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Mathematics > Algebraic Topology

arXiv:1807.11858 (math)
[Submitted on 31 Jul 2018 (v1), last revised 10 Dec 2018 (this version, v2)]

Title:Antipodes of monoidal decomposition spaces

Authors:Louis Carlier, Joachim Kock
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Abstract: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 Möbius function as $\mu = \zeta \circ S$, just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors $S_{\textrm{even}} - S_{\textrm{odd}}$, and it is a refinement of the general Möbius inversion construction of Gálvez-Kock-Tonks, but exploiting the monoidal structure.
Comments: 14 pages. Dedicated to the memory of Thomas Poguntke. v2: minor expository adjustments; final version to appear in Commun. Contemp. Math
Subjects: Algebraic Topology (math.AT); Combinatorics (math.CO); Category Theory (math.CT); Rings and Algebras (math.RA)
MSC classes: 16T10, 16T05, 16T30, 18D10, 18G30, 06A75
Cite as: arXiv:1807.11858 [math.AT]
  (or arXiv:1807.11858v2 [math.AT] for this version)
  https://doi.org/10.48550/arXiv.1807.11858
Journal reference: Commun. Contemp. Math. 22 (2020), 1850081
Related DOI: https://doi.org/10.1142/S0219199718500815

Submission history

From: Louis Carlier [view email]
[v1] Tue, 31 Jul 2018 15:08:48 UTC (17 KB)
[v2] Mon, 10 Dec 2018 13:48:21 UTC (17 KB)
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