nLab
homotopical category

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(∞,1)-Category theory

(∞,1)-category theory

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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 h∘g and g∘f are weak equivalences, then so are f, g, h and h∘g∘f.

Remarks

Simplicial localization

Every homotopical category C “presents” or “models” an (infinity,1)-category LC, a simplicially enriched category called the simplicial localization of C, 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)
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