nLab
strict 2-category

Idea

• A strict 2-category is a directed 2-graph equipped with a composition operation on adjacent 1-cells and 2-cells which is strictly unital and associative.

• The concept of a strict 2-category is the simplest generalization of a category to a higher category. It is the one-step categorification of the concept of a category.

The term 2-category implicitly refers to a globular structure. By contrast, double categories are based on cubes instead. The two notions are closely related, however: every strict 2-category gives rise to several strict double categories, and every double category has several underlying 2-categories.

Notice that double category is another term for 2-fold category. Strict 2-categories may be identified with those strict 2-fold/double categories whose category of vertical morphisms is discrete, or those whose category of horizontal morphisms is discrete.

(And similarly, strict globular n-categories may be identified with those n-fold categories for which all cube faces “in one direction” are discrete. A similar statement for weak $n$-categories is to be expected, but little seems to be known about this.)

Definition

A strict 2-category, often called simply a 2-category, is a category enriched over Cat, where $\mathrm{Cat}$ is treated as the 1-category of strict categories.

Similarly, a strict 2-groupoid is a groupoid enriched over groupoids. This is also called a globular strict 2-groupoid, to emphasise the underlying geometry. The category of strict 2-groupoids is eqivalent to the category of crossed modules over groupoids. It is also equivalent to the category of (strict) double groupoids with connections.

They are also special cases of strict globular omega-groupoids, and the category of these is equivalent to the category of crossed complexes.

Details

Working out the meaning of ’$\mathrm{Cat}$-enriched category’, we find that a strict 2-category $K$ is given by

• a collection $\mathrm{ob}K$ of objects $a,b,c,\dots$, together with
• a hom-category $K(a,b)$ for each $a,b$, and
• a functor $1_a:1\to K(a,a)$ and a functor $\mathrm{comp}:K(b,c)\times K(a,b)\to K(a,c)$ for each $a,b,c$

satisfying associativity and identity axioms (given here).

As for ordinary ($\mathrm{Set}$-enriched) categories, an object $f\in K(a,b)$ is called a morphism or 1-cell from $a$ to $b$ and written $f:a\to b$ as usual. But given $f,g:a\to b$, it is now possible to have non-trivial arrows $\alpha :f\to g\in K(a,b)$, called 2-cells from $f$ to $g$ and written as $\alpha :f\Rightarrow g$. Because the hom-objects $K(a,b)$ are by definition categories, 2-cells carry an associative and unital operation called vertical composition. The identities for this operation, of course, are the identity 2-cells $1_f$ given by the category structure on $K(a,b)$.

The functor $\mathrm{comp}$ gives us an operation of horizontal composition on 2-cells. Functoriality of $\mathrm{comp}$ then says that given $\alpha :f\Rightarrow g:a\to b$ and $\beta :f\prime \Rightarrow g\prime :b\to c$, the composite $comp(\beta ,\alpha )$ is a 2-cell $\beta \alpha :f\prime f\Rightarrow g\prime g:a\to c$. Note that the boundaries of the composite 2-cell are the composites of the boundaries of the components.

We also have the interchange law: because $\mathrm{comp}$ is a functor it commutes with composition in the hom-categories, so we have (writing vertical composition with $\circ$ and horizontal as juxtaposition):

$$(\beta \prime \circ \beta )(\alpha \prime \circ \alpha )=(\beta \prime \alpha \prime )\circ (\beta \alpha )$$(\beta' \circ \beta)(\alpha' \circ \alpha) = (\beta' \alpha') \circ (\beta \alpha)

The axioms for associativity and unitality of $\mathrm{comp}$ ensure that horizontal composition behaves just like composition of 1-cells in a 1-category. In particular, the action of $\mathrm{comp}$ on objects $f,g$ of hom-categories (i.e. 1-cells of $K$) is the usual composite of morphisms.

More details

In even more detail, a strict $2$-category $K$ consists of

• a collection $\mathrm{Ob}K$ or ${\mathrm{Ob}}_K$ of objects or $0$-cells,
• for each object $a$ and object $b$, a collection $K(a,b)$ or ${\mathrm{Hom}}_K(a,b)$ of morphisms or $1$-cells $a\to b$, and
• for each object $a$, object $b$, morphism $f:a\to b$, and morphism $g:a\to b$, a collection $K(f,g)$ or $2{\mathrm{Hom}}_K(f,g)$ of $2$-morphisms or $2$-cells $f\Rightarrow g$ or $f\Rightarrow g:a\to b$,

equipped with

• for each object $a$, an identity $1_a:a\to a$ or ${id}_a:a\to a$,
• for each $a,b,c$, $f:a\to b$, and $g:b\to c$, a composite $f;g:a\to c$ or $g\circ f:a\to c$,
• for each $f:a\to b$, an identity $1_f:f\Rightarrow f$ or ${Id}_f:f\to f$,
• for each $f,g,h:a\to b$, $\eta :f\Rightarrow g$, and $\theta :g\Rightarrow h$, a vertical composite $\theta •\eta :f\Rightarrow h$,
• for each $a,b,c$, $f:a\to b$, $g,h:b\to c$, and $\eta :g\Rightarrow h$, a left whiskering $\eta ◃f:g\circ f\Rightarrow h\circ f$, and
• for each $a,b,c$, $f,g:a\to b$, $h:b\to c$, and $\eta :f\Rightarrow g$, a right whiskering $h▹\eta :h\circ f\Rightarrow h\circ g$,

such that

• for each $f:a\to b$, the composites $f\circ {id}_a$ and ${id}_b\circ f$ each equal $f$,
• for each $a\stackrel{f}{\to }b\stackrel{g}{\to }c\stackrel{h}{\to }d$, the composites $h\circ (g\circ f)$ and $(h\circ g)\circ f$ are equal,
• for each $\eta :f\Rightarrow g:a\to b$, the vertical composites $\eta •{Id}_f$ and ${Id}_g•\eta$ both equal $\eta$,
• for each $f\stackrel{\eta }{\Rightarrow }g\stackrel{\theta }{\Rightarrow }h\stackrel{\iota }{\Rightarrow }i:a\to b$, the vertical composites $\iota •(\theta •\eta )$ and $(\iota •\theta )•\eta$ are equal,
• for each $a\stackrel{f}{\to }b\stackrel{g}{\to }c$, the whiskerings ${Id}_g◃f$ and $g▹{Id}_f$ both equal ${Id}_{g\circ f}$,
• for each $\eta :f\Rightarrow g:a\to b$, the whiskerings $\eta ◃{id}_a$ and ${id}_b▹\eta$ equal $\eta$,
• for each $f:a\to b$ and $g\stackrel{\eta }{\Rightarrow }h\stackrel{\theta }{\Rightarrow }i:b\to c$, the vertical composite $(\theta ◃f)•(\eta ◃f)$ equals the whiskering $(\theta •\eta )◃f$,
• for each $f\stackrel{\eta }{\Rightarrow }g\stackrel{\theta }{\Rightarrow }h:a\to b$ and $i:b\to c$, the vertical composite $(i▹\theta )•(i▹\eta )$ equals the whiskering $i▹(\theta •\eta )$,
• for each $a\stackrel{f}{\to }b\stackrel{g}{\to }c$ and $\eta :h\Rightarrow i:c\to d$, the left whiskerings $\eta ◃(g\circ f)$ and $(\eta ◃g)◃f$ are equal,
• for each $f:a\to b$, $\eta :g\Rightarrow h:b\to c$, and $i:c\to d$, the whiskerings $i▹(\eta ◃f)$ and $(i▹\eta )◃f$ are equal,
• for each $\eta :f\Rightarrow g:a\to b$ and $b\stackrel{h}{\to }c\stackrel{i}{\to }d$, the right whiskerings $i▹(h▹\eta )$ and $(i\circ h)▹\eta$ are equal, and
• for each $\eta :f\Rightarrow g:a\to b$ and $\theta :h\Rightarrow i:b\to c$, the vertical composites $(i▹\eta )\circ (\theta ◃f)$ and $(\theta ◃g)\circ (h▹\eta )$ are equal.

The construction in the last axiom is the horizontal composite $\theta \circ \eta :h\circ f\to i\circ g$. It is possible (and probably more common) to take the horizontal composite as basic and the whiskerings as derived operations. This results in fewer, but more complicated, axioms.

Remarks

• A strict 2-category is the same as a strict omega-category which is trivial in degree $n\ge 3$.

• This is to be contrasted with a weak 2-category called a bicategory. In a strict 2-category composition of 1-morphisms is strictly associative and comoposition with identity morphisms strictly satisfies the required identity law. In a weak 2-category these laws may hold only up to coherent 2-morphisms.