homotopy invariance
Let be a functor from the category of topological spaces to some category . Then is called homotopy invariant if for any two homotopic maps between topological spaces and the morphisms and in induced by are identical.
Suppose is a homotopy invariant functor, and and are homotopy equivalent topological spaces. Then there are continuous maps and such that and (i.e. and are homotopic to the identity maps on and , respectively). Assume that is a covariant functor. Then the homotopy invariance of implies
and
From this we see that and are isomorphic in . (The same argument clearly holds if is contravariant instead of covariant.)
An important example of a homotopy invariant functor is the fundamental group ; here is the category of groups.
Title | homotopy invariance |
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Canonical name | HomotopyInvariance |
Date of creation | 2013-03-22 14:24:51 |
Last modified on | 2013-03-22 14:24:51 |
Owner | pbruin (1001) |
Last modified by | pbruin (1001) |
Numerical id | 4 |
Author | pbruin (1001) |
Entry type | Definition |
Classification | msc 55Pxx |
Related topic | HomotopyEquivalence |
Defines | homotopy invariant |