4
PRELIMINARIES
(3) We define a functor S{ :
2n~l

2n
by 6i(a) = d^a) U {i}. This corre
sponds just to omitting the ith subset.
(4) Given / as in (2), we can take the inverse image of subsets. In particular,
di defines Si : 2
n
— 2
n _ 1
(1 ^ i ^ n). This corresponds to introducing
151 as new ith subset.
(5) In a category with initial object (the empty set or a base point), we can
introduce this object as new ith subset. We will denote the corresponding
operation on nads by Oi: we can define on simply by
anS(X) = 0 ]
« C { l , . . . , n  l } .
anS(a U {n}) = S{a) J
The operations %, Si, Si, &i satisfy four analogues of the usual semi
simplicial identities, with minor changes.
(6) Given an (ra + l)ad S and an (n + l)ad T we define an (ra + n + l)ad
SxT. If (for simplicity) we adjust notation so that T is defined on subsets
of {ra + 1,..., ra + n}, then for a C { 1 , . . . , ra}, (3 C {ra + 1,..., ra + n},
we define simply
S x T(a U 13) = S(a) x T{0).
One can regard sn and an as the operations of multiplying by the 2ads
(pairs) ({P}, {P}) and ({P},0) respectively, where P is a point.
(7) If S is an (n + l)ad, we can form an nad by (e.g.) amalgamating the last
two subspaces. We will only use this construction in the case where these
two subspaces are disjoint, and will denote the nad by cS.
There are many other operations, and many further relations between these:
we will not attempt to list them here.
We next give a straightforward analogue of the usual mapping cylinder con
struction for converting maps into inclusion maps. Suppose X a functor from 2
n
to the category of topological spaces : we will define the mapping cube, M(X),
a topological (n + l)ad. Begin with the disjoint union
( J { X ( o ) x /
a
' : a € { l , 2
)
. . . , n } }
)
where a! denotes the complement of a in { 1 , . . . , n}. Now for i : a C /?, we have
ft C a' and will identify each (x, t) e X(a) x P' with (X(i)(x),t) eX(/3)x P'.
Let M(X) be the identification space. There is a welldefined projection
^ i l M p O l ^ 1 ' 2 '    ^ .
For t e
/i1'2'™},
we set t
_1
{0} = {i : 1 i ^ n and t(i) = 0}, and similarly
for ^_1{1} Then p^it) can be identified with X ^ r 1 ^ } ) . Define M(X)(a) =