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homotopy (as an operation)

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  • This page is about homotopy as an operation. For homotopy as a transformation, see homotopy.

Homotopy sets in homotopy categories

Idea

The following abstract nonsense is a bit tentative. See warning below.

Urs Schreiber: the more coherent picture seems to be that developing at

I think we should eventually make that the main page on homotopy and merge into it from the material here only what deserves being kept.

Ordinary homotopy is a way to probe objects in an (∞,1)-topos H by mapping spheres into them:

the ordinary homotopy group π n(X,x) of an object X∈H is the fiber over x∈X of the morphism

H(S n,X)→H(*,X)≃π 0(X)H(S^n, X) \to H({*},X) \simeq \pi_0(X)

induced on H, the homotopy category of an (infinity,1)-category associated to H.

H(S n,X):=π 0(H(S n,X)) *.H(S^n ,X) := \pi_0(\mathbf{H}(S^n,X))_* \,.

In this sense homotopy is the notion that is Eckmann-Hilton dual to cohomology.

For a detailed discussion see

Remark

This duality suggests that more generally we may be entitled to speak for B and X objects in H of

H(B,X):=π 0H(B,X) *H(B,X) := \pi_0 \mathbf{H}(B,X)_*

as the homotopy of X with co-coefficients in B (or efficients in B if you want to be funny).

Examples of such constructions exist, but are rarely thought of (or even recognized as) generalizations of the notion of homotopy. Rather, by the above duality, the same situation is usually regarded in the context of cohomology, which, still by the above duality, is just as well.

An experimental attempt to dualize the [[cohomology]] page

Examples

Revised on July 23, 2010 20:43:31 by Toby Bartels (75.88.98.75)
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