nLab
homotopical category

Context

Homotopy theory

(,1)(\infty,1)-Category theory

Contents

Idea

A homotopical category is a construction used in homotopy theory, related to but more flexible than a model category.

Definition

A homotopical category is a category with weak equivalences where on top of the 2-out-of-3-property the morphisms satisfy the 2-out-of-6-property:

  • If morphisms hgh \circ g and gfg \circ f are weak equivalences, then so are ff, gg, hh and hgfh \circ g \circ f.

Remarks

Simplicial localization

Every homotopical category CC “presents” or “models” an (infinity,1)-category LCL C, a simplicially enriched category called the simplicial localization of CC, which is in some sense the universal solution to inverting the weak equivalence up to higher categorical morphisms.

Related concepts

References

This definition is in paragraph 33 of

Revised on January 3, 2012 05:26:48 by Mike Shulman (173.8.161.189)