Homotopy homomorphisms and the classifying space functor

Abstract

We show that the classifying space functor $B: Mon \to Top*$ from the category of topological monoids to the category of based spaces is left adjoint to the Moore loop space functor $\Omega': Top*\to Mon$ after we have localized $Mon$ with respect to all homomorphisms whose underlying maps are homotopy equivalences and $Top*$ with respect to all based maps which are (not necessarily based) homotopy equivalences. It is well-known that this localization of $Top*$ exists, and we show that the localization of $Mon$ is the category of monoids and homotopy classes of homotopy homomorphisms. To make this statement precise we have to modify the classical definition of a homotopy homomorphism, and we discuss the necessary changes. The adjunction is induced by an adjunction up to homotopy between the category of well-pointed monoids and homotopy homomorphisms and the category of well-pointed spaces. This adjunction is shown to lift to diagrams. As a consequence, the well-known derived adjunction between the homotopy colimit and the constant diagram functor can also be seen to be induced by an adjuction up to homotopy before taking homotopy classes. As applications we among other things deduce a more algebraic version of the group completion theorem and show that the classifying space functor preserves homotopy colimits up to natural homotopy equivalences.