nLab
co-H-space

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

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