nLab free groupoid

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.

More generally, there is a free groupoid construction on (small) strict categories, given by freely adjoining inverse morphisms to all existing non-invertible morphisms, and the free groupoid on a directed graph is the pre-composition of that operation with the free category-construction on the graph:

$$Grpd^{strct}_{smll} \underoverset {\underset{U_{Grpd}}{\longrightarrow}} {\overset{F_{Grpd}}{\longleftarrow}} {\;\;\bot\;\;\;} Cat^{strct}_{smll} \underoverset {\underset{U_{Cat}}{\longrightarrow}} {\overset{F_{Cat}}{\longleftarrow}} {\;\;\bot\;\;\;} DiGrph$$

(Incidentally, the forgetful functor $F \,\colon\,Cat^{strct}_{smll} \longrightarrow Grpd^{strct}_{smll}$ also has a right adjoint, known as the core-construction).

Definition

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

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

The above sketched construction could be made more precise, but what really matters is the universal property it enjoys: the free groupoid $G(D)$ is the universal (initial) groupoid mapping out of $D$. By varying $D$, the free groupoid yields a functor $G$ 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.

The construction of free groupoids in “Topology and Groupoids” is by taking a disjoint union of copies of the unit interval groupoid $\mathbf I$ and then identifying the vertices according to the scheme given by the directed graph.

See the paper by Crisp and Paris for an application of free groupoids.

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 vertex is the free group on the set of edges of the graph. A result relevant to the Jordan Curve Theorem and the Phragmen-Brouwer Property is given in the Corrigendum referenced below. It gives conditions on a pushout of groupoids to contain a free groupoid as a retract.

Related concepts

• free category, core groupoid

• Nielsen-Schreier theorem

• Dwyer-Kan simplicial path groupoid

References

On the construction of free groupoids on a graph:

• Philip Higgins, §4 of: Categories and groupoids, Van Nostrand Reinhold (1971); Reprints in Theory and Applications of Categories 7 (2005) 1-195 [tac:tr7, pdf]

• Saunders MacLane, p. 51 Ex. 3 in: Categories for the Working Mathematician, Springer (1971) [doi:10.1007/978-1-4757-4721-8]

On the construction of free groupoids on a category (localization at all morphisms):

• Pierre Gabriel, Michel Zisman, §1.5.4 of: Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer (1967) [doi:10.1007/978-3-642-85844-4, pdf]

and in the generality of sSet-enriched categories and sSet-enriched groupoids (“simplicial groupoids”):

• William Dwyer, Daniel Kan, §5.5 of: Simplicial localizations of categories, J. Pure Appl. Algebra 17 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3]

• André Joyal, Myles Tierney, pp. 75 of: On the theory of path groupoids, Journal of Pure and Applied Algebra 149 1 (2000) 69-100 [doi:10.1016/S0022-4049(98)00164-9]

• Goerss & Jardine (2009), V.7

  (highlighting relation to the Milnor construction)

See also:

• Lauren Cote, Free groups and graphs: the Hanna Neumann theorem (2008) [pdf]

• Ronnie Brown, Topology and Groupoids (2006) [web]

• Omar Antolin Camarena and Ronnie Brown, “Corrigendum to ”Groupoids, the Phragmen-Brouwer Property, and the Jordan Curve Theorem“, J. Homotopy and Related Structures 1 (2006) 175-183.” J. Homotopy and Related Structures (pdf)

• J. Crisp, L. Paris, The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group, Invent. math. 145, 19–36 (2001).