Category theory; homological algebra Definition

Weak equivalence

Builds on Category Weak homotopy equivalence Quasi-isomorphism

Definition

Let $\mathcal{C}$ be a category. A class of weak equivalences in $\mathcal{C}$ is a class $W$ of morphisms of $\mathcal{C}$ such that:

every isomorphism of $\mathcal{C}$ is in $W$ ;
(two out of three) if $f \colon X \to Y$ and $g \colon Y \to Z$ are morphisms and two of $f$ , $g$ , $g \circ f$ are in $W$ , then so is the third.

A pair $(\mathcal{C}, W)$ is a category with weak equivalences, and the members of $W$ are its weak equivalences. The weak equivalences of a model category are the members of the class $W$ of its model structure, which satisfies these two clauses.

Remarks

Example: in the category of topological spaces, the weak homotopy equivalences form a class of weak equivalences. Homeomorphisms induce isomorphisms on all homotopy groups, and two out of three follows from functoriality of $\pi_0$ and of $\pi_n(-, x)$ at all base points. Example: in the category of chain complexes of modules over a ring, the quasi-isomorphisms form a class of weak equivalences, by functoriality of homology.

Non-example: in the category of sets, the class of injective maps is closed under composition and contains the isomorphisms, but it fails two out of three. With $f \colon \set{0} \to \set{0, 1}$ the inclusion and $g \colon \set{0, 1} \to \set{0}$ , both $f$ and $g \circ f = \id$ are injective, but $g$ is not.

Some authors, for homotopical categories, ask the stronger two out of six property.

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Tag	`NVXMVBS1ES`
URI	`m2://NVXMVBS1ES`
Kind	definition
Writing	W0 stub
MSC	18N40, 55U35
Defines	Weak equivalence
Uses	Category, Weak homotopy equivalence, Quasi-isomorphism

Weak equivalence

Definition

Remarks

Reading path

Further reading

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