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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