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Antipodes of monoidal decomposition spaces. (English) Zbl 1454.18013

The author introduces a notion of antipode for monoidal complete decomposition spaces, which, in the connected case, recovers the usual notion of antipode in Hopf algebras (Proposition 3.5); upgrading the results by Gálvez-Kock-Tonks introducing a useful weaker version of antipode for bialgebras. There is necessary some finiteness condition, such that the category should be Mobius. It is possible to compute the Möbius functor (Corollary 4.1) and inverses of certain multiplicative functors (Theorem 4.3).

MSC:

18M05 Monoidal categories, symmetric monoidal categories
16T05 Hopf algebras and their applications
16T10 Bialgebras
16T30 Connections of Hopf algebras with combinatorics
18N50 Simplicial sets, simplicial objects
06A75 Generalizations of ordered sets
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References:

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