In this chapter, we study the homotopy invariance of the functors SR(M) : RE → E associated to right modules over an operad R. In summary, we aim to prove that: – A weak-equivalence of R-algebras f : A ~ -→ B induces a weak-equivalence SR(M,f) : SR(M,A) ~ -→ SR(N,B) under reasonable assumptions on M, A and B, – A weak-equivalence of right R-modules f : M ~ -→ N induces a pointwise weak-equivalence of functors SR(f,A) : SR(M,A) ~ -→ SR(N,A), under reasonable assumptions on M, N and A. In §15.1, we assume that the right R-modules are cofibrant. In this context, the homotopy invariance properties hold under very mild assumptions on the R-algebras.
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© 2009 Springer-Verlag Berlin Heidelberg
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Fresse, B. (2009). Modules and homotopy invariance of functors. In: Modules over Operads and Functors. Lecture Notes in Mathematics(), vol 1967. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89056-0_15
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DOI: https://doi.org/10.1007/978-3-540-89056-0_15
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