Pushout (category Theory) - Construction Via Coproducts and Coequalizers

Construction Via Coproducts and Coequalizers

Pushouts are equivalent to coproducts, and coequalizers (if there is an initial object) in the sense that:

• Coproducts are a pushout from the initial object, and the coequalizer of f, g : X → Y is the pushout of and, so if there are pushouts (and an initial object), then there are coequalizers and coproducts;
• Pushouts can be constructed from coproducts and coequalizers, as described below (the pushout is the coequalizer of the maps to the coproduct).

All of the above examples may be regarded as special cases of the following very general construction, which works in any category C satisfying:

• For any objects A and B of C, their coproduct exists in C;
• For any morphisms j and k of C with the same domain and target, the coequalizer of j and k exists in C.

In this setup, we obtain the pushout of morphisms f : Z → X and g : Z → Y by first forming the coproduct of the targets X and Y. We then have two morphisms from Z to this coproduct. We can either go from Z to X via f, then include into the coproduct, or we can go from Z to Y via g, then include. The pushout of f and g is the coequalizer of these new maps.