## Strongly homotopy-commutative monoids revisited

### Summary

Summary: We prove that the delooping, i. e., the classifying space, of a grouplike monoid is an $H$-space if and only if its multiplication is a homotopy homomorphism, extending and clarifying a result of Sugawara. Furthermore it is shown that the Moore loop space functor and the construction of the classifying space induce an adjunction of the according homotopy categories.

### Mathematics Subject Classification

55P45, 55P35, 55R35

### Keywords/Phrases

H-spaces, classifying space, monoid, strongly homotopy commutative, homotopy homomorphism