Contents

group theory

# Contents

## Idea

The free group on a given set $S$ is the free object on $S$ in the category of groups. The elements of $S$ are called the generators of this group.

## Definition

### In type theory

In dependent type theory, the free group on a type $A$ is the 0-truncation of the underlying type of free infinity-group of $A$, $\mathrm{FreeGroup}(A) \coloneqq [\mathrm{UTFIG}(A)]_0$. This is because in any dependent type theory with uniqueness of identity proofs, where every type is 0-truncated, the underlying type of the free infinity-group of $A$ is the free group of $A$.

In addition, one could construct the free group directly from the traditional axioms of a group, as detailed below:

$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{FreeGroup}(A) \; \mathrm{type}}$

$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \eta:A \to \mathrm{FreeGroup}(A) \; \mathrm{type}} \quad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mu:\mathrm{FreeGroup}(A) \times \mathrm{FreeGroup}(A) \to \mathrm{FreeGroup}(A)}$
$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \epsilon:\mathrm{FreeGroup}(A)} \quad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \iota:\mathrm{FreeGroup}(A) \to \mathrm{FreeGroup}(A)}$
$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash x:\mathrm{FreeGroup}(A) \quad \Gamma \vdash y:\mathrm{FreeGroup}(A) \quad \Gamma \vdash z:\mathrm{FreeGroup}(A)}{\Gamma \vdash \alpha(x, y, z):\mathrm{Id}_{\mathrm{FreeGroup}(A)}(\mu(x, \mu(y, z)),\mu(\mu(x, y), z))}$
$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash x:\mathrm{FreeGroup}(A)}{\Gamma \vdash \lambda_\epsilon(x):\mathrm{Id}_{\mathrm{FreeGroup}(A)}(\mu(\epsilon, x), x)} \quad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash x:\mathrm{FreeGroup}(A)}{\Gamma \vdash \rho_\epsilon(x):\mathrm{Id}_{\mathrm{FreeGroup}(A)}(\mu(x, \epsilon), x)}$
$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash x:\mathrm{FreeGroup}(A)}{\Gamma \vdash \lambda_\iota(x):\mathrm{Id}_{\mathrm{FreeGroup}(A)}\mu(\iota(x), x), \epsilon)} \quad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash x:\mathrm{FreeGroup}(A)}{\Gamma\vdash \rho_\iota(x):\mathrm{Id}_{\mathrm{FreeGroup}(A)}(\mu(x, \iota(x)), \epsilon)}$
$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \tau:\mathrm{isSet}(\mathrm{FreeGroup}(A))}$

## References

Early discussion of (subgroups of) free groups (proof of the Nielsen-Schreier theorem) and generalization to the amalgamated free product of groups:

Expository review:

• Abhay Chandel, Free groups and amalgamated product, BSc thesis (2013) [pdf, pdf]

Discussion of free groups in homotopy type theory:

Last revised on May 6, 2023 at 12:34:14. See the history of this page for a list of all contributions to it.