What is an internal groupoid?

TL;DR

This paper offers a new characterization of internal groupoids that does not rely on finite limits, viewing them as involutive tri-graphs or simplified structures with interlinked involutions.

Contribution

It introduces a novel perspective on internal groupoids, representing them as involutive tri-graphs and simpler morphism structures, expanding the theoretical understanding.

Findings

Characterization holds without assuming finite limits.

Internal groupoids are equivalent to involutive tri-graphs.

Simplified structures with interlinked involutions are possible.

Abstract

An answer to the question investigated in this paper brings a new characterization of internal groupoids such that: (a) it holds even when finite limits are not assumed to exist; (b) it is a full subcategory of the category of involutive-2-links, that is, a category whose objects are morphisms equipped with a pair of interlinked involutions. This result highlights the fact that even thought internal groupoids are internal categories equipped with an involution, they can equivalently be seen as tri-graphs with an involution. Moreover, the structure of a tri-graph with an involution can be further contracted into a simpler structure consisting of one morphism with two interlinked involutions. This approach highly contrasts with the one where groupoids are seen as reflexive graphs on which a multiplicative structure is defined with inverses.