Localization of a category

Remarks

By the universal property, a localization is unique up to a unique isomorphism of categories compatible with $\gamma$. It always exists if size issues are ignored. Its objects are those of $\mathcal{C}$, and its morphisms are equivalence classes of zigzags of morphisms of $\mathcal{C}$ whose backward arrows are in $W$. The hom-sets of this construction may fail to be small; conditions such as a calculus of fractions or a model structure give control over them. Many authors use a weaker universal property, asking that precomposition with $\gamma$ be an equivalence between functor categories onto the functors that invert $W$.

Example: let $\mathcal{C}$ have two objects $0$ and $1$ and one non-identity morphism $a \colon 0 \to 1$, and let $W = \set{a}$. The category $\mathcal{L}$ with objects $0, 1$ and exactly one morphism between any two objects is a localization, with $\gamma$ the identity on objects. Indeed $\gamma(a)$ is invertible, and a functor $F$ inverting $a$ factors uniquely by sending the morphism $1 \to 0$ of $\mathcal{L}$ to $F(a)^{-1}$.

Non-example: the identity functor $\mathcal{C} \to \mathcal{C}$ satisfies clause 2 for the functors that invert $a$, but it fails clause 1, since $a$ is not an isomorphism in $\mathcal{C}$.