Coequalizer - Imported Knowlpedia Content

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

f,g : A \to B

be parallel morphisms (same domain and codomain).

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

(Coequalizing condition) $q\circ f = q\circ g$ (using composition), and
(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 $u\circ q = h.$

Equivalent characterizations

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

Remarks

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

f(a)\sim g(a)\quad \text{for all } a\in A.

Then the coequalizer is the quotient map

q:B \longrightarrow B/{\sim},

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

q:B\to B/N.

Example ( $R$ -Mod)

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

q:B \to B/\operatorname{im}(f-g).

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

Example (Set)

Example (Grp)

Example (RRR-Mod)

Example ( $R$ -Mod)