Monoidal abelian envelopes with a quotient property. (English) Zbl 1504.18014

A fully faithful embedding of a pseudo-tensor category into a tensor category with an appropriate universal property is known as an abelian envelope of the pseudo-tensor category. The question of which pseudo-tensor categories do admit an abelian envelope has attracted a lot of attention recently [D. Benson and P. Etingof, Adv. Math. 351, 967–999 (2019; Zbl 1430.18013); D. Benson et al., “New incompressible symmetric tensor categories in positive characteristic”, Preprint, arXiv:2003.10499; J. Comes and V. Ostrik, Algebra Number Theory 8, No. 2, 473–496 (2014; Zbl 1305.18019); K. Coulembier et al., Algebra Number Theory 16, No. 9, 2099–2117 (2022; Zbl 1503.18006); K. Coulembier, “Additive Grothendieck pretopologies and presentations of tensor categories”, Preprint, arXiv:2011.02137; P. Deligne, in: Algebraic groups and homogeneous spaces. Proceedings of the international colloquium, Mumbai, India, January 6–14, 2004. New Delhi: Narosa Publishing House/Published for the Tata Institute of Fundamental Research. 209–273 (2007; Zbl 1165.20300); I. Entova-Aizenbud et al., Int. Math. Res. Not. 2020, No. 15, 4602–4666 (2020; Zbl 1477.18034)], leading to various interesting applications. It is observed empirically that every object in the tensor category is a quotient of an object in the pseudo-tensor category. On the other hand, it was observed in [D. Benson et al., “New incompressible symmetric tensor categories in positive characteristic”, Preprint, arXiv:2003.10499] that when a pseudo-tensor category \(\boldsymbol{D}\) can be embedded in a tensor category \(\boldsymbol{T}\) such that every object in \(\boldsymbol{T}\) is a quotient of an object in \(\boldsymbol{D}\), \(\boldsymbol{T}\) must be the abelian envelope of \(\boldsymbol{D}\).
This paper aims to further investigate abelian envelopes, in particular in relation to the above quotient property. The first main result establishes an intrinsic criterion on a pseudo-tensor category determining whether there exists an abelian envelope with the quotient property. The other main results establish new ways to interpret known tensor categories as abelian envelopes. It is also investigated whether the Deligne product and extension of scalars are always tensor categories. It is finally demonstrated that if the extension of scalars is always a tensor category, then so is Deligne’s product of tensor categories.


18M05 Monoidal categories, symmetric monoidal categories
17B37 Quantum groups (quantized enveloping algebras) and related deformations
14A20 Generalizations (algebraic spaces, stacks)
14L15 Group schemes
20G05 Representation theory for linear algebraic groups
Full Text: DOI arXiv


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