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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2-categorical universal property of the classifying category of a type theory
For example let us say we are in the setting of cartesian closed categories and the simply typed $\lambda$-calculus. Let $\mathtt{strCCCat}$ denote the $2$-category of strict cartesian closed ...
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Universal characterization of the arrow (interval, ordinal, 1-simplex) object? (Is $2$-category theory actually needed?)
In the "category of categories" $Cat$ (see e.g. here for a rigorous definition), there is a category $A$ with two objects and exactly one non-identity morphism. (Cf. "two-valued object&...
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The functor that maps a (co-)monad to its (co-)Eilenberg-Moore category
I have noticed that the function that maps a monad $T : C \to C$ to the Eilenberg-Moore category $C^T$ can easily be extended into a functor $E_C$ from the category of monads $\textbf{Mnd}_C$ to $\...
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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 $D$.
...
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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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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 ...
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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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The homotopy 2-category - Are homotopies unique up to homotopy?
Consider a homotopy $h: I \times X \rightarrow Y$ and denote
$$\begin{array}{rcl}
h_1: X & \rightarrow & Y\\
x & \mapsto & h(1,x).
\end{array}$$
Define the constant homotopy
$$\begin{...
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Is there a concept of "homotopy category of a 2-category"?
I apologize in advance for my naiveté. I'm completely missing the formal background to ask this question, but am very curious about one particular point.
I recently learned that there's the concept of ...
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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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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 ...
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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'...
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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 ...
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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,...
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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 ...
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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 (...
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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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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 &...
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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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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 ...
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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 $\...
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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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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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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:
...
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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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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 $M + N$ ...
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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 ...
user175304
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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 ...
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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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Coslice category in 2-categories
This question deals with certain morphisms and 2-morphisms in the definition of coslice categories in 2-categories. Though I'm aware of the notion of lax (co)slice categories, I believe the following ...
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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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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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