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By Myles Tierney

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Additional resources for Categorical Constructions in Stable Homotopy Theory: A Seminar given at the ETH, Zürich, in 1967

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O o(Ign-I ker (0 i : (SX)n (SX) n_l} = N o(i4n-I ker {X~i Xn_2 } = (NX) n-1 the : Xn-i need This is the clear w h y we have of a chain it w o u l d be n e c e s s a r y we n e v e r these to i n t r o d u c e signs, (obviously inductive N ker using n we have and for > o On n Xn_ 1 agreement signs simpler : FD f o that of w to h a v e into equivalence shows the c o n s t r u c t i o n s in b o t h cases In o r d e r it s e e m e d natural) s~e ' sign n~ely shows and w the above relation of an that ) FD f~ m form the p a i r for and this FD-module.

1 Note a simplicial spectrum has infinitely many faces and degeneracies in each dimension, of ~ As with simplicial sets, a simplex be the full subcategory of that since Let o the subcategory with the same objects, but whose morphisms are injective in each dimension. be deqenerate if such that i > m(~). that commutes with the faces and degeneracies. category, ~ o ~i' ~' and which we call We denote also by S S, given by the restriction ~''i We prove first a result for spectra whose analogue for simplicial sets was quoted in w We give the proof in detail, simplicial sets.

O Thus, (DX) n-I etc. Thus, for T u ~ o. define since q S > Jl D~ (DX) n Jl S > Xn = But contradicting < m form Jr we n s]qu u G where for . J1 for = 0 o O / u ~ X Suppose n > (FDf+,s) if k and ) m(u) since then r is u ~ Nkx. - in e x a c t l y the same w a y as w FD-modules of degree Namely, Consider if X ~ FD f~ there is a n a t u r a l i n j e c t i o n of n ) SX b y the i n c l u s i o n of the d i r e c t s u m m a n d X of n (SX) n+l" the s e q u e n c e Y Y X and put, - I y : X g i v e n in d i m e n s i o n 53 ) SX ) S2X ~ .

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