nLab
free groupoid

Context

Category theory

Homotopy theory

Contents

Idea

The free groupoid on a directed graph is the groupoid whose objects are the vertices of the graph and whose morphisms are finite concatenations of the edges in the graph and formal inverses to them.

This construction is the left adjoint free construction to the forgetful functor that sends a groupoid to its underlying directed graph.

Definition

Given a graph DD, that is, a collection of vertices and of labeled arrows between them, the free groupoid G(D)G(D) on DD is the groupoid that has the vertices of DD as objects, and whose morphisms are constructed recursively by formal composition (i.e., juxtaposition) from identity maps, the arrows of DD and formal inverses for the arrows of DD.

The only relations between morphisms of G(D)G(D) are the necessary ones defining the identity of each object, the inverse of each arrow in DD and the associativity of composition. This is clearly a groupoid, which comes with an evident morphism DG(D)D \to G(D) of quivers.

The above sketched construction could be made more precise, but what really matters is the universal property it enjoys: the free groupoid G(D)G(D) is the universal (initial) groupoid mapping out of DD. By varying DD, the free groupoid yields a functor GG from directed graphs to groupoids, left adjoint to the forgetful functor.

This last conceptual characterization is best taken as the definition. Similarly, it is possible to construct the left adjoint to the forgetful functor from groupoids to categories, that is the free groupoid over a category.

Properties

Fundamental group

Proposition

The fundamental group of a free groupoid on a countable directed graph (for any basepoint) is a free group.

For instance (Cote, theorem 2.3).

Example

The fundamental group of the free groupoid of a graph with a single vertect is the free group on the set of edges of the graph.

References

  • Lauren Cote, Free groups and graphs: the Hanna Neumann theorem (pdf)

Revised on October 9, 2012 01:18:40 by Urs Schreiber (82.169.65.155)