nLab pointed abelian group

Definition

Definition

A pointed abelian group is an abelian group $(A, +, 0, -)$ equipped with the choice of an element $1 \in A$. A homomorphism of pointed abelian groups is a homomorphism of underlying groups which preserves the choice of the extra points.

Remark

Pointed abelian groups are “pointed” in the sense of pointed objects in a monoidal category, in that the choice of an element $1 \in A$ is equivalently the choice of a homomorphism $\mathbb{Z} \longrightarrow A$ out of the additive group of integers (which is fixed by its image of $1 \in \mathbb{Z}$).

Therefore the category of pointed abelian groups is equivalently the coslice category $Ab^{\mathbb{Z}/}$ of Ab under $\mathbb{Z}$.

Remark

The additive neutral element $0 \in A$ is not actually needed in the definition of apointed abelian groups (Def. ): Pointed abelian groups could equally be defined as pointed commutative invertible semigroups, hence as commutative semigroups $(A, +)$ equipped with an element $1 \in A$ and a function $- \colon A \to A$ such that for all elements $a \in A$ and $b \in A$, $a + b + (-b) = a$.

From commutativity and associativity one may derive the other three invertibility properties for an invertible semigroup: $a + (-b) + b = a$, $b + (-b) + a = a$, and $(-b) + b + a = a$. The element $1 + (-1)$ is both left unital and right unital with respect to the binary operation $+$, so by defining $0 \coloneqq 1 + (-1)$, $A$ becomes an abelian group $(A, +, 0, -)$ with an additional point $1 \in A$; hence a pointed abelian group.

Examples

Example

Every abelian group becomes a pointed abelian group (Def. ) by taking the point to be the neutral element $0$. Since the neutral element $0$ is necessarily preserved by an group homomorphism, this constitutes a full subcategory-inclusion

$$Ab \hookrightarrow Ab^{\mathbb{Z}/} \,.$$

On the other hand:

Example

The notation $1 \in A$ is motivated from the case of rings $(R,0,+,1,\cdot)$, underlying which is the pointed abelian group $(R,0,+,1)$ with the “point” being the ring’s unit element $1 \in R$.

Properties

Remark

Despite the inclusion Exp. , the category of pointed abelian groups (Def. ) is not equivalent to the category of abelian groups ($\mathrm{Ab}$):

In Ab, the initial object and terminal objects are the same, both are given by the trivial group. However, in the category of pointed abelian groups, while the terminal object is still the trivial group, the initial object is the additive group integers $\mathbb{Z}$ (manifestly so from Rem. ). In this, the category of pointed abelian groups has more in common with the categories Ring and CRing of rings and of commutative rings, respectively.

Related concepts

• pointed module

• pointed object in a monoidal category

• commutative invertible semigroup

• abelian group

• Tarski group

• lattice

References

• W. Edwin Clark, Xiang-dong Hou, Galkin Quandles, Pointed Abelian Groups, and Sequence A000712 [arXiv:1108.2215]

• A. M. Nurakunov, Quasivariety Lattices of Pointed Abelian Groups, Algebra and Logic 53 (2014) 238–257 [doi:10.1007/s10469-014-9286-5]