nLab
homotopy (as an operation)

• for disambiguation see homotopy

Idea

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

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

the ordinary homotopy group ${\pi }_n(X,x)$ of an object $X\in H$ is the fiber over $x\in X$ of the morphism

$$H(S^n,X)\to H(*,X)\simeq {\pi }_0(X)$$H(S^n, X) \to H({*},X) \simeq \pi_0(X)

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

$$H(S^n,X):={\pi }_0(H(S^n,X)).$$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 more precise statement of homotopy in $(\mathrm{\infty }\mathrm{,1})$-toposes see section 6.5.1 of

• Jacob Lurie, Higher Topos Theory .

Warning: notice that in that section 6.5.1 homotopy groups in an $(\mathrm{\infty }\mathrm{,1})$-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.

Remark

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

$$H(B,X):={\pi }_{0}H(B,X)$$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

• classes of special cases of homotopies with their own entries include (create page if you think it might work)

  • group homotopy
  • generalized (Eilenberg-Steenrod) homotopy
  • abelian cosheaf homotopy
  • nonabelian homotopy
  • Čech homotopy
  • differential homotopy
  • twisted homotopy