GAfor (idi(G))oAand
fiF for /xo(idi(F)).
These two derived operations are so useful that they have been respectively dubbed
left and right whiskering by Street [Str96]. Indeed it is possible (and quite infor-
mative) to re-cast the theory of 2-categories purely in terms of these whiskering
operations and vertical composition alone. That this is the case follows directly
from the observation that the identity and middle four interchange rules imply that
for horizontally composable 2-cells A: F F' and ji: G G' we have
(G'X) (fiF) = fio\ = (nF') (GA)
13 (Cat as a 2-category). We know, from Kelly [Kel82], that
me may immediately enrich the cartesian closed category Cat over itself, by making
[B,C] the enriched homset between categories B and C. This 2-category structure
is often best described explicitly in terms of whiskering operations, it has:
O-cells (large) categories, 1-cells functors between these and 2-cells nat-
ural transformations between those.
vertical composition of 2-cells given by the usual "point-wise" composite
of natural transformations.
left whiskering of a natural transformation A: F F': B C by
a functor G: C V is formed by applying G "point-wise" to the com-
ponents of A, that is (GA)& ^ G(A&) for each b G obj(i3).
right whiskering of a natural transformation \i: G == G': C V
by a functor F: B C is obtained by re-indexing the components of \i
using the action of the functor F on objects, that is (/xF)&
each b G obj(S).
14 (A
as a 2-category). Since each object of A
is an ordered
set it follows that each of its homsets A+([n], [m]) possesses a natural partial order
, given by
a (3 if and only if (Vj G [n])(a(j) P(j))
Furthermore, since each map in A
is order preserving, we also know that compo-
sition preserves these partial orders, in the sense that if a (5 in A+([ra], [m]) and
a' (3f in A+([m], [r]) then (a' o a) ((3' o (3) in A+([n], [r]).
It follows that, under these partial orders on its homsets, A
becomes a partial
order enriched category or, in other words, a 2-category each homset of which is a
partial order.
In fact, this is simply the full sub-2-category of Cat on those O-cells obtained
by considering each totally ordered set [n] (n G N) as a category in the usual way.
That is to say, think of [n] as a category with objects {0,1,...,n} and a unique
arrow i j for each pair of integers with i j .
It is worth noting that the following useful inequalities (2-cells) hold between
(composites of) face and degeneracy operators:
for n 1 and j G [n 1] we have 8™
id[n] and 5^+1
if n 1 and j k G [n] then 8% 8™ and a^'1 cr£_1.
15 (adjoints in the 2-category A+). In the sequel we will have
occasion to consider adjoint pairs a H (3 of simplicial operators. These can be
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