For a Kan complex $X$ with basepoint $x$, the simplicial homotopy groups $\pi_n(X, x)$ are isomorphic to the topological homotopy groups $\pi_n(|X|, x)$:

$\pi_n^{\mathrm{simp}}(X, x) \cong \pi_n^{\mathrm{top}}(|X|, x)$

This means the algebraic invariants of spaces are completely captured by the combinatorial structure of Kan complexes.

The Postnikov tower of a Kan complex $X$ (constructed simplicially via coskeleton functors) corresponds exactly to the Postnikov tower of $|X|$ (constructed topologically). The $k$-invariants match under the correspondence $H^{n+1}_{\mathrm{simp}} \cong H^{n+1}_{\mathrm{top}}$.

Kan fibrations of simplicial sets correspond to Serre fibrations of topological spaces. The long exact sequence of homotopy groups for a Kan fibration $F \to E \to B$ corresponds to the long exact sequence for the Serre fibration $|F| \to |E| \to |B|$.

Eilenberg--MacLane spaces $K(A, n)$ can be constructed either as simplicial abelian groups (via Dold--Kan) or as topological spaces (via cell attachments). The resulting cohomology theories agree:

$H^n_{\mathrm{simp}}(X; A) = [X, K(A,n)]_{\mathrm{simp}} \cong [|X|, |K(A,n)|]_{\mathrm{top}} = H^n_{\mathrm{top}}(|X|; A)$

In his 1983 manuscript "Pursuing Stacks," Grothendieck proposed that homotopy types should be modeled by purely algebraic objects -- $\infty$-groupoids -- without reference to topology. He envisioned an equivalence:

$\{\text{homotopy types}\} \xleftrightarrow{\;\sim\;} \{\infty\text{-groupoids}\}$

The difficulty was making the notion of $\infty$-groupoid precise. Various definitions were proposed (globular, simplicial, operadic), and the homotopy hypothesis became a test: any good definition of $\infty$-groupoid should be equivalent to topological spaces.

In the simplicial setting (Kan complexes), the homotopy hypothesis was already established by Quillen (1967) and Milnor (1957), predating Grothendieck's formulation.

The homotopy hypothesis has been verified for various models of $\infty$-groupoids:

1. Kan complexes (Quillen): Classical Quillen equivalence $\mathbf{sSet} \simeq_Q \mathbf{Top}$.
2. Globular sets with composition (Batanin, Leinster): Shown to model homotopy types.
3. Algebraic Kan complexes (Nikolaus): Kan complexes with chosen fillers, equivalent to spaces.
4. Topological categories (groupoid-enriched): Equivalent via the nerve construction.

Each verification confirms Grothendieck's insight that algebra and topology are fundamentally interchangeable at the level of homotopy types.

The homotopy hypothesis allows us to work entirely within the world of Kan complexes (or more generally, simplicial sets) without ever mentioning topological spaces. All of homotopy theory -- homotopy groups, fibrations, spectral sequences, obstruction theory -- can be developed purely combinatorially.

This is practically important: simplicial sets are more amenable to algebraic manipulations and computations than topological spaces, and they interface better with algebraic structures (chain complexes, ring spectra, etc.).

In homotopy type theory (HoTT), the homotopy hypothesis takes a different form: types (in the type-theoretic sense) are interpreted as $\infty$-groupoids via their identity types. The univalence axiom (Voevodsky) states that equivalence of types is equivalent to identity of types, which is closely related to the structure of the $\infty$-category of spaces.

The homotopy hypothesis thus connects three worlds: topology, higher category theory, and foundations of mathematics.

The homotopy hypothesis gives a clean interpretation of classifying spaces:

• $BG = K(G, 1)$ for a discrete group $G$: the $\infty$-groupoid with one object and automorphism group $G$.
• $B^n A = K(A, n)$ for an abelian group $A$: the $\infty$-groupoid with one object, trivial automorphisms, and $\pi_n = A$.
• The classifying space of a topological group: the delooping of $G$ viewed as an $\infty$-group.

These are naturally $\infty$-groupoids, and the homotopy hypothesis tells us they are "the same" as topological spaces.

A key consequence is that descent data for $\infty$-groupoids (homotopy-coherent gluing data for spaces) is the same as descent data for topological spaces. This justifies the use of stacks and higher stacks: a higher stack is a functor valued in $\infty$-groupoids, which is the same as a functor valued in spaces.