Expanding on the comprehensive factorization of functors internal to a category C, under fairly mild conditions on a monad T on C we establish that this orthogonal factorization system exists even in Burroni's category Cat(T) of (internal) T-categories and their functors. This context provides for some expected applications and some unexpected connections. For example, it lets us deduce that the comprehensive factorization is also available for functors of Lambek's multicategories. In topology, it leads to the insight that the role of discrete cofibrations is played by perfect maps, with the comprehensive factorization of a continuous map of Tychonoff spaces given by its fibrewise compactification.
Keywords: T-graph, T-category, T-functor, discrete cofibration, comprehensive factorization, multicategory, perfect map, wide pullback, small-topological functor
2020 MSC: 18A32, 18A30, 18C15, 18D30, 54C10
Theory and Applications of Categories, Vol. 36, 2021, No. 8, pp 206-249.