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Questions tagged [2-categories]

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

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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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1answer
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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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Categorical pull-backs and pasting laws

Let $\cal C$ be a category with pull-backs. Is it always true that given two objects $X$,$Y\in \cal C$ then $X\times_{Y} Y\cong X$? My guess is that we have to require at least that $\cal {C}$$(X,Y)$ ...
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2-Functor preserves left-exactness of localization

Let $C$, $D$ be categories and let $F\dashv G:C\rightleftarrows D$ be an adjunction such that the left adjoint $F$ preserves finite limits. Suppose $J:Cat\rightarrow Cat$ is a 2-functor. Does it hold ...
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The one-arrow category as a weighted limit in Cat

Many categories can be defined as weighted limits or colimits in the 2-category of categories Cat. For example the category 1 (one object with its identity) is the terminal object of Cat, the category ...
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About the Definition of Strict $2$-Natural Transformations between Strict $2$-Functors?

A strict $2$-category $\mathcal{C}$ consists of: (i) A horizontal category $\mathcal{C}^h:=(\mathcal{C}_0, \mathcal{C}_2, s_h, t_h, u_h, \circ_h)$; (ii) A vertical category $\mathcal{C}^v:=(\mathcal{...
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F-algebras in 2-categories

Given a functor $F : C \to C$, one can usually study the $F$-algebras: morphisms $\alpha : F X \to X$. Where can I read about its generalisation to 2-categories? I think that one can consider now "lax"...
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Weighted limits in the $Cat$-category of categories

What is a weighted limit in the $Cat$-category of categories, functors and natural transformations? I can find the general definition of a weighted limit for enriched categories in Kelly's book or ...
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How to define a weighted cone?

Let $F : I \to C$ be a diagram in $C$, and $N$ an object of $C$. A cone from $N$ to $F$ is a family of morphism $P_X : N \to F(X)$ such that for every morphism $f : i1 \to i2$ in $I$, $F(f) \circ P_X =...
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Contruction of Weighted Colimit in a 2-category

On page 306 of Kelly's Elementary Observations on 2-categorical Limits, it is explained that a weighted limit $\{F, G\}$ in a 2-category can be constructed as the equalizer of $v$ and $w$ in $$ (3.2) \...
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Construction of 2-limits in 2-categories

Limits in a category can be built as a combination of the basic limits that are products and equalizers. Is there a similar construction for 2-limits in a 2-category? If yes, is it from products and ...
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On the definition of 2-rigs

I am reading the nLab entry on 2-rigs. In its list of definitions, it says that a 2-rig category can be defined as a $Ab$-enriched category which is enriched monoidal. Why is the enrichment in $Ab$? ...
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Can every monad give rise to a monad transformer?

Can every monad give rise to a monad transformer? In the paper Calculating monad transformers with category theory by Oleksandr Manzyuk, one finds a construction of monad transformers as ...
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What is a Monad in the two category $Rel$?

The 2-category $Rel$ is a category with sets as $0$-cells, relations as $1$-cells (with relation composition as composition), and inclusions as $2$-cells (with vertical composition being the fact that ...
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Yoneda Lemma for 2-categories - lax version

Is there a sort of lax Yoneda Lemma for 2-categories? Here is what I seem to have proven (although I have not checked all the details): If $\mathcal{C}$ is a (weak) 2-category, $A$ is an object of $\...
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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 ...