Equalizer

An equalizer is the limit of a diagram indexed by the parallel pair category.

The Equalizer as a Limit Cone

Consider a diagram \(F: J \to C\), where \(J\) is the parallel pair category. \(F\) is simply a choice of two parallel morphisms, \((f, g: c \to c') \in C\). A cone over such a diagram with summit \(c''\) is a pair of morphisms \(a: c'' \to c\), \(b: c'' \to c'\) such that \(fa = ga = b\).

The Category of Groups

Consider the category of groups. Let \(G\) and \(H\) be groups, \(\phi: G \to H\) be a generic homomorphism, and \(e: G \to H\) be the trivial homomorphism. The equilizer of these two homomorphisms is the kernel of \(\phi\), and the leg of the limit cone is the inclusion map \(\ker \phi \hookrightarrow G\). Generically, the equalizer of two parallel homomorphisms \(\phi, \psi: G \to H\) is the subgroup of elements \(g \in G\) such that \(\phi(g) = \psi(g)\).