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 is a pointed space, , together with a map (the wedge sum), such that is homotopic to , where , are the projections . Alternatively, is a co-H-space if and only if is homotopic to , where is the inclusion and is the diagonal map.
The importance of the notion is that is a co-H-space if and only if for every space , has a binary operation with unit. Further properties of are of interest, in particular being (co)associative and having right and left (co)inverses. In this case is a cogroup. The suspension of a topological space is a cogroup.