Internalization and enrichment via spans and matrices in a tricategory

We introduce categories M and S internal in the tricategory Bicat 3 of bicategories, pseudofunctors, pseudonatural transformations and modifications, for matrices and spans in a 1-strict tricategory V. Their horizontal tricategories are the tricategories of matrices and spans in V. Both the internal and the enriched constructions are tricategorifications of the corresponding constructions in 1-categories. Following Fiore et al. (J Pure Appl Algebra 215(6):1174–1197, 2011), we introduce monads and their vertical morphisms in categories internal in tricategories. We prove an equivalent condition for when the internal categories for matrices M and spans S in a 1-strict tricategory V are equivalent, and deduce that in that case their corresponding categories of (strict) monads and vertical monad morphisms are equivalent, too. We prove that the latter categories are isomorphic to those of categories enriched and discretely internal in V, respectively. As a by-product of our tricategorical constructions, we recover some results from Femić (Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories. arXiv:2101.01460v2). Truncating to 1-categories, we recover results from Cottrell et al. (Tbilisi Math J 10(3):239–254, 2017) and Ehresmann and Ehresmann (Cah Topol Géom Differ Catég 19/4:387–443, 1978) on the equivalence of enriched and discretely internal 1-categories.