  A moment category is endowed with a distinguished set of split idempotents,
called moments, which can be transported along morphisms. Equivalently, a
moment category is a category with an active/inert factorisation system
fulfilling two simple axioms. These axioms imply that the moments of a fixed
object form a monoid, actually a left regular band.
  Each locally finite unital moment category defines a specific type of operad
which records the combinatorics of partitioning moments into elementary ones.
In this way the notions of symmetric, non-symmetric and $n$-operad correspond
to unital moment structures on $\Gamma$, $\Delta$ and $\Theta_n$ respectively.
  There is an analog of Baez-Dolan's plus construction taking a unital moment
category $\mathbb{C}$ to a unital hypermoment category $\mathbb{C}^+$. Under
this construction, $\mathbb{C}$-operads get identified with
$\mathbb{C}^+$-monoids, i.e. presheaves on $\mathbb{C}^+$ satisfying Segal-like
conditions strictly. We show that the plus construction of Segal's category
$\Gamma$ fully embeds into the dendroidal category $\Omega$ of Moerdijk-Weiss.
