Hopf monoids in the category of species
Marcelo Aguiar, Swapneel Mahajan
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Source: Crossref
Published: Jan 1, 2013
DOI: 10.1090/conm/585/11665
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A Hopf monoid (in Joyal’s category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a self-contained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We study several examples of this nature. We emphasize the central role played in the theory by the Tits algebra of set compositions. Its product is tightly knit with the Hopf monoid axioms, and its elements constitute universal operations on connected Hopf monoids. We study analogues of the classical Eulerian and Dynkin idempotents and discuss the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems for Hopf monoids.
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