The following abstract nonsense is a bit tentative. See warning below.
Ordinary homotopy is a way to probe objects in an (infinity,1)-topos by mapping spheres into them:
the ordinary homotopy group of an object is the fiber over of the morphism
induced on , the homotopy category of an (infinity,1)-category associated to .
In this sense homotopy is the notion that is Eckmann-Hilton dual to cohomology.
For a more precise statement of homotopy in -toposes see section 6.5.1 of
Warning: notice that in that section 6.5.1 homotopy groups in an -topos are at least not manifestly defined in this way, though it should come close. Somebody should have a close look and sort this out.
This duality suggests that more generally we may be entitled to speak for and objects in of
as the homotopy of with co-coefficients in (or efficients in 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.
classes of special cases of homotopies with their own entries include (create page if you think it might work)