DEFINITION 10 (2-category). A 2-category is a category enriched in the carte-
sian closed category Cat Correspondingly, a 2-functor is simply an Cat-enriched
functor between 2-categories.
The canonical reference for the theory of enriched categories is, of course,
Kelly's comprehensive book [Kel82] - to which we recommend the reader.
11 (2-categories explicitly). Fundamentally, a 2-category is sim-
ply a category C_ in which each homset C(C, D) is itself a category (object of Cat).
This immediately implies that our 2-category contains 3 sorts of objects:
0-cells which are the objects of C (for which we use symbols A, B, G,...),
1-cells which are the objects of the homsets C_(A, B) (for which we use
symbols F, G, if,...),
2-cells which are the arrows of the homsets C(-A, B) (for which we use
Greek symbols A, /i, z/,...).
Commensurate with the two layers of "category-ness" involved in C_ we get two
distinct category structures on these cells:
horizontal with objects which are 0-cells, arrows which are 1-cells and
2-cells and compositional structure given by (o, domo, codo, ido), whose
composition we often simply write as juxtaposition.
vertical with objects which are 1-cells, arrows which are 2-cells and com-
positional structure given by (•, domi, codi, idi).
In diagrams and running text we will tend to use single arrows F: A - B
to denote 1-cells (with A =
and B = codo(F)) and double arrows
: A : F =^ G to denote 2-cells (with F = domi(A) and G = codi(A)).
These category structures must satisfy a number of compatibility conditions
which bind them together intimately. Most important amongst these are:
globularity: for 2-cells A G C_ we have domo(domi(A)) = domo(codi(A))
and codo(domi(A)) = codo(codi(A)).
middle four interchange: if we have 2-cells A, A',/i and // then the
equality (// o A') (/JL o A) = (// /i) o (A' A) holds whenever the various
composites involved are defined. Notice that the composite on the left of
this equality is defined iff that on its right is defined.
The globularity condition on a 2-cell A implies that we may picture it as follows:
where F = domi(A), G = codi(A), A = domo (A) and B = codo (A).
Finally, a 2-functor T: C_ V_ may be thought of explicitly as a map which
maps the cells of C to those of P in a way which acts functorially on both horizontal
and vertical category structures
12 (whiskering). We commonly identify 0-cells with their corre-
sponding identity 1-cells (under ido),
a n
d 1-cells in turn with their identity 2-cells
(under idi). For instance, we will often use this convention, and the one that allows
us to replace o by juxtaposition, to write things like
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