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
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