The standard model structure on $\mathbf{Top}$ (for compactly generated weak Hausdorff spaces) has:

• Weak equivalences: weak homotopy equivalences (maps inducing isomorphisms on all homotopy groups).
• Cofibrations: retracts of relative CW inclusions.
• Fibrations: Serre fibrations (maps with the RLP against $D^n \times \{0\} \hookrightarrow D^n \times [0,1]$).

Every object is fibrant. The cofibrant objects are retracts of CW complexes.

A cofibration in $\mathbf{sSet}$ is a monomorphism $A \hookrightarrow B$. Since geometric realization sends $\partial\Delta[n] \hookrightarrow \Delta[n]$ to $S^{n-1} \hookrightarrow D^n$ (the inclusion of the sphere into the disk), and every monomorphism is built from these generators by pushout and transfinite composition, $|A| \hookrightarrow |B|$ is a relative CW inclusion, hence a cofibration in $\mathbf{Top}$.

A trivial cofibration in $\mathbf{sSet}$ is an anodyne extension (a monomorphism that is also a weak equivalence). Since $|\cdot|$ sends horn inclusions $\Lambda^n_k \hookrightarrow \Delta[n]$ to the topological horn inclusion $|\Lambda^n_k| \hookrightarrow |\Delta^n|$, which is a homotopy equivalence (the topological horn deformation-retracts onto the full simplex), the result follows by closure properties.

If $p: Y \to Z$ is a Serre fibration in $\mathbf{Top}$, then $\operatorname{Sing}(p): \operatorname{Sing}(Y) \to \operatorname{Sing}(Z)$ is a Kan fibration. This is because lifting a horn $\Lambda^n_k \to \operatorname{Sing}(Y)$ over $\operatorname{Sing}(Z)$ corresponds to extending a map $|\Lambda^n_k| \to Y$ to $|\Delta^n| \to Y$ over $Z$, which is possible since $|\Lambda^n_k| \hookrightarrow |\Delta^n|$ is an NDR pair and $p$ is a Serre fibration.

For any space $Y$, the counit $\varepsilon_Y: |\operatorname{Sing}(Y)| \to Y$ is a weak homotopy equivalence. This is a classical theorem: $|\operatorname{Sing}(Y)|$ is a CW approximation of $Y$.

The proof uses the fact that $\pi_n(|\operatorname{Sing}(Y)|) \cong \pi_n(\operatorname{Sing}(Y)) \cong \pi_n(Y)$, where the first isomorphism comes from the compatibility of simplicial homotopy groups with geometric realization, and the second is essentially the definition of simplicial homotopy groups for $\operatorname{Sing}(Y)$.

For a Kan complex $X$, the unit $\eta_X: X \to \operatorname{Sing}(|X|)$ is a weak equivalence of simplicial sets. This follows from the fact that for Kan complexes, the simplicial homotopy groups of $X$ agree with the topological homotopy groups of $|X|$, and $\operatorname{Sing}(|X|)$ has the same homotopy groups as $|X|$.

Combined with the counit being a weak equivalence, this shows the adjunction is a Quillen equivalence.

The Quillen equivalence gives:

$\operatorname{Ho}(\mathbf{sSet}) \simeq \operatorname{Ho}(\mathbf{Top})$

Objects of $\operatorname{Ho}(\mathbf{sSet})$ are simplicial sets (localized at weak equivalences), and objects of $\operatorname{Ho}(\mathbf{Top})$ are topological spaces (localized at weak homotopy equivalences).

In $\operatorname{Ho}(\mathbf{sSet})$, every object is isomorphic to a Kan complex (its fibrant replacement). In $\operatorname{Ho}(\mathbf{Top})$, every object is isomorphic to a CW complex (its cofibrant replacement).

Morphisms in the homotopy category are homotopy classes of maps: $[X, Y]$ for Kan complexes $X, Y$ in $\mathbf{sSet}$ corresponds to $[|X|, |Y|]$ for topological spaces.

The Quillen equivalence justifies computing homotopy invariants of spaces using simplicial sets. For instance:

• $\pi_n(S^m)$ can be computed using simplicial models of spheres.
• Eilenberg--MacLane spaces $K(G, n)$ have natural simplicial models (the Dold--Kan correspondence gives chain complexes, which give simplicial abelian groups).
• Postnikov towers, fibration sequences, and spectral sequences all have simplicial analogues.

The combinatorial nature of simplicial sets makes them amenable to computation in ways that point-set topology is not.

Milnor's theorem states that the geometric realization of a simplicial set is a CW complex, and the geometric realization of a Kan fibration $p: X \to Y$ (where $Y$ has countably many non-degenerate simplices) is a Serre fibration $|p|: |X| \to |Y|$.

This is a crucial technical result: it ensures that the homotopy-theoretic properties visible in $\mathbf{sSet}$ (Kan fibrations, weak equivalences) translate faithfully to topological properties.

Given a continuous map $f: |X| \to |Y|$ between geometric realizations of finite simplicial sets, there exists a subdivision $\mathrm{sd}^N(X)$ and a simplicial map $g: \mathrm{sd}^N(X) \to Y$ such that $|g|$ is homotopic to $f \circ |\mathrm{sd}^N(X) \to X|$.

This "simplicial approximation" ensures that every topological map can be approximated by a combinatorial one after sufficient subdivision, another manifestation of the Quillen equivalence at the level of individual maps.

The internal hom in $\mathbf{sSet}$ gives the function complex (or mapping space):

$\operatorname{Map}(X, Y)_n = \operatorname{Hom}_{\mathbf{sSet}}(X \times \Delta[n], Y).$

When $Y$ is a Kan complex, $\operatorname{Map}(X, Y)$ is also a Kan complex, and its geometric realization is weakly equivalent to the topological mapping space $\operatorname{Map}(|X|, |Y|)$ with the compact-open topology (when $X$ is finite).

This shows that the simplicial enrichment of $\mathbf{sSet}$ is compatible with the topological enrichment of $\mathbf{Top}$ via the Quillen equivalence.

The Quillen equivalence can be upgraded to a Dwyer--Kan equivalence (equivalence of simplicial categories): the simplicial localizations $L^H(\mathbf{sSet})$ and $L^H(\mathbf{Top})$ are equivalent as simplicially enriched categories. This is a stronger statement than the equivalence of homotopy categories, capturing the full "higher homotopy" information of the mapping spaces.

In modern language, this says the $\infty$-categories $\mathbf{sSet}[W^{-1}]$ and $\mathbf{Top}[W^{-1}]$ are equivalent, where $W$ denotes weak equivalences.