Coequalizer

Definition

Let $\mathcal{C}$ be a category and let

be parallel morphisms (same domain and codomain).

A coequalizer of $(f,g)$ is a morphism $q:B\to Q$ such that:

1. (Coequalizing condition) $q\circ f = q\circ g$ (using composition ), and
2. (Universal property) for every object $X$ and morphism $h:B\to X$ with $h\circ f = h\circ g$, there exists a unique morphism $u:Q\to X$ such that $u\circ q = h.$

Equivalently, $q$ is universal among morphisms out of $B$ that identify $f$ and $g$.

If a coequalizer exists, it is unique up to unique isomorphism .

Relationship to other constructions

• A coequalizer is a special case of a colimit : it is the colimit of the diagram $A \rightrightarrows B$.
• It is dual to an equalizer .

Examples

Example (Set)

In $\mathbf{Set}$, given functions $f,g:A\to B$, form the smallest equivalence relation $\sim$ on $B$ generated by relations

Then the coequalizer is the quotient map

where $B/{\sim}$ is the quotient set .

Example (Grp)

In $\mathbf{Grp}$, for homomorphisms $f,g:A\to B$, let $N\trianglelefteq B$ be the normal subgroup generated by all elements $f(a)g(a)^{-1}$ ($a\in A$).
Then the coequalizer is the quotient homomorphism

Example ($R$-Mod)

In $R\text{-}\mathbf{Mod}$, for $R$-linear maps $f,g:A\to B$, the coequalizer is

Equivalently, it is the cokernel of $f-g$ (computed in this abelian category).