Ordinary homotopy is a way to probe objects in an (∞,1)-topos $\mathbf{H}$ by mapping spheres into them:
the ordinary homotopy group $\pi_n(X,x)$ of an object $X \in \mathbf{H}$ is the fiber over $x \in X$ of the morphism
In this sense homotopy is the notion that is Eckmann-Hilton dual to cohomology.
For a detailed discussion see
This duality suggests that more generally we may be entitled to speak for $B$ and $X$ objects in $\mathbf{H}$ of
as the homotopy of $X$ with co-coefficients in $B$.
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.
abelian cosheaf homotopy
nonabelian homotopy
differential homotopy
twisted homotopy
homotopy | cohomology | homology | |
---|---|---|---|
$[S^n,-]$ | $[-,A]$ | $(-) \otimes A$ | |
category theory | covariant hom | contravariant hom | tensor product |
homological algebra | Ext | Ext | Tor |
enriched category theory | end | end | coend |
homotopy theory | derived hom space $\mathbb{R}Hom(S^n,-)$ | cocycles $\mathbb{R}Hom(-,A)$ | derived tensor product $(-) \otimes^{\mathbb{L}} A$ |