In $\mathbf{Top}$ (with the Quillen model structure), a map $f: X \to Y$ is a weak equivalence if it is a weak homotopy equivalence: for every basepoint $x \in X$ and every $n \geq 0$,

$f_*: \pi_n(X, x) \xrightarrow{\;\cong\;} \pi_n(Y, f(x))$

is an isomorphism. Note that this is weaker than being a homotopy equivalence: every homotopy equivalence is a weak homotopy equivalence, but not conversely (unless the spaces are CW complexes, by Whitehead's theorem).

For example, the Warsaw circle (which is not locally path-connected) has the same homotopy groups as $S^1$ but is not homotopy equivalent to $S^1$. However, they are weakly homotopy equivalent.

In $\mathbf{sSet}$ (Kan--Quillen), $f: X \to Y$ is a weak equivalence if $|f|: |X| \to |Y|$ is a weak homotopy equivalence. Equivalently (for Kan complexes), $f$ induces isomorphisms on all simplicial homotopy groups $\pi_n(X, x) \cong \pi_n(Y, f(x))$.

For general simplicial sets (not necessarily Kan), the definition must use fibrant replacement: $f$ is a weak equivalence iff the induced map on fibrant replacements $Rf: RX \to RY$ induces isomorphisms on homotopy groups.

In $\mathbf{Ch}(R)$, weak equivalences are quasi-isomorphisms: maps $f: C_\bullet \to D_\bullet$ inducing isomorphisms on all homology groups $H_n(f): H_n(C) \xrightarrow{\cong} H_n(D)$.

A quasi-isomorphism need not be a chain homotopy equivalence. For example, the augmentation $\cdots \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{0} \mathbb{Z} \to 0$ mapping to $\cdots \to 0 \to 0 \to \mathbb{Z}/2 \to 0$ is a quasi-isomorphism (both have homology $\mathbb{Z}/2$ in degree $0$) but not a chain homotopy equivalence.

In $\mathbf{Cat}$ with the canonical (folk) model structure:

• Weak equivalences are equivalences of categories: functors $F: \mathcal{C} \to \mathcal{D}$ that are fully faithful and essentially surjective.

In the Thomason model structure on $\mathbf{Cat}$:

• Weak equivalences are functors $F$ such that $|N(F)|$ is a weak homotopy equivalence.

These are very different: an equivalence of categories is always a Thomason weak equivalence, but not conversely. The Thomason model structure sees the "homotopy type" of a category (as a space), while the folk model structure sees its categorical structure.

If $f: X \xrightarrow{\sim} Y$ and $g: Y \xrightarrow{\sim} Z$ are both weak equivalences, then $g \circ f: X \xrightarrow{\sim} Z$ is a weak equivalence.

In $\mathbf{Top}$: if $f$ and $g$ induce isomorphisms on all $\pi_n$, then $g \circ f$ does too (composition of isomorphisms).

If $g \circ f$ and $f$ are weak equivalences, then $g$ is a weak equivalence. This is the "right cancellation" version.

In $\mathbf{Ch}(R)$: if $f: A \to B$ and $gf: A \to C$ are quasi-isomorphisms, then $g: B \to C$ is a quasi-isomorphism. This follows from the five lemma applied to the long exact sequences.

If $g \circ f$ and $g$ are weak equivalences, then $f$ is a weak equivalence. This is the "left cancellation" version.

This is the most subtle case. In $\mathbf{Top}$: if $\pi_n(g)$ and $\pi_n(g \circ f)$ are isomorphisms for all $n$, then $\pi_n(f) = \pi_n(g)^{-1} \circ \pi_n(g \circ f)$ is an isomorphism.

A morphism that is both a weak equivalence and a fibration is called a trivial fibration (or acyclic fibration). These have the right lifting property against all cofibrations.

In $\mathbf{sSet}$: a trivial fibration is a map with the RLP against all boundary inclusions $\partial\Delta[n] \hookrightarrow \Delta[n]$. Such maps are "surjective up to homotopy" -- they have contractible fibers over every simplex.

In $\mathbf{Top}$: a trivial Serre fibration is a Serre fibration that is also a weak homotopy equivalence. Fibers of such maps are weakly contractible.

A morphism that is both a weak equivalence and a cofibration is a trivial cofibration (or acyclic cofibration). These have the left lifting property against all fibrations.

In $\mathbf{sSet}$: trivial cofibrations are anodyne extensions (monomorphisms that are weak equivalences). They are built from horn inclusions $\Lambda^n_k \hookrightarrow \Delta[n]$.

Trivial cofibrations between cofibrant objects are particularly nice: they always have a homotopy inverse (any trivial cofibration between cofibrant objects is a strong deformation retract in the model-categorical sense).

In a simplicial model category $\mathcal{M}$, a map $f: X \to Y$ between cofibrant objects is a weak equivalence if and only if for every fibrant object $Z$, the induced map on mapping spaces

$f^*: \operatorname{Map}(Y, Z) \to \operatorname{Map}(X, Z)$

is a weak equivalence of simplicial sets. This is "Yoneda-like" detection of weak equivalences.

In the homotopy category $\operatorname{Ho}(\mathcal{M})$, a morphism is an isomorphism if and only if it is represented by a weak equivalence. The Whitehead theorem for model categories states:

A map $f: X \to Y$ between bifibrant objects is a weak equivalence if and only if it is a homotopy equivalence (i.e., has a homotopy inverse using cylinder/path objects).

This generalizes the classical Whitehead theorem: a weak homotopy equivalence between CW complexes is a homotopy equivalence.

In $\mathbf{Ch}(R)$, weak equivalences (quasi-isomorphisms) are detected by homology functors: $f$ is a weak equivalence iff $H_n(f)$ is an isomorphism for all $n$. These homology functors play the role of homotopy groups in topological model categories.

More generally, in any stable model category, weak equivalences can be detected by the homotopy groups $\pi_n(X) = [S^n, X]$ in the triangulated homotopy category.

The localization $\mathbf{Ch}(R)[\text{qis}^{-1}]$ is the derived category $D(R)$. This is the fundamental construction of homological algebra, and it was historically one of the motivations for model categories.

In $D(R)$, morphisms from $C$ to $D$ are computed as $\operatorname{Hom}_{D(R)}(C, D) \cong [PC, ID]$ where $PC$ is a projective resolution and $ID$ is an injective resolution.