Co-H-space
A topological space with a co-multiplication; the dual notion is an 
-space.
Comments
The sum of two objects 
and 
in the category of pointed topological spaces is the disjoint union of 
and 
with 
and 
identified, and this point serves as base point; it can be realized (and visualized) as the subset 
of 
. A co-
-space thus is a pointed topological space with a continuous mapping of pointed spaces 
, termed co-multiplication, such that the composites 
and 
are homotopic to the identity. Here 
is the mapping which sends all of 
to the base point 
. If the two composites 
and 
are homotopic to each other, the co-multiplication is called homotopy co-associative (or homotopy associative). A continuous mapping of pointed spaces 
is a homotopy co-inverse for 
if the two composites 
and 
are both homotopic to 
. Here for 
, 
, 
is the mapping determined by the defining property of the sum in the category of pointed topological spaces, i.e. 
restricted to 
is equal to 
, and 
restricted to 
is equal to 
. A co-
-space with co-associative co-multiplication which admits a homotopy co-inverse is called an 
-co-group. Thus, an 
-co-group is a co-group object in the category 
of pointed topological spaces and homotopy classes of mappings.
References
| [a1] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. I, Sect. 6 |
Co-H-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Co-H-space&oldid=46369