In mathematics, a quotient category is a category obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in the categorical setting.

Let ${\displaystyle \mathbf {C} }$ be a category. A congruence relation ${\displaystyle {\mathcal {R}}}$ on ${\displaystyle \mathbf {C} }$ is given by: for each pair of objects ${\displaystyle X}$, ${\displaystyle Y}$ in ${\displaystyle \mathbf {C} }$, an equivalence relation ${\displaystyle {\mathcal {R}}_{X,Y}}$ on ${\displaystyle \mathrm {Hom} (X,Y)}$, such that the equivalence relations respect composition of morphisms. That is, if

are related in ${\displaystyle \mathrm {Hom} (X,Y)}$ and

are related in ${\displaystyle \mathrm {Hom} (Y,Z)}$, then ${\displaystyle g_{1}f_{1}}$ and ${\displaystyle g_{2}f_{2}}$ are related in ${\displaystyle \mathrm {Hom} (X,Z)}$.

Given a congruence relation ${\displaystyle {\mathcal {R}}}$ on ${\displaystyle \mathbf {C} }$ we can define the quotient category ${\displaystyle \mathbf {C} /{\mathcal {R}}}$ as the category whose objects are those of ${\displaystyle \mathbf {C} }$ and whose morphisms are equivalence classes of morphisms in ${\displaystyle \mathbf {C} }$. That is,

Composition of morphisms in ${\displaystyle \mathbf {C} /{\mathcal {R}}}$ is well-defined since ${\displaystyle {\mathcal {R}}}$ is a congruence relation.

There is a natural quotient functor from ${\displaystyle \mathbf {C} }$ to ${\displaystyle \mathbf {C} /{\mathcal {R}}}$ which sends each morphism to its equivalence class. This functor is bijective on objects and surjective on Hom-sets (i.e. it is a full functor).

Every functor ${\displaystyle F\colon \mathbf {C} \to \mathbf {D} }$ determines a congruence on ${\displaystyle \mathbf {C} }$ by saying ${\displaystyle f\sim g}$ iff ${\displaystyle F(f)=F(g)}$. The functor ${\displaystyle F}$ then factors through the quotient functor ${\displaystyle \mathbf {C} \to \mathbf {C} /\sim }$ in a unique manner. This may be regarded as the "first isomorphism theorem" for categories.