Abstract
In this paper we generalize Tannakian formalism to fiber functors over general tensor categories. We will show that (under some technical conditions) if the fiber functor has a section, then the source category is equivalent to the category of comodules over a Hopf algebra in the target category. We will also give a description of this Hopf algebra using the notion of framed objects.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The notion of framed objects appeared first in [3].
For definition of Tannakian categories, see Sect. 5.
For definition and basic properties of extension of scalars in Tannakian categories, see [6], §4.
We used this notation to distinguish it from our notation \(\otimes \) for product of categories.
For basic algebraic geometry notions in tensor categories, see [5], 7.8
The fundamental group has also a natural action on every algebra in the category and because the category of affine schemes is the opposite category of the category of algebras, it has also a right action on every affine scheme. But we use the word “natural action” in the case of affine schemes for the inverse of this action which is also an action from left.
The example is taken from Derek Holt’s answer in http://math.stackexchange.com/a/1668652/202376.
References
D. Arapura, The Hodge Theoretic Fundamental Group and Its Cohomology, The Geometry of Algebraic Cycles (American Mathematical Society, Providence, 2010)
F. Borceux, Handbook of Categorical Algebra 2, Categories and Structures (Cambridge University Press, Cambridge, 1994)
A. Beilinson, A. Goncharov, V. Schechtman, A. Varchenko, Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of Pairs of triangles on the plane. The Grothendieck Festchrift 1, 135–172 (1990). (Birkhuser)
B. Day, Enriched Tannaka reconstruction. J. Pure Appl. Algebra 108(1), 17–22 (1996)
P. Deligne, Categories Tannakiennes. The Grothendieck Festchrift 2, 111–195 (1990). (Birkhuser)
P. Deligne, Semi-simplicit de produits tensoriels en caractristique p. Invent. Math. 197(3) (2014)
P. Deligne, J. Milne, Tannakian categories, Lecture notes in Mathematics, Volume 900. Springer (1982)
P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor Categories (American Mathematical Society, Providence, 2015)
A. Goncharov, Mixed Elliptic Motives, Galois Representations in Arithmetic Algebraic Geometry (Cambridge University Press, Cambridge, 1998)
J. Greenough, Monoidal 2-structure of bimodule categories. J. Algebra 324(8), 1818–1859 (2010)
S. Mac Lane, Categories for the Working Mathematician, 2nd edn. (Springer, Berlin, 1998)
V. Ostrik, Module categories, weak Hopf algebras and modular invariants. Transf. Groups 8(2), 177–206 (2003)
R. Saavedra, Categories Tannakiennes, Lecture notes in Mathematics, vol. 265 (Springer, 1972)
D. Schäppi, A characterization of categories of coherent sheaves of certain algebraic stacks, ArXiv e-prints (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jafari, A., Einollahzadeh, M. Tannakian formalism for fiber functors over tensor categories. Period Math Hung 77, 1–26 (2018). https://doi.org/10.1007/s10998-017-0213-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-017-0213-0