  This paper introduces and studies a categorical analogue of the familiar
monoid semiring construction. By introducing an axiomatisation of summation
that unifies notions of summation from algebraic program semantics with various
notions of summation from the theory of analysis, we demonstrate that the
monoid semiring construction generalises to cases where both the monoid and the
semiring are categories. This construction has many interesting and natural
categorical properties, and natural computational interpretations.
