural transformations

In the definition of a monad, there are two ways to specify the equations: equalities between natural transformations, or equalities between morphisms as is done there on Wikipedia. In the usual ...
Bob's user avatar
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The homotopy 2-category - Are homotopies unique up to homotopy?

Consider a homotopy $h: I \times X \rightarrow Y$ and denote $$\begin{array}{rcl} h_1: X & \rightarrow & Y\\ x & \mapsto & h(1,x). \end{array}$$ Define the constant homotopy $$\begin{...
Jonas Linssen's user avatar
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Is there a concept of "homotopy category of a 2-category"?

I apologize in advance for my naiveté. I'm completely missing the formal background to ask this question, but am very curious about one particular point. I recently learned that there's the concept of ...
user913156's user avatar
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localization in 2-categories with 2-morphisms

I know almost nothing about higher categories. In a 2-category, is there a way to localize on a set of 2-morphisms rather than 1-morphisms ?
user2478159's user avatar
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Free completion for a 2-category with Eilenberg-Moore objects

For a $2$-category $\mathcal{K}$, the free completion of $\mathcal{K}$ under Eilenberg-Moore-objects (EM-objects) is a $2$-category which has $0$-cells the (internal) monads $(A, t, \mu^t, \eta^t)$ in ...
user846915's user avatar
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0 answers
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Horizontal composition of pseudonatural transformations

Let $A$, $B$, and $C$ be 2-categories (or bicategories, etc), and let $F,F':A\to B$ and $G,G':B\to C$ be 2-functors (or pseudofunctors, etc). Now let $\alpha:F\Rightarrow F'$ and $\beta:G\Rightarrow G'...
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Restricting to structured right homotopies

Let $\mathbf{Cat}$ be the 2-category of categories, functors, natural transformations. If $F, G : \mathscr C \to \mathscr D$ are functors, then natural transformations $\eta : F \Rightarrow G$ may be ...
varkor's user avatar
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Property of 2-natural transformations

Let $C$ and $D$ be (strict) 2-categories, let $F,G:C\to D$ be (strict) 2-functors, and let $\alpha:F\Rightarrow G$ be a (strict) 2-natural transformation. Let now $X$ and $Y$ be objects of $C$, let $f,...
geodude's user avatar
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Proving a commutative diagram involving the associator and two 2-cells in a bicategory

Let $f \; : \; A \to A$, $g \; : \; A \to B$ and $h \; : \; B \to B$ be 1-cells, and $\sigma \; : \; g \cdot f \; \to \; g$ and $\theta \; : \; g \; \to \; h \cdot g$ be 2-cells in a bicategory. I ...
Bob's user avatar
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3 votes
1 answer
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2-commutative diagram in 2-category

Definition 3.27: A diagram in a 2-category is called 2-commutative, if its 1- morphisms commute up to given 2-isomorphisms and these 2-isomorphisms commute in the induced diagram taking 1-morphisms (...
metalder9's user avatar
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Adjunctions from triangle isomorphisms

A very elementary result in category theory allow us to promote equivalences of categories to adjoint equivalences by changing one of the two natural isomorphisms. Generalizing this, consider the ...
Nicola Di Vittorio's user avatar
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0 answers
115 views

Universal property of 2-pullback

According to the n-lab article on 2-pullbacks, A 2-pullback in a 2-category is a square $$ \begin{array}{ccc} P & \xrightarrow{p} & A \\\ q\downarrow & & \downarrow{f}\\\ B &...
sysyphusV's user avatar
3 votes
1 answer
133 views

Non-evil definition of a Kleisli object in a weak 2-category

Let $t : a \to a$ be a monad in a weak 2-category. According to nLab, the 1-dimensional universal property of a Kleisli object $(f_t : a \to a_t, \lambda : f_t t \to f_t)$ is that for any right $t$-...
Bob's user avatar
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7 votes
0 answers
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Monoidal structure preserving order

I am working with a category $\mathcal{C}$ in which the hom-sets are orders. Alternatively, we could look at it as a bicategory in which hom-sets for 2-cells are thin. Is there a notion of monoidal ...
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