category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a branch of mathematics, Grothendieck's homotopy hypothesis states that the

∞-groupoid In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets (with the standard model structure). It ...

s are

space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...

s. If we model our ∞-groupoids as

Kan complex In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...

es, then the homotopy types of the geometric realizations of these sets give models for every homotopy type. It is conjectured that there are many different "equivalent" models for ∞-groupoids all which can be realized as homotopy types.

References

*John Baez
The Homotopy Hypothesis
* *

External links

*
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?Jacob Lurie's Home Page
Homotopy theory Higher category theory Hypotheses Conjectures {{categorytheory-stub

See also

References

External links