2 1. SIMPLICIAL OPERATOR S AND SIMPLICIAL SETS

• For each n G N and i G [n] the operator a™ : [0] - [n] given by £™(0) = i

is called the

ith

vertex operator of [n].

• For each n G N we use the notation rf1 to denote the unique operator from

[n] to [0].

Unless doing so would introduce an ambiguity, we will tend to reduce notational

clutter by dropping the superscripts of these elementary operators.

OBSERVATION

4 (the simplicial identities). The following classical relationships

hold in A+ and are sufficient to fully characterise equalities between composites of

elementary face and degeneracy operators in A

+

:

• for any pair j i G [n + 1] we have 6™+1 o 5" = 5™+1 o 5f_x, and

• for any pair j i G [n — 1] we have r™_1 o cr ™ = &™~l o a^_x.

• for all j G [n] and i G [n — 1] we have

id[n_i]

en—1 _ _n—2

a"

1

o 5] = { id if j = i or j - i + 1,

oa^-1 if j i + l.

NOTATION

5 (partition operators). We say that a pair p, q G N is a partition

o f n G N i f p + g = n. For each such partition we have:

• face operators Jif,q: \p] • [n] given by ^q{i) = i and iL^'9: [#] - [n]

given by Jlf^(j) = j + p, and

• degeneracy operators it™: [n] • \p] given by

„

n

, v f i when i r and

\p when z p.

and

IT^'9 :

[n] - [q] given by:

p q

j 0 when i p and

i — p when i p.

We call these partition operators and, as is easily verified, they satisfy the

following partition identities:

¥ f 9 o TT?+9'r = IT?'9+r TT™ o ¥? + 9 ' r = TTfr o ¥ ™ + r Tlfr o 1T^9+r = Tl*+q'r

(1)

OBSERVATION

6 (duals of simplicial operators). There exists a canonical func-

tor (—)° from A+ to itself which "maps each ordinal to its dual as an ordered set".

Explicitly, [n]° = [n] and if a: [n] • [m] is a simplicial operator then, for each

i G [n], a°(i) = m — a(n — i). Clearly this dual functor is strictly involutive in the

sense that the diagram