  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.
