Contents

topos theory

# Contents

## Idea

The theory of abelian groups $T$ is the logical theory whose models in a cartesian category are the abelian group objects.

## Definition

The theory of abelian groups $T$ is the theory over the signature with one sort $X$, one constant $0$, two function symbols $+$ (of sort $X\times X\to X$) and $-$ (of sort $X\to X$) and equality with axioms:

• $\top\vdash_x x+0=x \quad$
• $\top\vdash_{x,y,z} x+(y+z)=(x+y)+z\quad$,
• $\top\vdash_x x+(-x)=0\quad$.
• $\top\vdash_{x,y} x+y=y+x\quad$.

The theory of abelian groups is algebraic. In the following we list some extensions that use increasingly stronger fragments of geometric logic.

## Extensions of the theory of abelian groups

The theory of torsion-free abelian groups $T^\infty$ results from $T$ by addition of the following axioms:

• For all $n\geq 2$: $((nx=0))\vdash_x (x=0))\quad$.

The resulting theory is a Horn theory.

The theory of divisible abelian groups $T^\backslash$ results from $T$ by addition of the following regular axioms:

• For all $n\geq 2$: $\top\vdash_x (\exists y) (ny=x)\quad$.

The theory of divisible torsion-free abelian groups $T^{\backslash\infty}$ results from $T^\backslash$ by adding the above axioms for torsion-freeness. The resulting theory is cartesian.

The theory of torsion abelian groups $T^~$ is an infinitary geometric theory resulting from $T$ by addition of:

• $\top\vdash_x \bigvee_{n\geq 2} (nx=0)\quad$.