# The Eilenberg-Watts theorem in homotopical algebra - Mathematics > Algebraic Topology

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Abstract: The object of this paper is to prove that the standard categories in whichhomotopy theory is done, such as topological spaces, simplicial sets, chaincomplexes of abelian groups, and any of the various good models for spectra,are all homotopically self-contained. The left half of this statementessentially means that any functor that looks like it could be a tensor productor product, or smash product with a fixed object is in fact such a tensorproduct, up to homotopy. The right half says any functor that looks like itcould be Hom into a fixed object is so, up to homotopy. More precisely, supposewe have a closed symmetric monoidal category resp. Quillen model category M.Then the functor T {B} that takes A to A tensor B is an M-functor and a leftadjoint. The same is true if B is an E-E-bimodule, where E and E- are monoidsin M, and T {B} takes an E-module A to A tensored over E with B. Define aclosed symmetric monoidal category resp. model category to be leftself-contained resp. homotopically left self-contained if every functor Ffrom E-modules to E-modules that is an M-functor and a left adjoint resp. anda left Quillen functor is naturally isomorphic resp. naturally weaklyequivalent to T {B} for some B. The classical Eilenberg-Watts theorem inalgebra then just says that the category of abelian groups is leftself-contained, so we are generalizing that theorem.

Author: ** Mark Hovey**

Source: https://arxiv.org/