(i) lay the foundation for a homotopy theory of cosimplicial spaces, i.e. we show that it is possible to define, for cosimplicial spaces, notions of function space, weak equivalence, cofibration and fibration, which satisfy Quillen’s axioms for a closed simplicial model category (see Ch. VIII, 4.9), and then
(ii) combine this with the results of Chapter IX and obtain, for every cosimplicial space, an extended homotopy spectral sequence, which is an important tool in our study of the R-completion of a space in Part I.
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