Computational Paths and the Fundamental Groupoid of a Type

Using computational paths as the fundamental concept, we show that we can leverage Category Theory to propose the concept of fundamental groupoid of a type.

Groupoids and the algebra of rewriting in group presentations

Presentations of groups by rewriting systems (that is, by monoid presentations), have been fruitfully studied by encoding the rewriting system in a 2-complex—the Squier complex—whose fundamental groupoid then describes the derivation of consequences of the rewrite rules. We describe a reduced form of the Squier complex, investigate the structure of its fundamental groupoid, and show that key properties of the presentation are still encoded in the reduced form.

Algebras of Quantum Monodromy Data and Character Varieties

This chapter examines the Poisson structure of the representation variety of the fundamental groupoid of a Riemann surface with punctures and cusps, and the associated decorated character variety.

The fundamental groupoid as a terminal costack

In this paper we prove that for good topological spaces the assignment

Covering groupoids of categorical rings

A categorical group is a kind of categorization of group and similarly a categorical ring is a categorization of ring. For a topological group X, the fundamental groupoid ?X is a group object in the category of groupoids, which is also called in literature group-groupoid or 2-group. If X is a path connected topological group which has a simply connected cover, then the category of covering groups of X and the category of covering groupoids of ?X are equivalent. Recently it was proved that if (X, x0) is an H-group, then the fundamental groupoid ?X is a categorical group and the category of the covering spaces of (X, x0) is equivalent to the category of covering groupoids of the categorical group ?X. The purpose of this paper is to present similar results for rings and categorical rings.