Higher homotopy groupoids and Toda brackets

Abstract

We describe a category htTop whose objects are pointed continuous maps and whose morphisms are generated under composition by the tracks (relative homotopy classes) of homotopies. For example, if $m_t:hk\to *$ is a nullhomotopy then its track is a morphism from $k$ to $h$. The composition of tracks in htTop amounts to a sharpening of the classical secondary composition operation (Toda bracket). Standard properties of the Toda bracket can be derived in this setting. Moreover we show that htTop is itself the homotopy category of a bicategory htTop and so admits also a secondary composition operation.