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 M\"obius 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 M\"obius inversion construction of G\'{a}lvez-Kock-Tonks, but exploiting the monoidal structure.