In this talk we will present a new characterization of internal groupoids using involutive-2-links. An involutive-2-link consists of a single morphism equipped with a pair of interlinked involutions on its domain [1]. We will see that there exists a full and faithful functor from the category of internal groupoids to the category of involutive-2-links that can be used to establish a categorical equivalence between internal groupoids and a suitable subcategory of involutive-2-links. This approach is different from the one that considers an internal groupoid as having an explicit underlying reflexive graph in which a multiplicative (or composition) structure is defined. That point of view is advantageous, for example, in the context of Mal'tsev categories, where the forgetful functor from groupoids to reflexive graphs is full and faithful. However, this is not the case in general. Analysing internal groupoids through involutive-2-links is an attempt to extend some results obtained in the restricted context of Mat'tsev categories to more general situations. This suggests a change in the way we look at internal groupoids: instead of focusing our attention on the underlying reflexive graph of a groupoid, we can also look at its multiplicative structure along with an appropriate pair of interlinked involutions, from which the underlying reflective graph appears as soon as it exists.

[1] N. Martins-Ferreira, What is an internal groupoid?, https://arxiv.org/abs/2211.12531.