Grothendieck’s Homotopy Hypothesis: Equivalence of n-groupoids and n-types

Establish an equivalence between n-groupoids and n-types such that, for every topological space X, the fundamental n‑groupoid π≤n(X) corresponds to the n‑type of X; moreover, establish an equivalence between all homotopy types and ∞‑groupoids such that the homotopy type of X corresponds to the fundamental ∞‑groupoid π≤∞(X).

Background

Classical results identify low-dimensional homotopy types with algebraic structures: sets with 0‑types, groupoids with 1‑types, and strict 2‑groupoids with 2‑types. Grothendieck proposed extending this correspondence to all dimensions, asserting that higher homotopy-invariant information of a space is captured by higher groupoids.

The conjecture has guided modern homotopical approaches to higher categories, motivating definitions of ∞‑groupoids via homotopy types and, in turn, development of (∞,n)‑categories. Despite many partial confirmations in low dimensions and model equivalences, a general proof of the full hypothesis remains open and central to higher category theory.