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 ? ...
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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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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 ...
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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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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 ...
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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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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 $D$. ...
• 1,548
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 ...
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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 ...
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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 ...
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1 vote
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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 ...
1 vote
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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 ...
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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 ...
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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 ...
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1 vote
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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 $t$-...
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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 ...