Internal factorisation systems - Ranchod, Sanjiv (Department of Mathematics and Applied Mathematics, 2023)
We introduce internal factorisation systems for internal categories. We recall the definitions and theory of internal categories and factorisation systems. We develop a diagrammatic calculus of pullbacks for ease of internal calculation. To define an internal factorisation system we define and study the subobjects of isomorphisms, an internalisation of the class of isomorphisms of a category. We provide an abstract example of an internal factorisation system. We then internalise various properties of factorisation systems, such as the two components determining each other, the cancellation properties and the essential uniqueness of factorisations, and show that an internal factorisation system satisfies these internal conditions.