Operadické kategorie

• Název práce v češtině: Operadické kategorie
• Název v anglickém jazyce: Operadic Categories
• Klíčová slova: operády|teorie kategorií
• Klíčová slova anglicky: operads|category theory
• Akademický rok vypsání: 2023/2024
• Typ práce: disertační práce
• Ústav: Matematický ústav UK (32-MUUK)
• Vedoucí / školitel: RNDr. Martin Markl, DrSc.

Zásady pro vypracování

In their influential 2015 paper, Batanin and Markl introduced operadic categories as a versatile tool for describing the most common mathematical structures appearing in geometry, homological algebra, and mathematical physics. Despite the progress made since then, several unresolved issues persist within this field. This doctoral thesis project aims to address some of these challenges.

[1] Standard operadic categories are equipped with the cardinality functor landing in the skeletal category of finite sets. The skeletality guarantees the existence of unique pullbacks. However, for certain practical applications, allowing arbitrary finite sets as cardinalities would be preferable. Achieving this seemingly straightforward modification poses unexpected coherence problems,
which will be a the first focus of this research.

[2] Batanin proposed the concept of distributive operadic categories, inspired by distributive laws for operads introduced by Fox and Markl in the nineties of the last century. The second theme of the thesis is to further develop this notion and explore its potential applications, such as in the study of incidence bialgebras and related combinatorial structures.

[3] Under appropriate conditions, each operadic category admits its plus-construction. This construction yields an operadic category where algebras for its terminal operad precisely correspond to operads over the original category. Developing the general theory of this construction, akin to Baez-Dolan opetopes, is another objective of the thesis.

[4] Finally, the research will investigate the parallels between Batanin's wreath product of operadic categories and the notoriously difficult Boardman-Vogt product of operads, based on the operadic Grothendieck construction and its adjoint. Here the aim is to elucidate this parallel and explore potential implications in homotopy theory and combinatorial algebra.

Seznam odborné literatury

[1] J. Kock, A. Joyal, M. Batanin, J.-F. Mascari: Polynomial functors and opetopes. Adv. Math.224(2010), no.6, 2690-2737.

[2] M. Batanin, C. Berger: Homotopy theory for algebras over polynomial monads, Theory Appl. Categ. 32 (2017), Paper No. 6, 148-253.

[3] M. Markl, M. Batanin: Operadic categories as a natural environment for Koszul duality, Compositionality (5)3, 2023.

[4] M. Markl, M. Batanin: Koszul duality for operadic categories, Compositionality (5)4, 2023.

[5] M. Markl, M. Batanin: Operadic categories and duoidal Deligne's conjecture, Advances in Mathematics 285(2015), 1630-1687.

plus the informal notes by Markl, Batanin, Kock and Weber, available on request from the authors.