We propose definitions of regular and exact (virtual) double categories, proving a number of results which parallel many basic results in the theory of regular and exact categories. We show that any regular virtual double category admits a factorization system which generalizes the factorization of a functor between categories into a bijective-on-objects functor followed by a fully-faithful functor. Finally, we show that our definition of exact double category is equivalent to an axiom proposed… Expand

This work proposes an extension of the earlier set-valued functor model, making use of multi-sorted algebraic theories to incorporate concrete data in a principled way, and shows how all of the components of this model fit into a single double categorical structure called a proarrow equipment.Expand

A double category of relations is essentially a cartesian equipment with strong, discrete and functorial tabulators and for which certain local products satisfy a Frobenius Law. A double category of… Expand

Introduction 1. The language of categories 2. Limits 3. Adjoint functors 4. Generators and projectives 5. Categories of fractions 6. Flat functors and Cauchy completeness 7. Bicategories and… Expand

In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors,… Expand

Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere… Expand