  We consider sets with infinite addition, called $\Sigma$-monoids, and
contribute to their literature in three ways. First, our definition subsumes
those from previous works and allows us to relate them in terms of adjuctions
between their categories. In particular, we discuss $\Sigma$-monoids with
additive inverses. Second, we show that every Hausdorff commutative monoid is a
$\Sigma$-monoid, and that there is a free Hausdorff commutative monoid for each
$\Sigma$-monoid. Third, we prove that $\Sigma$-monoids have well-defined tensor
products, unlike topological abelian groups.
