# Quotient category

A construct analogous to that of a quotient set or quotient algebra. Let $\mathfrak K$ be an arbitrary category, and suppose that an equivalence relation $\sim$ is given on its class of morphisms $\mathop{\rm Mor} \mathfrak K$, satisfying the following conditions: 1) if $\alpha \sim \beta$, then the sources and targets of the morphisms $\alpha$ and $\beta$ are the same; and 2) if $\alpha \sim \beta$, $\gamma \sim \delta$ and if the product $\alpha \gamma$ is defined, then $\alpha \gamma \sim \beta \delta$. Let $[ \alpha ]$ denote the equivalence class of $\alpha$. The quotient category of $\mathfrak K$ by $\sim$ is the category (denoted by $\mathfrak K / \sim$) with the same objects as $\mathfrak K$, and for any pair of objects $A$, $B$ the set of morphisms $H ( A , B )$ in $\mathfrak K / \sim$ consists of the equivalence classes $[ \alpha ]$, where $\alpha : A \rightarrow B$ in $\mathfrak K$; multiplication of two morphisms $[ \alpha ]$ and $[ \beta ]$ is defined by the formula $[ \alpha ] [ \beta ] = [ \alpha \beta ]$( when the product $\alpha \beta$ is defined).
Any equivalence relation satisfying the conditions above is commonly called a congruence on $\mathfrak K$( cf. Congruence (in algebra)).