Antipodes of monoidal decomposition spaces

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 $S_{\text{even}}-S_{\text{odd}}$, and it is a refinement of the general Möbius inversion construction of Gálvez-Kock-Tonks, but exploiting the monoidal structure.

14 pages. Dedicated to the memory of Thomas Poguntke. v2: minor expository adjustments; final version to appear in Commun. Contemp. Math

References in corpus (4)

• Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
• Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness
• Decomposition spaces, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals
• Homotopy linear algebra