Algebraic theories by Wraith, Gavin

By Wraith, Gavin

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The v e r i f i c a t i o n i s again immediate. sets 61" = a@ a 4 0 term i s QO. i n with GbLn, un. O= GJ. 7, and Ggkn U onto t h e f i r s t summand. The f i r s t equivalence is ~ * ( E A Ys) EhCrnY by an easy d i r e c t inspection of there results a unit When Y = SO, d e f i n i t i o n s on t h e prespectrum l e v e l . give t h i s category a closed symmetric monoidal s h c t u r e . Clbserve t h a t , by v i r t u e of our conventions on G-universes, t h e r e is a t l e a s t one G-linear isometry un + U f o r each n and t h u s each is 9 ( u n ,u) Define a n a t u r a l commutativity isomorphism (ii) t h i s point of view w i l l be given i n [107], and we t u r n here t o t h e i n t e r n a l i z a t i o n process.

5 t h a t It i s obvious t h a t (i) r. E"E*x E*E"X f o r a space X. Applying t h i s t o X = YG and passing t o a d j o i n t s , $4. Change of groups functors and isomorphisms we obtain a n a t u r a l map Let a: H + G a G-universe U H-universe v i a be a homomorphism of compact Lie groups. may a l s o be regarded a s an indexing s e t i n a, An indexing s e t U a regarded a s an and t h e r e r e s u l t s a f o r g e t f u l functor c": 3 + 4uG on t h e l e f t and 1": G I iGdU i n t h e middle and on t h e r i g h t .

By analogy, we say t h a t an 3 (N)-cw spectrum i s an N-free equivalence GJU be an i n c l u s i o n of G-universes. 8. Moreover, E' e E admits a U'-representation by If A'Z"X (and thus t o E E GIU f i l t r a t i o n of i*Eh = En K hence We c o n s t r u c t E, E' and an equivalence s t a r t i n g with Eb = have been constructed. sfl ~3 e *. E' 6 Let GBU c G ~ U ' and note t h a t be i t s c o f i b r e . G-CW spectrum, then G-CW spectrum E'. i s a f i n i t e N-free Moreover, E admits a uN-represen- i s unique up t o equivalence E' E spectrum, then G-CW X-~X"X) f o r some f i n i t e N-free G-CW E i s equivalent t o complex and some X Z.

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