# 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 |

**How to Cite This Entry:**

Co-H-space.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Co-H-space&oldid=13962