# 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 |

**How to Cite This Entry:**

Quotient category.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Quotient_category&oldid=48407