External derivations of internal groupoids

Groupoids: Unifying Internal and External Symmetry

744 NOTICES OF THE AMS VOLUME 43, NUMBER 7 Introduction Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps largely because of the well-known Erlanger program of F. Klein and the related work of S. Lie, which virtually defined geometric structures by their groups of automorphisms. (See, for example, Yaglom’s a...

Groupoids, Imaginaries and Internal Covers

The questions this manuscript addresses arose in the course of an investigation of the imaginary sorts in ultraproducts of p-adic fields. These were shown to be understandable given the imaginary sorts of certain finite-dimensional vector spaces over the residue field. The residue field is pseudo-finite, and the imaginary elements there were previously studied, and shown in fact to be eliminabl...

On the Monad of Internal Groupoids

We deeply analyse the structural organisation of the fibration of points and of the monad of internal groupoids. From that we derive: 1) a new characterization of internal groupoids among reflexive graphs in the Mal’cev context; 2) a setting in which a Mal’cev category is necessarily a protomodular category.

Groupoids: Unifying Internal and External Symmetry a Tour through Some Examples

External and Internal Relations ∗

In the index to Appearance and Reality (First Edition) Mr. Bradley declares that all relations are “intrinsical”; and the following are some of the phrases by means of which he tries to explain what he means by this assertion. “A relation must at both ends affect, and pass into, the being of its terms” (p. 364). “Every relation essentially penetrates the being of its terms, and is, in this sens...