# Co-H-space

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