2. TH E ALGEBRAIST'S A AND 2-CATEGORIES 3

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

(JL?'9)°

- Jil'p

(JJ™)°

= u?'p

(TT?'9)°

=

TT*'P

(ir^)° - irfp {)

hold between elementary operators and their duals.

OBSERVATION

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

jf

G [n] we have a(f) — a(j)

jf

— 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].

OBSERVATION

8 (face-degeneracy factorisation). We may uniquely factor each

simplicial operator a: [n] - [m] into a composite a? o a

d

where a^ is a face

operator and

ad

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.

OBSERVATION

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-

tions.

By definition, this function space is characterised by the adjunction

— [c,*] —

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: