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=13962