The Internal Logic and Finite Colimits

We describe how finite colimits can be described using the internal lanuage, also known as the Mitchell-Benabou language, of a topos, provided the topos admits countably infinite colimits. This description is based on the set theoretic definitions of colimits and coequalisers, however the translation is not direct due to the differences between set theory and the internal language, these differences are described as internal versus external. Solutions to the hurdles which thus arise are given.

Automated summary

This article describes how finite colimits can be described using the internal language of a topos, provided the topos allows countably infinite colimits. The article points out the differences between set theory and the internal language, and provides solutions to these differences.