# 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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### 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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### 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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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 $\... • 1,260 1 vote 1 answer 193 views ### Construction of the coproduct of two monoidal categories Consider the 2-category$\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$... • 334 1 vote 1 answer 125 views ### 2-functor and CAT Let$\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}$naturally defined as a ... 3 votes 1 answer 146 views ### Why Strict 2 group is equivalent to Group object in Category of Categories? Standard definition of Strict 2-Group says that it is a Strict Monoidal Category in which every morphism is invertible and each object has a strict inverse. Also it is a well known fact that Strict ... 2 votes 1 answer 114 views ### Ind-completion of a 2-category If$\mathcal{C}$is a category, there is a well known construction called the Ind-completion of$\mathcal{C}$, indicated by$\text{Ind}(\mathcal{C})$. This can be equivalently defined in several ways: ... • 581 5 votes 1 answer 161 views ### Books/Lecture notes about 2-categories. Are there good books or lecture notes just about 2-categories?(not about higher categories nor$\infty$-categories) (I'm studying fibered categories for the descent theory of quasi-coherent sheaves. ... • 311 0 votes 1 answer 129 views ### Understanding 2-category theory There are a lot of examples of categories, functors and natural transformations — one can find them anywhere. On the contrary (weak) 2-categorical stuff seems to be more subtle. I have comprehended ... • 389 4 votes 1 answer 104 views ### Right (bi)adjoint of the inclusion of$\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}$has ... • 581 4 votes 1 answer 123 views ### Is there such thing as a morphism of natural maps and what does it look like? I'm writing software to be general, so right now I'm writing a NaturalMap class which will be a graphical arrow that goes between any two ... • 19.5k -1 votes 1 answer 33 views ### Can we see FdHlb as a 2Category of groupoids? Can we see a finite dimensional Hilbert space,$H$as a groupoid if we include the unitary endomorphisms of$H$? It would be like a category with a single object and just isos. If so, can we take a ... • 1,079 2 votes 1 answer 109 views ### Description of the 2-monads for (strict) monoidal categories The nLab article on 2-monads says: For example, ordinary (non-strict) monoidal categories are the strict algebras for a strict 2-monad$T_{MC}$on$Cat$, but usually we care about pseudo, lax, and ... • 164 2 votes 0 answers 135 views ### Uniqueness up to unique isomorphism of quasi inverse of an arrow in a strict 2-category I just started to learn the basics of strict 2-categories, and reading some notes I've found the following simple proposition: Let C be a strict 2-category, and$F:X \rightarrow Y$a 1-cell (i.e.... • 581 1 vote 1 answer 85 views ### Confusion with (strict) 2-adjoints This is an elementary question on 2-categories, namely on the naturally arising notion of 2-adjunction when strict 2-functors are involved. Perhaps I don't understand some "internal" universal ... • 12.7k 2 votes 1 answer 94 views ### How to precisely state how "strict 2-category" relates to "bicategory in which all coherence 2-cells are identities"? Question. It is widely known that strict 2-categories "are" bicategories for which all coherence 2-cells are identities. However, how to make the "are" precise here? Remarks. I have seen ... • 1,228 1 vote 0 answers 124 views ### Coslice category in 2-categories This question deals with certain morphisms and 2-morphisms in the definition of coslice categories in 2-categories. Though I'm aware of the notion of lax (co)slice categories, I believe the following ... • 225 1 vote 0 answers 44 views ### What's the name for this kind of 2-categorical unsymmetric pullback? What is the name for this weak version of a 2-categorical pullback? The solid square commutes up to a not necessarily invertible 2-morphism$\eta$and has the following universal property: For any ... • 2,555 1 vote 1 answer 69 views ### Proof that the left or right adjoint composed after a diagonal morphism in a 2-category is the identity Suppose I have some strict 2-category$\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$. ... • 558 0 votes 1 answer 99 views ### Categorical pull-backs and pasting laws Let$\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)$... • 323 4 votes 0 answers 64 views ### 2-Functor preserves left-exactness of localization Let$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 ... • 644 4 votes 1 answer 278 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 ... • 1,260 1 vote 0 answers 95 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{...
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"...