Antipodes of monoidal decomposition spaces

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 [Formula: see text], just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors [Formula: see text], and it is a refinement of the general Möbius inversion construction of Gálvez–Kock–Tonks, but exploiting the monoidal structure.