Grothendieck topologies from unique factorisation systems

This work presents a way to associate a Grothendieck site structure to a category endowed with a unique factorisation system of its arrows. In particular this recovers the Zariski and Etale topologies and others related to Voevodsky’s cd-structures. As unique factorisation systems are also frequent outside algebraic geometry, a construction applies to some new contexts, where it is related with known structures defined otherwise. The paper details algebraic geometrical situations and sketches only the other contexts.

💡 Research Summary

The paper develops a systematic method for constructing Grothendieck sites and associated spectra from a category equipped with a unique factorisation system (UFS). Starting from the classical notion of lifting, the author defines a unique lifting system as a pair of classes (A, B) of morphisms such that A consists exactly of those maps left‑orthogonal to every map in B, and B consists of those right‑orthogonal to every map in A. When the category is locally presentable and the lifting system is generated by a small set of maps, one can promote it to a unique factorisation system: every morphism f factors uniquely (up to a unique isomorphism) as an A‑map followed by a B‑map.

Given a UFS (A, B) on the opposite of a locally presentable category (for example, the opposite of commutative rings), the author introduces several key notions:

1. Etale maps – the right class B, interpreted as “open” or “etale” morphisms depending on the concrete situation.
2. Points – an A‑map p : P → X is a point of X if for every B‑map U → X there exists a unique lift N → U completing the usual lifting square. This captures the intuition that a point lies in every neighbourhood.
3. Local objects – objects L such that every point of any object admits a morphism to L; they play the role of “germs” or “local rings”.
4. Spectra – two Grothendieck topologies are built: the factorisation topology whose covering families are families of B‑maps jointly surjective on points, and the Nisnevich forcing which restricts coverings to those that lift maps from a chosen class of objects (typically fields). The small spectrum of an object A is the topos of sheaves for the factorisation topology; the big spectrum is the corresponding classifying topos for all A‑algebras.

The paper works out four principal examples on the category of commutative rings:

• Zariski system (Loc, Cons) – A‑maps are localisations, B‑maps are conservative ring homomorphisms. Points are spectra of residue fields, the small spectrum recovers the usual Zariski spectrum, and the big spectrum classifies all localisations of a ring.
• Etale system (Ind‑Et, Hens) – A‑maps are ind‑etale morphisms, B‑maps are henselian maps. Points are spectra of fields, the small spectrum is the classical étale topos, and the big spectrum classifies all strict henselisations.
• Domain system (Surj, Mono) – A‑maps are surjections, B‑maps are monomorphisms. Points correspond to integral domains, yielding a “domain topology” whose small spectrum classifies quotient domains.
• Finite system (Ind‑Fin, IntCl) – A‑maps are ind‑finite morphisms, B‑maps are integrally closed maps. Points are spectra of strictly integrally closed fields; the resulting topology matches Voevodsky’s lower cd‑structure on affine schemes.

A novel contribution is the notion of Nisnevich forcing: given a class C of objects (typically fields), one forces the covering families of the factorisation topology to lift all maps from objects of C. Applying this to the étale topology reproduces the classical Nisnevich topology; applying it to the finite topology yields Voevodsky’s cd‑structure restricted to affine schemes. The author calls the data (UFS, C) a Nisnevich context and shows that the construction of spectra and the computation of their global points (Theorem 44) extend verbatim to any such context.

Beyond algebraic geometry, the paper sketches several non‑geometric examples where similar factorisation systems appear: the “Töen–Vaqi” setting for spectra over ℤ, factorisation systems arising from (Epi, Mono) in any abelian category, factorisation systems on small categories given by initial/final functors, and a simplicial set example where points are vertices and local objects are forced to be all simplices. In each case the same pattern—UFS → points → local objects → spectra—produces a Grothendieck topos that encodes the underlying combinatorial or algebraic structure.

Overall, the work provides a unifying categorical framework that explains why classical Grothendieck topologies (Zariski, étale, Nisnevich, finite) possess both “small” and “big” spectra and why distinguished classes of morphisms act as open embeddings. By abstracting the construction to any uniquely factorisable category, it opens the door to new Grothendieck topologies in diverse mathematical contexts, while retaining a concrete computational handle via points and local objects.