Hereditary Species as Monoidal Decomposition Spaces, Comodule Bialgebras, and Operadic Categories

DOI10.1093/IMRN/RNAA260zbMATH Open1489.18010arXiv1903.07964OpenAlexW2921959845MaRDI QIDQ5075175

Louis Carlier

Publication date: 10 May 2022

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: We show that Schmitt's hereditary species induce monoidal decomposition spaces, and exhibit Schmitt's bialgebra construction as an instance of the general bialgebra construction on a monoidal decomposition space. We show furthermore that this bialgebra structure coacts on the underlying restriction-species bialgebra structure so as to form a comodule bialgebra. Finally, we show that hereditary species induce a new family of examples of operadic categories in the sense of Batanin and Markl.

Full work available at URL: https://arxiv.org/abs/1903.07964

zbMATH Keywords

incidence coalgebraoperadic categorycomodule bialgebrashereditary spacesmonoidal decomposition spaceunital 2-Segal space

Mathematics Subject Classification ID

Groupoids (i.e. small categories in which all morphisms are isomorphisms) (20L05) Bialgebras (16T10) 2-categories, bicategories, double categories (18N10) Monoidal categories, symmetric monoidal categories (18M05)

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