coequal

Coequal, short for coequalizer, is a fundamental construction in category theory. Given two parallel morphisms f and g from X to Y, a coequalizer consists of an object Q and a morphism q: Y → Q such that q f = q g and, moreover, if h: Y → Z satisfies h f = h g, then there exists a unique u: Q → Z with h = u q. In other words, Q is the universal target through which f and g become equal.

The coequalizer is dual to the equalizer, which is defined using arrows X → Y and seeks a universal morphism into X that factors through any morphism making the two arrows agree. Coequalizers are a type of colimit, and many categories used in mathematics have them, including Set, Grp, Ab, Ring, and Top. They are preserved by functors that preserve colimits, such as left adjoints.

Concrete descriptions in common categories help intuition. In Set, the coequalizer of f,g: X → Y is the quotient of Y by the smallest equivalence relation ~ such that f(a) ~ g(a) for every a ∈ X. For example, if X = {0} and Y = {a,b} with f(0) = a and g(0) = b, the coequalizer identifies a and b, yielding a singleton set. In Grp, the coequalizer of f,g: X → Y is Y modulo the normal subgroup generated by f(a) g(a)^{-1} for all a ∈ X. In Ab (an abelian category), the coequalizer of f,g: X → Y is the cokernel of f − g: X → Y, often written coker(f − g).

In summary, the coequalizer provides the universal way to force two parallel arrows to coincide, encapsulating a central notion of quotienting by a relation induced by the arrows.