Approximate categorical structures

TL;DR

This paper explores the concept of approximate categorical structures, generalizing 2-metric spaces, and provides an embedding theorem that links these structures to metrized categories under certain conditions.

Contribution

It introduces a new framework for approximate categorical structures and establishes conditions for their embedding into metrized categories.

Findings

Defined notions of metrized categories and approximate categorical structures

Proved an embedding theorem connecting approximate structures to metrized categories

Provided sufficient conditions for such embeddings

Abstract

We consider notions of metrized categories, and then approximate categorical structures defined by a function of three variables generalizing the notion of $2$-metric space. We prove an embedding theorem giving sufficient conditions for an approximate categorical structure to come from an inclusion into a metrized category.