of functors commutes (on the nose). In other words, the functor obtained by com-
posing (—)° with itself is the identity on A+. Notice also that the identities
- Jil'p
= u?'p
(ir^)° - irfp {)
hold between elementary operators and their duals.
7 (a useful characterisation of face and degeneracy operators).
The following facts about simplicial operators a: [n] [m] are sometimes of use:
(i) a is a degeneracy operator iff a(0) = 0, a(n) = m and for all j j ' G [n] we
have a(j') - a(j) j ' - j .
(ii) a is a face operator iff for all j
G [n] we have a(f) a(j)
j .
(iii) a is a face operator iff there exists a simplicial operator a: [n] [m n]
such that a(j) = j + a(j) for all j G [n].
8 (face-degeneracy factorisation). We may uniquely factor each
simplicial operator a: [n] - [m] into a composite a? o a
where a^ is a face
operator and
is a degeneracy. Furthermore, let im(a) denote the subset of [m]
given by im(a) = {i G [m] \ (3j G [^])ce(i) = i} then it is a trivial, but nonetheless
useful, fact that one simplicial operator a: [m] [n] factors though another
(3: [r] [n], that is to say there is some 7: [m] [r] with /? o 7 = a, if and
only if im(a) C im(/3).
2. The Algebraist's A and 2-Categories
We quickly review the theory of 2-categories, which will be useful for expressing
some of the "meta-theory" developed in the remainder of this chapter. Later on,
in chapter 3, we will take a second bite at the 2-category cherry and consider them
"in the small" when we review the algebraic theory of ^-categories.
9 (the cartesian closed category Cat). We will let Cat denote
the (huge) category of all (large) categories and functors between them. This is
cartesian closed, where the "function space" from a category C to a category V is
the functor category [C,V] which has:
Objects functors from C to X,
Maps natural transformations between such functors,
Composition the usual, point-wise, composition of natural transforma-
By definition, this function space is characterised by the adjunction
where x denotes the cartesian product of categories. In other words, there exists a
natural bijection between functors F : B x C V and F: B [C, V]. It follows
(see [Kel82]) that Cat is a rich "universe" over which we might enrich the homsets
of other categories. This observation leads to the following definition:
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