# Questions tagged [2-categories]

A 2-category is a category with "morphisms between morphisms".

32 questions
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### Ind-completion of a 2-category

If $\mathcal{C}$ is a category, there is a well known construction called the Ind-completion of $\mathcal{C}$, indicated by $\text{Ind}(\mathcal{C})$. This can be equivalently defined in several ways: ...
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### Books/Lecture notes about 2-categories.

Are there good books or lecture notes just about 2-categories?(not about higher categories nor $\infty$-categories) (I'm studying fibered categories for the descent theory of quasi-coherent sheaves. ...
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### Understanding 2-category theory

There are a lot of examples of categories, functors and natural transformations — one can find them anywhere. On the contrary (weak) 2-categorical stuff seems to be more subtle. I have comprehended ...
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### Right (bi)adjoint of the inclusion of $\mathbf{Grpd}$ in $\mathbf{Cat}$

Let $\mathbf{Grpd}$ and $\mathbf{Cat}$ be respectively the 2-categories of small groupoids and of small categories. At the 1-categorical level, the inclusion $\mathbf{Grpd}\rightarrow\mathbf{Cat}$ has ...
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### Is there such thing as a morphism of natural maps and what does it look like?

I'm writing software to be general, so right now I'm writing a NaturalMap class which will be a graphical arrow that goes between any two ...
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### Can we see FdHlb as a 2Category of groupoids?

Can we see a finite dimensional Hilbert space, $H$ as a groupoid if we include the unitary endomorphisms of $H$? It would be like a category with a single object and just isos. If so, can we take a ...
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### Description of the 2-monads for (strict) monoidal categories

The nLab article on 2-monads says: For example, ordinary (non-strict) monoidal categories are the strict algebras for a strict 2-monad $T_{MC}$ on $Cat$, but usually we care about pseudo, lax, and ...
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### Uniqueness up to unique isomorphism of quasi inverse of an arrow in a strict 2-category

I just started to learn the basics of strict 2-categories, and reading some notes I've found the following simple proposition: Let C be a strict 2-category, and $F:X \rightarrow Y$ a 1-cell (i.e....
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### Confusion with (strict) 2-adjoints

This is an elementary question on 2-categories, namely on the naturally arising notion of 2-adjunction when strict 2-functors are involved. Perhaps I don't understand some "internal" universal ...
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### How to precisely state how “strict 2-category” relates to “bicategory in which all coherence 2-cells are identities”?

Question. It is widely known that strict 2-categories "are" bicategories for which all coherence 2-cells are identities. However, how to make the "are" precise here? Remarks. I have seen ...
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### What's the name for this kind of 2-categorical unsymmetric pullback?

What is the name for this weak version of a 2-categorical pullback? The solid square commutes up to a not necessarily invertible 2-morphism $\eta$ and has the following universal property: For any ...
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### Proof that the left or right adjoint composed after a diagonal morphism in a 2-category is the identity

Suppose I have some strict 2-category $\mathcal{C}$ that's cotensored over $\mathcal{C}at$. Then for $A\in \mathcal{C}$ and $X\in \mathcal{C}at$, there's the diagonal morphism $\Delta:A\to A^X$. ...
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### Does the bicategory of bimodules really have left Kan extensions?

Let $\mathsf{Bimod}$ denote the bicategory whose $0$-cells are rings $A$, a $1$-cell $A \longrightarrow B$ is an $(A,B)$-bimodule and a $2$-cell is a bimodule map. The composition of $1$-cells is ...
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### What are 2-morphisms in the category of proofs?

After reading through "Categories for the practicing physicist" I came to learn there is a category whose objects are propositions $A,B,...$ and whose morphisms are proofs $f:A\rightarrow{B}$ that ...
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### Does the 2-functor $PsAlg\to \mathfrak{X}$ reflect equivalences?

Consider a $2$-monad $T: \mathfrak{X}\to \mathfrak{X}$ and consider its 2-category of pseudoalgebras $PsAlg$. There is a forgetful functor $U: PsAlg\to \mathfrak{X}$. Does this forgetful functor ...
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### 2-natural equivalence

Let $F, G: X\to Y$ be 2-functors. Is the following statement true? " A 2-natural transformation $\alpha : F\to G$ is a 2-natural equivalence if and only if each component $\alpha _ K$ is an ...
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### Catsters Video Question

The first Catsters video on adjunctions has just finished, at this time, describing adjunctions in 2-categorical terms. Basically, the idea is to whisker the adjoint functors and the (co)unit of ...
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### Strong epimorphic counit iff conservative right adjoint?

On page 13 of Lack and Street's Combinatorial Categorical Equivalences, it is written (but not proven) that: A right adjoint is conservative if and only if the components of the counit are strong ...
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### Duality 2-functor on adjunctions

This question is about the definition of the duality 2-functor in Hovey's book on Model categories, Section 1.4. There he defines the 2-category of categories with adjunctions as follows: objects ...