0577958 - MÚ 2024 RIV CH eng J - Článek v odborném periodiku
Batanin, Michael - Davydov, A.
Cosimplicial monoids and deformation theory of tensor categories.
Journal of Noncommutative Geometry. Roč. 17, č. 4 (2023), s. 1167-1229. ISSN 1661-6952. E-ISSN 1661-6960
Grant ostatní: AV ČR(CZ) AP1801
Program: Akademická prémie - Praemium Academiae
Institucionální podpora: RVO:67985840
Klíčová slova: cosimplicial algebras * Steenrod products * operads * tensor categories
Obor OECD: Pure mathematics
Impakt faktor: 0.7, rok: 2023 ; AIS: 0.736, rok: 2023
Způsob publikování: Open access
Web výsledku:
https://doi.org/10.4171/jncg/512DOI: https://doi.org/10.4171/JNCG/512

We introduce the notion of n-commutativity (0 ≤ n ≤ ∞) for cosimplicial monoids in a symmetric monoidal category V, where n = 0 corresponds to just cosimplicial monoids in V, while n=∞ corresponds to commutative cosimplicial monoids. When V has a monoidal model structure, we endow (under some mild technical conditions) the total object of an n-cosimplicial monoid with a natural and very explicit En+1-algebra structure. Our main applications are to the deformation theory of tensor categories and tensor functors.We show that the deformation complex of a tensor functor is a total complex of a 1-commutative cosimplicial monoid and, hence, has an E2-algebra structure similar to the E2-structure on Hochschild complex of an associative algebra provided by Deligne’s conjecture. We further demonstrate that the deformation complex of a tensor category is the total complex of a 2-commutative cosimplicial monoid and, therefore, is naturally an E3-algebra. We make these structures very explicit through a language of Delannoy paths and their noncommutative liftings. We investigate how these structures manifest themselves in concrete examples.
Trvalý link: https://hdl.handle.net/11104/0347033