Antipodes of monoidal decomposition spaces
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We introduce a notion of antipode for monoidal (complete) decomposition spaces, inducing a notion of weak antipode for their incidence bialgebras. In the connected case, this recovers the usual notion of antipode in Hopf algebras. In the non-connected case, it expresses an inversion principle of more limited scope, but still sufficient to compute the Mobius function as mu = zeta o S, just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors S-even - S-odd, and it is a refinement of the general Mobius inversion construction of Galvez-Kock-Tonks, but exploiting the monoidal structure.
Original language | English |
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Article number | 1850081 |
Journal | Communications in Contemporary Mathematics |
Volume | 22 |
Issue number | 2 |
Number of pages | 15 |
ISSN | 0219-1997 |
DOIs | |
Publication status | Published - Mar 2020 |
Externally published | Yes |
- Bialgebra, antipode, decomposition space, 2-Segal space, incidence algebra, BIALGEBRAS
Research areas
ID: 331497757