2-category
A small 2-category,
, is the first of higher order categories
constructed as follows.
- define Cat as the category of small categories
and functors
- define a class of objects
in
called `0
- cells'
- for all `0
-cells'
,
, consider a set denoted as “
” that is defined as
, with the elements of the latter set being the functors between the 0
-cells
and
; the latter is then organized as a small category whose
-`morphisms', or `
-cells' are defined by the natural transformations
for any two morphisms of
, (with
and
being functors between the `0
-cells'
and
, that is,
); as the `
-cells' can be considered as `2-morphisms' between 1-morphisms, they are also written as:
, and are depicted as labelled faces in the plane determined by their domains and codomains
- the
-categorical composition
of
-morphisms is denoted as “
” and is called the vertical composition
- a horizontal composition, “
”, is also defined for all triples of 0
-cells,
,
and
in
as the functor
which is associative
- the identities
under horizontal composition are the identities of the
-cells of
for any
in
- for any object
in
there is a functor from the one-object/one-arrow category
(terminal object) to
.
- The
-category
of small categories, functors, and natural transformations;
- The
-category
of internal categories in any category
with
finite limits, together with the internal functors and the internal natural transformations between such internal
functors;
- When
, this yields again the category
, but if
, then one obtains the 2-category of small double categories;
- When
, one obtains the
-category of crossed modules.
Remarks:
Contributors to this entry (in most recent order):
As of this snapshot date, this entry was owned by bci1.