TY - JOUR

T1 - The fundamental pro-groupoid of an affine 2-scheme

AU - Chirvasitu, Alex

AU - Johnson-Freyd, Theo

N1 - Funding Information:
Acknowledgements We would like to thank D. Ben-Zvi, A. Geraschenko, M. Olsson, N. Reshetikhin, N. Rozenblyum, V. Serganova, C. Teleman, and H. Williams for their many helpful comments and discussions. M. Brandenburg caught a number of errors in an early version of this paper. We would particularly like to thank the anonymous referee for the detailed and engaging review. The correct definition of torsors in a general commutative 2-ring (and in particular the definition of the morphisms τx defined above Lemma 2.4.4) is due to the referee, and we have also included a few comments and examples the referee suggested. Some of this work was completed while the second author was a visitor at Northwestern University, whom he thanks for the hospitality. This work is supported by the NSF grant DMS-0901431.

PY - 2013/10

Y1 - 2013/10

N2 - 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.

AB - 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.

KW - Affine 2-schemes

KW - Categorification

KW - Fundamental groupoids

KW - Galois theory

KW - Higher category theory

KW - Presentable categories

KW - Tannakian reconstruction

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U2 - 10.1007/s10485-011-9275-y

DO - 10.1007/s10485-011-9275-y

M3 - Article

AN - SCOPUS:84884284622

SN - 0927-2852

VL - 21

SP - 469

EP - 522

JO - Applied Categorical Structures

JF - Applied Categorical Structures

IS - 5

ER -