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HIGHER HOMOTOPY GROUPOIDS AND TODA BRACKETS

K.A. Hardie, K.H. Kamps and R.W. Kieboom

We describe a category {\bf 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 {\bf 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 {\bf htTop$^*$} is itself the homotopy category of a bicategory {\bf bTop$^*$} and so admits also a secondary composition operation.

Homology, Homotopy and Applications, Vol. 1, 1999, No. 4, pp 117-134
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