nLab
homotopy (as an operation)

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

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Idea

Ordinary homotopy is a way to probe objects in an (∞,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

$$[S^n,X{]}_H\to {\tau }_{0}X\simeq {\pi }_0(X)$$[S^n, X]_{\mathbf{H}} \to \tau_0 X \simeq \pi_0(X)

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

For a detailed discussion see

• homotopy groups in an (∞,1)-topos

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 $B$.

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.

Examples

• group homotopy

• generalized (Eilenberg-Steenrod) homotopy

• abelian cosheaf homotopy

• nonabelian homotopy

• Čech homotopy

• differential homotopy

• twisted homotopy

Related concepts

	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 $ℝ\mathrm{Hom}(S^n,-)$	cocycles $ℝ\mathrm{Hom}(-,A)$	derived tensor product $(-){\otimes }^{𝕃}A$

• homotopy, homotopy group,

• homotopy equivalence, weak homotopy equivalence

• homotopy groups in an (∞,1)-topos .