Internalization and enrichment via spans and matrices in a tricategory

TL;DR

This paper develops tricategorical frameworks for matrices and spans within a tricategory, establishing equivalences between internal, enriched, and discretely internal categories, and generalizing known 1-category results.

Contribution

It introduces categories internal in a tricategory for matrices and spans, proving their equivalences and extending classical 1-category results to tricategories.

Findings

Categories of monads and monad morphisms are equivalent in internal and enriched contexts.

Internal and enriched constructions are tricategorifications of classical 1-category concepts.

Recovered known results on equivalences of enriched and discretely internal categories from prior literature.

Abstract

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 \cite{FGK} 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 byproduct of our tricategorical constructions we recover some results from \cite{Fem}. Truncating to 1-categories we recover results from \cite{CFP} and \cite{Ehr} on the equivalence of enriched and discretely internal 1-categories.