The fundamental pro-groupoid of an affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don’t remember $\Forget$? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the 'etale fundamental groupoid $\pi_1(\spec(R))$, i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for 'etale $\pi_1(\spec(R))$ in terms of the category of $R$-modules rather than the category of 'etale covers. More generally, we introduce a new notion of “commutative 2-ring” that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of $\pi_1$ for the corresponding “affine 2-schemes.” These results help to simplify and clarify some of the peculiarities of the 'etale fundamental group. For example, 'etale fundamental groups are not “true” groups but only profinite groups, and one cannot hope to recover more: the “Tannakian” functor represented by the 'etale fundamental group of a scheme preserves finite products but not all products.

💡 Research Summary

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The paper “The fundamental pro‑groupoid of an affine 2‑scheme” addresses a natural question arising from Tannakian reconstruction: what can be recovered from a symmetric monoidal category of representations if the forgetful functor to the underlying module category is not recorded? Working over an arbitrary commutative ring (R), the authors show that the answer is given by the étale fundamental groupoid (\pi_{1}(\operatorname{Spec}R)), i.e. the separable absolute Galois group when (R) is a field. This provides a new definition of the étale fundamental groupoid purely in terms of the category of (R)-modules, bypassing the traditional construction via étale covers.

To formulate the result in a broad categorical context, the authors introduce commutative 2‑rings: symmetric closed monoidal presentable categories. This class includes Grothendieck topoi (as categories of Set‑valued sheaves) and module categories (\operatorname{Mod}_{R}) (as categories of Ab‑valued functions). Morphisms of commutative 2‑rings generalize geometric morphisms of topoi, and the opposite 2‑category is denoted (\mathbf{Af2Sch}), the category of affine 2‑schemes.

Given a small groupoid (G) and a commutative 2‑ring (A), the functor category (\operatorname{Fun}(G,A)) is again a commutative 2‑ring, denoted (G A). The Tannakian‑Krein category (\operatorname{TK}(G,A)) is defined as the category of commutative (A)-algebra maps (G A \to A). By a careful analysis (Lemma 2.3.12) this is equivalent to the category of morphisms of commutative 2‑rings (G!\operatorname{Set} \to A). The authors then give an internal description: such morphisms are precisely right (G)-torsors in (A) (Proposition 3.2.5). They prove that for any small groupoid (G), the category (\operatorname{Tors}(G,A)) is essentially small (Theorem 3.2.14) and behaves well under base change.

The central technical achievement is the pro‑representability of the functor \