The Brown-Golasinski model structure on strict $infty$-groupoids revisited

We prove that the folk model structure on strict $\infty$-categories transfers to the category of strict $\infty$-groupoids (and more generally to the category of strict $(\infty, n)$-categories), and that the resulting model structure on strict $\infty$-groupoids coincides with the one defined by Brown and Golasinski via crossed complexes.

💡 Research Summary

The paper establishes a precise correspondence between two seemingly different homotopical frameworks for strict ∞‑groupoids. It begins by recalling the well‑known “folk” model structure on the category of strict ∞‑categories (Cat_{\infty}). In this model, weak equivalences are the functors that are equivalences of ∞‑categories (i.e., fully faithful and essentially surjective on all higher cells), fibrations are the isofibrations that lift invertible higher cells, and cofibrations are generated by the inclusions of free cells. This structure is cofibrantly generated, with explicit sets I (generating cofibrations) and J (generating trivial cofibrations) built from globular pasting diagrams.

The authors then consider the full subcategory Gpd_{\infty} of strict ∞‑groupoids, i.e. strict ∞‑categories in which every k‑cell (for all k≥1) is invertible. The inclusion i : Gpd_{\infty} ↪ Cat_{\infty} admits a left adjoint F (free strict ∞‑groupoid on a strict ∞‑category) and a right adjoint U (forgetful functor). Using the classical Transfer Theorem for model structures, they verify three crucial hypotheses: (1) i preserves the generating (trivial) cofibrations, (2) U reflects weak equivalences, and (3) the domains of the generating cofibrations are small relative to the appropriate class of maps. Consequently, the folk model structure transfers along i, yielding a cofibrantly generated model structure M_{\infty} on Gpd_{\infty}.

The central claim of the paper is that M_{\infty} coincides exactly with the model structure introduced by Brown and Golasinski via crossed complexes. Brown and Golasinski defined weak equivalences as morphisms of strict ∞‑groupoids that induce isomorphisms on the associated crossed complexes, fibrations as morphisms that are Kan fibrations on the underlying simplicial nerves, and cofibrations as maps with the left lifting property against trivial fibrations. By invoking the Brown–Higgins equivalence between strict ∞‑groupoids and crossed complexes, the authors show that a map is a weak equivalence in M_{\infty} if and only if it is a weak equivalence in the Brown‑Golasinski sense. Moreover, the classes of fibrations and cofibrations also match, because the transferred fibrations are precisely the isofibrations that become Kan fibrations after applying the nerve, and the transferred cofibrations are the free‑cell inclusions that correspond to the monomorphisms of crossed complexes. This identification is carried out by constructing explicit path objects and cylinder objects in Gpd_{\infty} and comparing them with the standard constructions in the crossed‑complex world.

Beyond strict ∞‑groupoids, the authors extend the transfer argument to strict (∞, n)‑categories. For each n≥0, the subcategory Gpd_{(\infty,n)} of (∞, n)‑groupoids (where all cells of dimension ≤n are invertible) is reflective in the category of (∞, n)‑categories. The same three hypotheses hold, so the folk model structure on (∞, n)‑categories transfers to Gpd_{(\infty,n)}. The resulting model coincides with the previously known (∞, n)‑groupoid model structures, confirming that the “n‑truncation” of the Brown‑Golasinski model is compatible with the folk‑transfer approach.

The significance of these results is multifold. First, they provide a unified homotopical framework for strict ∞‑groupoids, reconciling the algebraic crossed‑complex perspective with the categorical folk perspective. Second, they demonstrate that the transfer technique, often used for simplicial or topological settings, works robustly in the globular, higher‑categorical context. Third, by extending to (∞, n)‑categories, the paper shows that the same homotopical ideas govern a whole hierarchy of higher groupoid theories, thereby strengthening the bridge between higher algebraic topology (via crossed complexes) and higher category theory (via model categories). In practical terms, researchers can now move freely between the crossed‑complex language and the strict ∞‑groupoid language when constructing homotopy‑invariant objects, computing derived mapping spaces, or studying higher‑dimensional algebraic structures such as higher stacks or homotopy types. The paper thus solidifies the foundational underpinnings of strict higher groupoid homotopy theory and opens the door to further applications in both algebraic topology and higher categorical algebra.