n-category = (n,n)-category
n-groupoid = (n,0)-category
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 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 -categories is to be expected, but little seems to be known about this.)
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.
Working out the meaning of ’-enriched category’, we find that a strict 2-category is given by
satisfying associativity and identity axioms (given here).
As for ordinary (-enriched) categories, an object is called a morphism or 1-cell from to and written as usual. But given , it is now possible to have non-trivial arrows , called 2-cells from to and written as . Because the hom-objects 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 given by the category structure on .
The functor gives us an operation of horizontal composition on 2-cells. Functoriality of then says that given and , the composite is a 2-cell . 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 is a functor it commutes with composition in the hom-categories, so we have (writing vertical composition with and horizontal as juxtaposition):
(\beta' \circ \beta)(\alpha' \circ \alpha) = (\beta' \alpha') \circ (\beta \alpha)
The axioms for associativity and unitality of ensure that horizontal composition behaves just like composition of 1-cells in a 1-category. In particular, the action of on objects of hom-categories (i.e. 1-cells of ) is the usual composite of morphisms.
In even more detail, a strict -category consists of
The construction in the last axiom is the horizontal composite . 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.
A strict 2-category is the same as a strict omega-category which is trivial in degree .
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.