The fundamental pro-groupoid of an affine 2-scheme

Alex Chirvasitu*, Theo Johnson-Freyd

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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 étale fundamental groupoid π 1(spec(R)), i.e. the separable absolute Galois group of R when it is a field. This gives a new definition for étale π 1(spec(R)) in terms of the category of R-modules rather than the category of étale 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 π 1 for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the étale fundamental group of a scheme preserves finite products but not all products.

Original languageEnglish (US)
Pages (from-to)469-522
Number of pages54
JournalApplied Categorical Structures
Issue number5
StatePublished - Oct 2013


  • Affine 2-schemes
  • Categorification
  • Fundamental groupoids
  • Galois theory
  • Higher category theory
  • Presentable categories
  • Tannakian reconstruction

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Algebra and Number Theory
  • General Computer Science


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