cohomology

# Contents

## Idea

Where homotopy groups are groups of homotopy classes of maps out spheres, $\pi_n(X)\coloneqq [S^n \to X]$, cohomotopy groups are groups of homotopy classes into spheres, $\pi^n(X) \coloneqq [X \to S^n]$.

## Relation to cohomology

The notion of cohomology – in its generalization beyond generalized (Eilenberg-Steenrod) cohomology to nonabelian cohomology – is related by abstract duality to homotopy. In the (infinity,1)-topos Top this is traditionally called Eckmann–Hilton duality.

Therefore it does make sense to speak of general cohomology as co-homotopy.

Indeed, cohomology is a concept dual to homology only in the very restrictive simple case of chain homology and cohomology. And this is really just because in this simple case it so happens that homotopy of chain complexes is their homology. This is part of the statement of the Dold-Kan correspondence: the nerve operation on chain complexes $N : Ch_+ \to SimpAb$ identifies chain homology groups with simplicial homotopy groups.

So on general grounds the word “cohomotopy” is actually better suited than “cohomology” for a concept of such fundamental importance.

## References

Revised on June 6, 2015 06:12:47 by Urs Schreiber (50.207.161.2)