Remark 4.27.3. The motivation for the construction of $S^{-1} \mathcal{C}$ is to “force” the morphisms in $S$ to be invertible by artificially creating inverses to them (at the cost of some existing morphisms possibly becoming identified with each other). This is similar to the localization of a commutative ring at a multiplicative subset, and more generally to the localization of a noncommutative ring at a right denominator set (see [Section 10A, Lam]). This is more than just a similarity: The construction of $S^{-1} \mathcal{C}$ (or, more precisely, its version for additive categories $\mathcal{C}$) actually generalizes the latter type of localization. Namely, a noncommutative ring can be viewed as a pre-additive category with a single object (the morphisms being the elements of the ring); a multiplicative subset of this ring then becomes a set $S$ of morphisms satisfying LMS1 (aka RMS1). Then, the conditions RMS2 and RMS3 for this category and this subset $S$ translate into the two conditions (“right permutable” and “right reversible”) of a right denominator set (and similarly for LMS and left denominator sets), and $S^{-1} \mathcal{C}$ (with a properly defined additive structure) is the one-object category corresponding to the localization of the ring.

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