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).

Every small category can be represented as a quotient category of the category of paths over an appropriate directed graph.

Comments

Any equivalence relation satisfying the conditions above is commonly called a congruence on $ \mathfrak K $( cf. Congruence (in algebra)).

References

[a1]	B. Mitchell, "Theory of categories" , Acad. Press (1965) pp. 4