Questions tagged [2-categories]
A 2-category is a category with "morphisms between morphisms".
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questions
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a parallel notion to bicategory and 2category [duplicate]
How is called the general 2category which has as associator arbitrary natural transformation rather than a natural isomorphism as for bicategory or idenity for 2category ?
I know that this notion for ...
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0answers
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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 ?
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Natural transformations and 2-cells
I have a strict $2$-category $\mathcal{C}$ having $2$-cells $\sigma : gh \Longrightarrow fh$, and $\delta : h \Longrightarrow h^2$, where $f : A \longrightarrow B$, $g : A \longrightarrow B$ and $h: A ...
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35 views
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 ...
4
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0answers
33 views
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'...
3
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0answers
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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 ...
0
votes
1answer
36 views
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,...
5
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1answer
47 views
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 ...
3
votes
1answer
121 views
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 (...
4
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1answer
70 views
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 ...
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0answers
33 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 &...
3
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0answers
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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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0answers
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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 ...
4
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1answer
233 views
Kleisli adjunction in a (weak) 2-category
In the 2-category of 1-categories, each monad $T$ on a category $\mathcal C$ determines a Kleisli category $\mathcal{C}_T$ and the so-called Kleisli adjunction between categories $\mathcal C$ and $\...
5
votes
2answers
149 views
Composition of adjunctions in a (weak) 2-category
Let $(f_1, g_1, \varepsilon_1, \eta_1)$ and $(f_2, g_2, \varepsilon_2, \eta_2)$ be adjunctions in a (weak) 2-category. Then there is an adjunction $(f_2 \circ f1, g1 \circ g2, \varepsilon, \eta)$. I ...
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1answer
53 views
Dualizing 2-categorical results in the context of locally small categories
Monad, comonad and adjunction are 2-categorical notions. Results about them can be dualized as shown in this answer.
In the second part the answer, the dualization is successfully applied to the ...
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1answer
116 views
Proving that a diagram involving associators and unitors commutes in a 2-category
In a 2-gategory, by coherence, the following equation holds for all $f : x \to y$ and $g : y \to z$:
$$
\rho_{g * f}
\; = \;
\textit{id}_g * \rho_f
\; \circ \;
\alpha_{g, f, I_x}
$$
Diagrammatically:
...
1
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1answer
63 views
An elementary question about horizontal composition in (weak) 2-categories
Let:
$\mathcal{K}$ be a (weak) 2-category
$A$, $B$ and $C$ be objects of $\mathcal{K}$
$f_1, f_2 : A \to B$ and $g : B \to C$ be 1-cells
$\alpha, \beta : f_1 \to f_2$ be 2-cells
Assuming that $\...
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1answer
157 views
Construction of the coproduct of two monoidal categories
Consider the 2-category $\mathbf{MonCat}$ of monoidal categories with strong monoidal functors and monoidal natural transformations. Where can I find an explicit construction of the coproduct $M + N$ ...
1
vote
1answer
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2-functor and CAT
Let $\cal K$ be a small category.
Let $\cal A$ be a subcategory of $\mathbf {CAT}$ and $U:{\cal A}\hookrightarrow{\mathbf {CAT}}$ the underlying functor. Now how is
$U^{\cal K}$ naturally defined as a ...
3
votes
1answer
113 views
Why Strict 2 group is equivalent to Group object in Category of Categories?
Standard definition of Strict 2-Group says that it is a Strict Monoidal Category in which every morphism is invertible and each object has a strict inverse.
Also it is a well known fact that Strict ...
2
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1answer
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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: ...
5
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1answer
108 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
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1answer
116 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
75 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
101 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
33 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
74 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 ...
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0answers
108 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
70 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
89 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 ...
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0answers
43 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
63 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
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1answer
84 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
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0answers
58 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
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1answer
214 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 ...
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0answers
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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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0answers
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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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votes
1answer
278 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
233 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 =...
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1answer
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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) \...
3
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1answer
537 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
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0answers
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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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0answers
268 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 ...
1
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1answer
222 views
What is a Monad in the two category $\mathsf{Rel}$?
The 2-category $\mathsf{Rel}$ 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 ...
4
votes
1answer
404 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
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1answer
115 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
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1answer
335 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
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0answers
55 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
79 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 ...
