Contents

Idea

A category with weak equivalences is an ordinary category with a class of morphisms 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 gives only the 1-morphisms and the information about which of these 1morphisms should become equivalences in the full (∞,1)-category.

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

Definition

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

• which contains all isomorphisms of $C$;

• which satisfies “two-out-of-three”: for $f,g$ any two composable morphisms of $C$, if two of $\{f,g,g\circ f\}$ are in $W$, 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.

• In a homotopical category the condition on the weak equivalences is slightly stronger.

• In a category of fibrant objects there are additional auxiliary morphisms called fibrations.

• In a Waldhausen category there are additional auxiliary morphisms called cofibrations.

• In a model category there are both of these additional auxiliary classes of morphisms with special interrelation between them.

Remarks

• If we denote by $\mathrm{Core}(C)$ the maximal subgroupoid of $C$, then we have a chain of inclusions $\mathrm{Core}(C)\hookrightarrow W\hookrightarrow C$.

• Sometimes it is useful to ask further closure properties of the weak equivalences. One such is the “2-out-of-6” property (if $gf$ and $hg$ are weak equivalences, then so are $f$, $g$, $h$, and $hgf$) in which case one speaks of a homotopical category. Another such is closure under retracts in the arrow category of $C$. Both are satisfied automatically by any model category.

• 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. 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 $(\mathrm{\infty }\mathrm{,1})$-categories

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

The procedure (or one of its equivalent variants) that constructs the (∞,1)-category $C$ from the category with weak equivalences $C$ 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

• model structure on categories with weak equivalences .

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

Note that the category with weak equivalences which presents a given $(\mathrm{\infty }\mathrm{,1})$-category will not, in general, be the homotopy category of this $(\mathrm{\infty }\mathrm{,1})$-category; more “flab” must be built into it.

Discussion