Questions tagged [2-categories]
A 2-category is a category with "morphisms between morphisms".
2
votes
1answer
53 views
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: ...
3
votes
1answer
64 views
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. ...
0
votes
1answer
99 views
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 ...
4
votes
1answer
54 views
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 ...
3
votes
1answer
71 views
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 ...
-1
votes
1answer
30 views
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 ...
2
votes
1answer
47 views
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 ...
2
votes
0answers
75 views
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....
1
vote
1answer
57 views
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 ...
2
votes
1answer
77 views
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 ...
1
vote
0answers
31 views
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 ...
1
vote
1answer
50 views
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 $\Delta:A\to A^X$. ...
0
votes
1answer
67 views
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}$$(X,Y)$ ...
4
votes
0answers
55 views
2-Functor preserves left-exactness of localization
Let $C$, $D$ be categories and let $F\dashv G:C\rightleftarrows D$ be an adjunction such that the left adjoint $F$ preserves finite limits. Suppose $J:Cat\rightarrow Cat$ is a 2-functor. Does it hold ...
2
votes
1answer
137 views
The one-arrow category as a weighted limit in Cat
Many categories can be defined as weighted limits or colimits in the 2-category of categories Cat. For example the category 1 (one object with its identity) is the terminal object of Cat, the category ...
1
vote
0answers
51 views
About the Definition of Strict $2$-Natural Transformations between Strict $2$-Functors?
A strict $2$-category $\mathcal{C}$ consists of:
(i) A horizontal category $\mathcal{C}^h:=(\mathcal{C}_0, \mathcal{C}_2, s_h, t_h, u_h, \circ_h)$;
(ii) A vertical category $\mathcal{C}^v:=(\mathcal{...
2
votes
0answers
52 views
F-algebras in 2-categories
Given a functor $F : C \to C$, one can usually study the $F$-algebras: morphisms $\alpha : F X \to X$. Where can I read about its generalisation to 2-categories? I think that one can consider now "lax"...
0
votes
1answer
204 views
Weighted limits in the $Cat$-category of categories
What is a weighted limit in the $Cat$-category of categories, functors and natural transformations?
I can find the general definition of a weighted limit for enriched categories in Kelly's book or ...
2
votes
1answer
192 views
How to define a weighted cone?
Let $F : I \to C$ be a diagram in $C$, and $N$ an object of $C$. A cone from $N$ to $F$ is a family of morphism $P_X : N \to F(X)$ such that for every morphism $f : i1 \to i2$ in $I$, $F(f) \circ P_X =...
0
votes
1answer
91 views
Contruction of Weighted Colimit in a 2-category
On page 306 of Kelly's Elementary Observations on 2-categorical Limits, it is explained that a weighted limit $\{F, G\}$ in a 2-category can be constructed as the equalizer of $v$ and $w$ in
$$
(3.2) \...
3
votes
1answer
330 views
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 ...
3
votes
0answers
40 views
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 $Ab$? ...
1
vote
1answer
163 views
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 ...
0
votes
1answer
145 views
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 ...
3
votes
1answer
285 views
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 $\...
2
votes
1answer
110 views
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 ...
3
votes
1answer
308 views
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 ...
2
votes
0answers
52 views
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 ...
1
vote
1answer
72 views
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 ...
0
votes
1answer
79 views
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 ...
2
votes
1answer
143 views
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 ...
6
votes
2answers
138 views
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 ...
