nLab
category with weak equivalences

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. It is not entirely clear whether every $(\mathrm{\infty }\mathrm{,1})$-category admits at least one such presentation, although it seems not unlikely.

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. However, the homotopy category of an $(\mathrm{\infty }\mathrm{,1})$-category can be recovered directly from any presentation of it, by freely adjoining inverse isomorphisms to the weak equivalences, resulting in the homotopy category of the category with weak equivalences.

Surely you don't mean to suggest (with ’The $(\mathrm{\infty }\mathrm{,1})$-category is recovered …’) that the composite

$$(\mathrm{\infty }\mathrm{,1})\mathrm{Cat}\to \mathrm{Cat}\to (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$$(\infty,1) Cat \to Cat \to (\infty,1) Cat

is equivalent to the identity, do you?

Mike Shulman: I’ve tried to clarify the statements being made.

Urs Schreiber: Thanks. You mention homotopy categories. But I understood the question differently: supppose we start with an $(\mathrm{\infty }\mathrm{,1})$-category in its incarnations as a simplicially enriched category. Then form the ordinary category obtained by retaining of all simplicial hom-sets only the set of 0-cells, now the set of morphisms. Mark those morphisms as weak equivalences that were equivalences in the original $(\mathrm{\infty }\mathrm{,1})$-category. Then pass to the simplicial localization of the resulting category with weak equivalences. How does that relate to the original $(\mathrm{\infty }\mathrm{,1})$-category?

Mike Shulman: Also an interesting question! However, I believe the answer is “it’s not the same.” Consider simplicially enriched groupoids, meaning simplicially enriched categories in which each category $C_n$ defined by $C_n(x,y)=C(x,y{)}_n$ is a groupoid. The fundamental $\mathrm{\infty }$-groupoid of any space can be realized as such a simplicially enriched groupoid. However, when we discard the higher cells in a simplicially enriched groupoid, we obtain simply an ordinary groupoid, whose simplicial localization is just itself. So in this case, the composite operation in question is the 1-truncation.

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.

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.

Purpose

The idea is that $C$ is a presentation of a higher category with higher morphisms, and that the weak equivalences are those morphisms which would become true equivalences in this higher category.

This higher category may be reconstructed by Dwyer-Kan localization as an $(\mathrm{\infty }\mathrm{,1})$-category.

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 $W$.

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.