Quotient Object

A quotient object of a Congruence $X\xleftarrow{p_{1}}R\xrightarrow{p_{2}}X$ in a Category $\mathcal{C}$ is an object $X/R$ along with a map $\iota:X\to X/R$ such that the diagram

is a Coequaliser Diagram. If we unfold this definition using Generalized Elements, we see that it gives us the "usual" definition of a quotient; EG, the universal object

X\xrightarrow{\iota}X/R

with the property that for all

\Gamma

-generalized elements

x,y:\Gamma\to X

with

R(x,y)

\iota(x)=\iota(y)

Examples

Quotient Monoids and Quotient Semirings are quotient objects.
Quotient Groups and Quotient Rings are also quotient objects, though this requires us to unfold the correspondence between Ideals and Congruences in the respective categories.