CatGloss

An equalizer is a limit of a diagram indexed by the parallel pair, the category \bullet \rightrightarrows \bullet with two objects and two parallel non-identity morphisms. A diagram of this shape is simply a parallel pair of morphisms f,g ⁣:ABf,g \colon A \rightrightarrows B in the target category C\mathsf{C}. A cone over this diagram with summit CC consists of a pair of morphisms a ⁣:CAa \colon C \to A and b ⁣:CBb \colon C \to B so that fa=bfa = b and ga=bga = b; these two equations correspond to the naturality conditions \eqref{eq:cone-naturality} imposed by each of the two non-identity morphisms in the indexing category. Together, they assert that fa=gafa = ga; the morphism bb is necessarily equal to this common composite. Thus, a cone over a parallel pair f,g ⁣:ABf,g \colon A \rightrightarrows B is represented by a single morphism a ⁣:CAa \colon C \to A so that fa=gafa = ga.

The equalizer h ⁣:EAh \colon E \to A is the universal arrow with this property. The limit diagram \(\xymatrix{ E \ar[r]^h & A \ar@<.5ex>[r]^f \ar@<-.5ex>[r]_g & B}\) is called an equalizer diagram. The universal property asserts that given any a ⁣:CAa \colon C \to A so that fa=gafa = ga, there exists a unique factorization k ⁣:CEk \colon C \to E of aa through hh. \(\xymatrix{ C \ar[dr]^a \ar@{-->}[d]^{\exists !}_k \\ E \ar[r]_h & A \ar@<.5ex>[r]^f \ar@<-.5ex>[r]_g & B}\)