Restriction categories as enriched categories

TL;DR

This paper demonstrates that restriction categories, which model partially defined mappings, can be understood as enriched categories, unifying various notions like join and range restriction through a suitable enrichment base.

Contribution

It establishes an equivalence between restriction categories and enriched categories, providing a new perspective and tools for studying partial maps using enriched category theory.

Findings

Restriction categories are equivalent to certain enriched categories.

The framework captures join and range restriction categories.

Enriched category theory offers new insights into partial mappings.

Abstract

Restriction categories were introduced to provide an axiomatic setting for the study of partially defined mappings; they are categories equipped with an operation called restriction which assigns to every morphism an endomorphism of its domain, to be thought of as the partial identity that is defined to just the same degree as the original map. In this paper, we show that restriction categories can be identified with \emph{enriched categories} in the sense of Kelly for a suitable enrichment base. By varying that base appropriately, we are also able to capture the notions of join and range restriction category in terms of enriched category theory.