nLab co-H-space

Idea

The concept of co-H-space is the Eckmann-Hilton dual of that of H-spaces.

They are co-H-objects in the category of pointed topological spaces. Thus a co-H-space $(X, \phi)$ is a pointed space, $X$, together with a map $\phi: X \to X \vee X$ (the wedge sum), such that $p_i \circ \phi$ is homotopic to $1_X$, where $p_i, i = 1, 2$, are the projections $X \vee X \to X$. Alternatively, $(X, \phi)$ is a co-H-space if and only if $j \circ \phi$ is homotopic to $\Delta$, where $j: X \vee X \to X \times X$ is the inclusion and $\Delta: X \to X \times X$ is the diagonal map.

The importance of the notion is that $X$ is a co-H-space if and only if for every space $Y$, $[X, Y]$ has a binary operation with unit. Further properties of $\phi$ are of interest, in particular being (co)associative and having right and left (co)inverses. In this case $X$ is a cogroup. The suspension of a topological space is a cogroup.

Every co-H-space is path-connected, and its fundamental group is free.

Reference

• Martin Arkowitz, Co-H-spaces, chapter 23 of Ioan Mackenzie James (ed.), Handbook of Algebraic Topology