Classically we start with natural numbers and integers, then invert to obtain rational numbers. We can then complete the rational numbers to obtain the real numbers.
One categorical level up, natural numbers can be realised as the cardinalities of sets. Likewise we can obtain positive rational numbers as the cardinalities of groupoids. Here, the cardinality of a grouped $\mathcal{G}$ is $$ |\mathcal{G}|:= \sum_{g\in\mathrm{Ob{\mathcal{G}}}}\frac{1}{|\mathrm{Aut}(g)|}. $$ This kind of counting is very natural in the context of algebraic stacks, where standard integral invarients of schemes become rational.
My question is whether the completion of the rational numbers to the real numbers can be realised in a categorified way fitting into the above framework. In particular, is there some plausible algebraic geometric context in which standard integral invariants are replaced not by rational numbers but by real numbers?