nLab
category with weak equivalences

Context

Homotopy theory

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

A category with weak equivalences is an ordinary category with a class of morphisms singled out – called ‘weak equivalences’ – that include the isomorphisms, but also typically other morphisms. Such a category can be thought of as a presentation of an (∞,1)-category that defines explicitly only the 1-morphisms (as opposed to n-morphisms for all nn) and the information about which of these morphisms should become equivalences in the full (∞,1)-category.

The desired (,1)(\infty,1)-category in question can be constructed from such a “presentation” by “freely adjoining inverse equivalences” to the weak equivalences, in a suitable (,1)(\infty,1)-categorical sense. One way to make this precise is by the process of simplicial localization . A single (,1)(\infty,1)-category can admit many different such presentations. See the section Presentations of (∞,1)-categories below for more details.

Definition

A category with weak equivalences is a category CC equipped with a subcategory (in the naïve sense) WCW \subset C

  • which contains all isomorphisms of CC;

  • which satisfies two-out-of-three: for f,gf, g any two composable morphisms of CC, if two of {f,g,gf}\{f, g, g \circ f\} are in WW, then so is the third.

Examples and refinements

Often categories with weak equivalences are equipped with further extra structure that helps with computing the simplicial localization, the homotopy category and derived functors.

Other variants include

Additional conditions

Three additional conditions which categories with weak equivalences often satisfy are:

In fact, these three conditions are closely related.

  • Obviously, saturation implies closure under retracts and two-out-of-six, since the isomorphisms in any category satisfy both.

  • In any model category, all three conditions hold automatically.

  • If the weak equivalences admit a calculus of fractions, or a well-behaved class of cofibrations or fibrations, then the three conditions are equivalent. See two-out-of-six property for the proofs, which are from Categories and Sheaves (for the calculus of fractions) and Blumberg-Mandell (for the case of cofibrations, in the context of a Waldhausen category).

Remarks

  • If we denote by Core(C)Core(C) the core of CC – the maximal subgroupoid of CC – then we have a chain of inclusions Core(C)WC Core(C) \hookrightarrow W \hookrightarrow C .

  • Many categories with weak equivalences can be equipped with the further structure of a model category. On the other hand, some categories with weak equivalences can not be equipped with a useful structure of a model category. In particular, categories of diagrams in a model category do not always inherit a useful model structure (on the other hand often they do, see model structure on functors). Several concepts exist that weaken the axioms of a model category in order to still obtain useful results in such a case – for instance a category of fibrant objects.

Presentation of (,1)(\infty,1)-categories

A category CC with weak equivalences serves as a presentation of an (∞,1)-category C\mathbf{C} with the same objects and at least the 1-morphisms of CC, and such that every weak equivalence in CC becomes a true equivalence (a homotopy equivalence) in C\mathbf{C}.

The procedure (or one of its equivalent variants) that constructs the (∞,1)-category C\mathbf{C} from the category with weak equivalences CC is called Dwyer-Kan simplicial localization.

In fact, every (∞,1)-category may be presented this way (and indeed posets equipped with wide subcategories of morphisms called weak equivalences are sufficient). This is discussed at

Alternatively, we may further project to the 1-category in which all weak equivalences become true isomorphisms: this is the homotopy category of CC with respect to WW. Equivalently this is the homotopy category of an (∞,1)-category of C\mathbf{C}.

Note that the category with weak equivalences which presents a given (,1)(\infty,1)-category can not, in general, be taken to be the homotopy category of that (,1)(\infty,1)-category; more “flab” must be built into it.

It also cannot, in general, be the underlying 1-category of a simplicially enriched presentation of that (,1)(\infty,1)-category. For instance, every \infty-groupoid can be realized as a simplicially enriched groupoid, but the underlying 1-category of a simplicially enriched groupoid is a 1-groupoid, which cannot be localized any further to produce a non-1-truncated \infty-groupoid.

Revised on August 7, 2013 17:02:45 by Urs Schreiber (131.174.43.103)