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 n-ads 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 2-ads
(pairs) ({P}, {P}) and ({P},0) respectively, where P is a point.
(7) If S is an (n + l)-ad, we can form an n-ad 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 n-ad 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 well-defined 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) =
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