nLab homotopy (as an operation)

Theorems

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

Homotopy sets in homotopy categories

Idea

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

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

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

For a detailed discussion see

Remark

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

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

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

homotopycohomologyhomology
$[S^n,-]$$[-,A]$$(-) \otimes A$
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space $\mathbb{R}Hom(S^n,-)$cocycles $\mathbb{R}Hom(-,A)$derived tensor product $(-) \otimes^{\mathbb{L}} A$

Revised on June 29, 2012 16:51:26 by Urs Schreiber (89.204.138.61)