Co-H-space

A topological space with a co-multiplication; the dual notion is an $ H $- space.

Comments

The sum of two objects $ ( X, x _ {0} ) $ and $ ( Y, y _ {0} ) $ in the category of pointed topological spaces is the disjoint union of $ X $ and $ Y $ with $ x _ {0} $ and $ y _ {0} $ identified, and this point serves as base point; it can be realized (and visualized) as the subset $ X \times \{ y _ {0} \} \cup \{ x _ {0} \} \times Y $ of $ X \times Y $. A co- $ H $- space thus is a pointed topological space with a continuous mapping of pointed spaces $ \mu : Q \rightarrow Q \lor Q $, termed co-multiplication, such that the composites $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ { \mathop{\rm id} \lor \epsilon } Q \lor \{ q _ {0} \} \simeq Q $ and $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ {\epsilon \lor \mathop{\rm id} } Q $ are homotopic to the identity. Here $ \epsilon $ is the mapping which sends all of $ Q $ to the base point $ q _ {0} $. If the two composites $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ { \mathop{\rm id} \lor \mu } Q \lor Q \lor Q $ and $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ {\mu \lor \mathop{\rm id} } Q \lor Q \lor Q $ are homotopic to each other, the co-multiplication is called homotopy co-associative (or homotopy associative). A continuous mapping of pointed spaces $ r: Q \rightarrow Q $ is a homotopy co-inverse for $ \mu $ if the two composites $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ {( \mathop{\rm id} , r) } Q $ and $ Q \rightarrow ^ \mu Q \lor Q \rightarrow ^ {( r, \mathop{\rm id} ) } Q $ are both homotopic to $ \epsilon : Q \rightarrow Q $. Here for $ f: X \rightarrow Z $, $ g: Y \rightarrow Z $, $ ( f, g) $ is the mapping determined by the defining property of the sum in the category of pointed topological spaces, i.e. $ ( f, g) $ restricted to $ X $ is equal to $ f $, and $ ( f, g) $ restricted to $ Y $ is equal to $ g $. A co- $ H $- space with co-associative co-multiplication which admits a homotopy co-inverse is called an $ H $- co-group. Thus, an $ H $- co-group is a co-group object in the category $ {\mathcal H} {\mathcal t} {\mathcal p} $ of pointed topological spaces and homotopy classes of mappings.

References

[a1]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. I, Sect. 6