@Neil Strickland gave me all I needed in his comment (thank you!). Escardó and Simpson's paper is not what I was looking for, but they do cite it: A Universal Characterization of $[0,\infty]$, by Denis Higgs.
For those curious but not curious enough to find a text of the paper: a magnitude module is a variety of algebras with a constant $0$, a unary operation $h$, and an $\omega$-ary operation $\Sigma$ satisfying the identities
- $\Sigma(\Sigma(x_{00},x_{01},\dots),\Sigma(x_{10},x_{11},\dots),\dots) = \Sigma(\Sigma(x_{00},x_{10},\dots),\Sigma(x_{01},x_{11},\dots),\dots)$.
- $\Sigma(0,\dots,0,x,0,\dots) = x$.
- $h (\Sigma(x_0,x_1,\dots)) = \Sigma(h(x_0),h(x_1),\dots)$.
- $\Sigma(h(x),h^2(x),h^3(x),\dots) = x$.
Because $[0,\infty]$ is the free magnitude module on one point, every magnitude module $M$ admits a scalar multiplication-like action $[0,\infty] \times M \to M$, hence the name 'magnitude module'.