By L. Gaunce Jr. Lewis, J. Peter May, Mark Steinberger, J.E. McClure

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**Example text**

Have t h e homotopy type of a G-CW spectrum. 13, kdU. and ; G ~ ( @ u a'). This implies t h a t smash products of equivariant c e l l s behave a p p r o p r i a t e l y . P eH preserves homotopies i n both v a r i a b l e s and t h e r e f o r e preserves homotopy types. 9. t h e r e a r e n a t u r a l isomorphisms EAE' , E and E' t o transport EAE' and t o functors on s t a b l e c a t e g o r i e s . 10. isomorphism of p a i r s The r e s t r i c t i o n t o a s i n g l e group i n s i g h t i s t h e simplest way t o G s e t up n o t a t i o n s , but t h e r e i s an a l t e r n a t i v e viewpoint t h a t i s sometimes u s e f u l .

GXU + GAU' X: X determined by varying G-maps f The of weak To make t h i s p r e c i s e , l e t jlJ and i d b U denote t h e c a t e g o r i e s of G-spaces of t h e weak homotopy type of 3-CW complexes (with basepoint) and of G-spectra of t h e weak homotopy type of 3-CW spectra. 13. ( Z, - h) and l o c a l i z a t i o n s a t t h e i r h ) ; here weak equivalences coincide with weak become E ~ + A (? ) The f u n c t o r s (e 3 ) - l ( h ~ 3 )---9h33 Proof. 5, t h e homotopy type of a x E @ & ( ?

3. 4. Q@'). 5 give t h e following r e s u l t . 2. The functor DAD' ( b u t not i n c l u s i o n p r e s p e c t r a o r s p e c t r a ) . If E and E' for D e G 6 4 D1 e GPOL', for E e and E" a')(DAD',D") G6)a(D,F(D1,D")) and D"E G @ ( a @ a l ) and and E" e G 8 ( & @ a 1 ) * preserves i n j e c t i o n and C-inclusion p r e s p e c t r a Both t h e prespectrum and spectrum l e v e l f u n c t o r s preserve i n j e c t i o n s , closed i n c l u s i o n s , and i n t e r s e c t i o n s of closed subobjects.