homotopy (as an operation)


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


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

Homotopy sets in homotopy categories


Ordinary homotopy is a way to probe objects in an (∞,1)-topos H\mathbf{H} by mapping spheres into them:

the ordinary homotopy group π n(X,x)\pi_n(X,x) of an object XHX \in \mathbf{H} is the fiber over xXx \in X of the morphism

[S n,X] Hτ 0Xπ 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


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

H(B,X):=π 0H(B,X) * H(B,X) := \pi_0 \mathbf{H}(B,X)_*

as the homotopy of XX with co-coefficients in BB.

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.


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

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