Omitting the direct-summand requirement in axiom (G3) of Grothendieck's Galois categories (SGA 1, Exposé V, Remarques 4.2)

In Exposé V of SGA 1 (1971), Grothendieck axiomatizes a Galois category by conditions (G1)-(G6) on a category $\mathcal{C}$ equipped with a fiber functor $F$ to finite sets, and proves the structure theorem (Théorème 4.1): $\mathcal{C}$ is equivalent to the category of finite sets with a continuous action of a profinite group, $F$ becoming the forgetful functor. In Remarques 4.2 he observes that the apparently stronger variants (G'2) and (G'5) of two axioms are in fact consequences of the system, and then asks whether, likewise, the requirement in axiom (G3) that the monomorphism of the factorization be an isomorphism onto a direct summand could simply be omitted. No published treatment is known to settle Grothendieck's question for the nondegenerate weakened system posed below, which therefore remains open.

Background

In the 1960-61 seminar « Séminaire de Géométrie Algébrique du Bois Marie » (SGA 1), Grothendieck constructed the étale fundamental group of a scheme, the algebro-geometric analogue of the topological fundamental group, classifying the finite étale covers, which play for schemes the role that finite covering spaces play for topological spaces. Exposé V, « Le groupe fondamental : généralités », extracts the formal skeleton. A fiber functor on a category $\mathcal{C}$ is a covariant functor $F:\mathcal{C}\to\mathsf{FinSet}$ to the category of finite sets; the leading example evaluates a finite étale cover at a geometric point (a morphism of schemes from the spectrum of an algebraically closed field). A strict epimorphism is a morphism that is the coequalizer of its kernel pair, i.e. a categorical quotient map that identifies exactly the pairs it must; a subobject of an object $Y$ is a direct summand (one says the subobject is complemented) if it admits a complementary subobject $Y''$ with $Y$ isomorphic to the coproduct $Y'\amalg Y''$ . In §4, « Conditions axiomatiques d'une théorie de Galois », the axioms on such a pair $(\mathcal{C},F)$ are laid out: (G1) a final object and finite fiber products exist in $\mathcal{C}$ ; (G2) finite coproducts and categorical quotients by finite groups of automorphisms exist in $\mathcal{C}$ ; (G3) every morphism $u:X\to Y$ factors as $X\xrightarrow{u'}Y'\xrightarrow{u''}Y$ with $u'$ a strict epimorphism and $u''$ a monomorphism that is an isomorphism onto a direct summand of $Y$ ; (G4) $F$ is left exact (takes the final object to a one-point set and commutes with fiber products); (G5) $F$ commutes with finite coproducts, takes strict epimorphisms to epimorphisms, and commutes with passage to quotients by finite groups; (G6) $F$ is conservative, i.e. a morphism of $\mathcal{C}$ whose image under $F$ is a bijection is already an isomorphism. Théorème 4.1, the structure theorem, states that these conditions force $\mathcal{C}$ to be equivalent to $\mathsf{C}(\pi)$ , the category of finite discrete sets on which a profinite group $\pi$ (an inverse limit of finite groups, topologized as such) acts continuously, the functor $F$ being carried to the forgetful functor. For a connected scheme this recovers the étale fundamental group; the term « catégorie galoisienne » itself is introduced in §5 (Définition 5.1), for a category equivalent to some $\mathsf{C}(\pi)$ , which by the structure theorem is the same as satisfying (G1)-(G3) together with a functor $F$ satisfying (G4)-(G6). In Remarques 4.2, immediately after the theorem, Grothendieck observes that replacing (G2) and (G5) by the simpler-looking (G'2) (finite colimits exist in $\mathcal{C}$ ) and (G'5) ( $F$ is right exact) is harmless, since the replacements follow from the structure theorem; he then adds that he does not know whether the same holds for the direct-summand requirement inside (G3). The question is thus one of logical independence among the axioms: would requiring only a strict-epimorphism–monomorphism factorization (mere existence of images of morphisms) already force the image subobjects to be complemented, and hence deliver the full axiom system? The later literature keeps the clause: modern reference accounts restated the axioms in 2000 and 2013 with the direct-summand requirement intact, while recent classification programs weaken the system along other axes — replacing the single fiber functor by a finite jointly conservative family, which leads to profinite monoids and profinite categories rather than groups — and topos-theoretic treatments reconstruct the fundamental group of a connected Grothendieck topos. Whether Grothendieck's original question has a positive or negative answer does not appear to have been recorded.

Problem Statement

In SGA 1, Exposé V, §4, axiom (G3) requires of the underlying category $\mathcal{C}$ that every morphism $u:X\to Y$ factor as $X\xrightarrow{u'}Y'\xrightarrow{u''}Y$ where $u'$ is a strict epimorphism and $u''$ is a monomorphism that is an isomorphism onto a direct summand of $Y$ . In Remarques 4.2 Grothendieck asks: « J'ignore si dans la condition (G 3), le fait que $u''$ soit un isomorphisme sur un sommande direct de $Y$ ne pourrait être omis. » — whether the direct-summand requirement can be omitted from (G3). Precisely: let $\mathcal{C}$ be a category together with a covariant functor $F:\mathcal{C}\to\mathsf{FinSet}$ satisfying (G1), (G2), (G4), (G5) and (G6) exactly as formulated in SGA 1 Exposé V §4, together with the weakened factorization axiom (G3 $^-$ ): every morphism of $\mathcal{C}$ factors as a strict epimorphism followed by a monomorphism, with no complementedness requirement on the monomorphism. To exclude degenerate models, the weakened system is further required to carry the nondegeneracy hypothesis that every subobject of every connected object of $\mathcal{C}$ is complemented, where an object is connected when it admits no decomposition as a coproduct of two non-initial subobjects; this regularity condition holds for the Galois categories $\mathsf{C}(\pi)$ and is strictly weaker than (G3), which requires complementation of the subobjects of every object of $\mathcal{C}$ , not only of the connected ones. Must such a pair satisfy the original (G3) — equivalently, does the conclusion of Théorème 4.1 hold for it: an equivalence of $\mathcal{C}$ with the category $\mathsf{C}(\pi)$ of finite continuous sets of a profinite group $\pi$ , carrying $F$ to the forgetful functor? Throughout, the axioms are those of SGA 1 Exposé V §4 in their original strength (including the SGA 1 notion of strict epimorphism and the original forms of the (G2)/(G5) quotient clauses), rather than a modern reformulation of them.

References

Alexander Grothendieck, « Le groupe fondamental : généralités », Exposé V in Revêtements étales et groupe fondamental (Séminaire de Géométrie Algébrique du Bois Marie 1960-61, SGA 1), Lecture Notes in Mathematics 224, Springer, 1971, pp. 105-144. DOI: 10.1007/bfb0058661, https://doi.org/10.1007/bfb0058661
Alexander Grothendieck and Michele Raynaud (eds.), Revêtements étales et groupe fondamental (SGA 1), re-typeset edition with updating remarks added in 2003 by M. Raynaud, arXiv:math/0206203, 2004, https://arxiv.org/abs/math/0206203 (Exposé V §4; Théorème 4.1; Remarques 4.2; Exposés VI, VIII).
Eduardo J. Dubuc and Constanza S. de la Vega, « On the Galois Theory of Grothendieck », arXiv:math/0009145, 2000 (journal reference on arXiv: Bol. Acad. Nac. Cienc. Cordoba 65 (2000)), https://arxiv.org/abs/math/0009145 (axiom restatement in §4.1 and the remark after Proposition 4.8).
Anna Cadoret, « Galois categories », in Arithmetic and Geometry around Galois Theory, Progress in Mathematics 304, Birkhäuser/Springer, Basel, 2013, pp. 171-246. DOI: 10.1007/978-3-0348-0487-5_3, https://doi.org/10.1007/978-3-0348-0487-5_3 (author's preprint at https://webusers.imj-prg.fr/~anna.cadoret/FG.pdf; Definition 2.1, §2.1.1-§2.1.2, Theorem 2.8).
Nate Harman and Andrew Snowden, « Pre-Galois categories and Fraïssé's theorem », arXiv:2301.13784 (v2, 2024), https://arxiv.org/abs/2301.13784 (Definition 4.12, Proposition 4.13).
Remy van Dobben de Bruyn, « Grothendieck Galois theory and étale exodromy », arXiv:2410.06278, 2024, https://arxiv.org/abs/2410.06278 (Definition 2.5, Theorem 2.18 and introduction).
Clemens Berger and Victor Iwaniack, « On the profinite fundamental group of a connected Grothendieck topos », arXiv:2304.05338 (v6, 2025), https://arxiv.org/abs/2304.05338 (introduction).