Given a category $\mathsf C$ and a class of arrows $\Sigma$, we say the category of fractions $\mathsf C [\Sigma ^{-1}]$ exists when there's a functor $\varphi :\mathsf C \rightarrow \mathsf C [\Sigma ^{-1}]$ that makes any $f\in \Sigma$ into an isomorphism, and is universal with respect to this property, i.e any other functor $F:\mathsf C \rightarrow \mathsf D$ which makes the elements of $\Sigma$ into isomorphisms uniquely factors through $\varphi$.
This definition is simple and makes perfect sense to me, and so does its construction when $\Sigma$ is a set. However, the concept of (right) calculus of fractions eludes me completely. Here's the definition given in the first volume of Borceux's Handbook of Categorical Algebra:
Questions:
- What is the idea behind this definition?
- Why is it called a calculus of fractions?