The internal logic and finite colimits

Abstract

We describe how finite colimits can be described using the internal language, 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 coequalizers, 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.