nLab
Ab

Idea

$\mathrm{Ab}$ denotes the category of abelian groups: it has abelian groups as objects and group homomorphisms between these as morphisms.

The archetypical example of an abelian group is the group $\mathbb{Z}$ of integers, and for many purposes it is useful to think of $\mathrm{Ab}$ equivalently as the category of modules over $\mathbb{Z}$

$$\mathrm{Ab}\simeq \mathbb{Z}\mathrm{Mod}.$$Ab \simeq \mathbb{Z} Mod \,.

The category $\mathrm{Ab}$ serves as the basic enriching category in homological algebra. There Ab-enriched categories play much the same role as Set-enriched categories (locally small categories) play in general.

In this vein, the analog of $\mathrm{Ab}$ in homotopy theory – or rather in stable homotopy theory – is the category of spectra, either regarded as the stable homotopy category or rather refined to the stable (infinity,1)-category of spectra. A spectrum is much like an abelian group up to coherent homotopy and the role of the archetypical abelian group $\mathbb{Z}$ is the played by the sphere spectrum $𝕊$

Properties

Free abelian groups

Remark

The category $\mathrm{Ab}$ is a concrete category, the forgetful functor

$$U:\mathrm{Ab}\to \mathrm{Set}$$U : Ab \to Set

to Set sends a group, regarded as a set $A$ equipped with the structure $(+,0)$ of a chose element $0\in A$ and a binary, associative and 0-unital operation $+$ to its underlying set

$$(A,+,0)\mapsto A.$$(A, +, 0) \mapsto A \,.

This functor has a left adjoint $F:\mathrm{Set}\to \mathrm{Ab}$ which sends a set $S$ to the free abelian group $\mathbb{Z}[S]$ on this set: the group of formal linear combinations of elements in $S$ with coefficients in $\mathbb{Z}$.

Direct sum, direct product and tensor product

We discuss basic properties of binary operations on the category of abelian groups: direct product, direct sum and tensor product. Below in Monoidal and bimonoidal structure we put these structures into a more abstract context.

Proposition

For $A,B\in \mathrm{Ab}$ two abelian groups, their direct product $A\times B$ is the abelian group whose elements are pairs $(a,b)$ with $a\in A$ and $b\in B$, whose 0-element is $(0,0)$ and whose addition operation is the componentwise addition

$$(a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2).$$(a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 + b_2) \,.

This is at the same time the direct sum $A\oplus B$.

Similarly for $I\in$FinSet$\hookrightarrow$ Set a finite set, we have

$${\oplus }_{i\in I}{A}_i\simeq \prod _i{A}_i.$$\oplus_{i \in I} A_i \simeq \prod_i A_{i} \,.

But for $I\in \mathrm{Set}$ a set which is not finite, there is a difference: the direct sum ${\oplus }_{i\in I}{A}_i$ of an $I$-indexed family ${A_i}_{i\in I}$ of abelian groups is the sub-group of the direct product on those elements for which only finitely many components are non-0

$${\oplus }_{i\in I}{A}_i\hookrightarrow \prod _i{A}_i.$$\oplus_{i \in I} A_i \hookrightarrow \prod_i A_i \,.

Example

The trivial group $0\in \mathrm{Ab}$ (the group with a single element) is a unit for the direct sum: for every abelian group we have

$$A\oplus 0\simeq 0\oplus A\simeq A.$$A \oplus 0 \simeq 0 \oplus A \simeq A \,.

Example

In view of remark 1 this means that the direct sum of $\mid I\mid$ copies of the additive group of integers with themselves is equivalently the free abelian group on $I$:

$${\oplus }_{i\in I}\mathbb{Z}\simeq \mathbb{Z}[I].$$\oplus_{i \in I} \mathbb{Z} \simeq \mathbb{Z}[I] \,.

Definition

For $A$ and $B$ two abelian groups, their tensor product of abelian groups is the group $A\otimes B$ with the property that a group homomorphism $A\otimes B\to C$ is equivalently a bilinear map out of the set $A\times B$.

See at tensor product of abelian groups for details.

Example

The unit for the tensor produc of abelian groups is the additive group of integers:

$$A\otimes \mathbb{Z}\simeq \mathbb{Z}\otimes A\simeq A.$$A \otimes \mathbb{Z} \simeq \mathbb{Z} \otimes A \simeq A \,.

Proposition

The tensor product of abelian groups distributes over arbitrary direct sums:

$$A\otimes ({\oplus }_{i\in I}{B}_i)\simeq {\oplus }_{i\in I}A\otimes B_o.$$A \otimes (\oplus_{i \in I} B_i) \simeq \oplus_{i \in I} A \otimes B_o \,.

Example

For $I\in \mathrm{Set}$ and $A\in \mathrm{Ab}$, the direct sum of $\mid I\mid$ copies of $A$ with itself is equivalently the tensor product of abelian groups of the free abelian group on $I$ with $A$:

$${\oplus }_{i\in I}A\simeq ({\oplus }_{i\in I}\mathbb{Z})\otimes A\simeq (\mathbb{Z}[I])\otimes A.$$\oplus_{i \in I} A \simeq (\oplus_{i \in I} \mathbb{Z}) \otimes A \simeq (\mathbb{Z}[I]) \otimes A \,.

Monoidal and bimonoidal structure

With the definitions and properties discussed above in Direct sum, etc. we have the following

Proposition

The category $\mathrm{Ab}$ becomes a monoidal category

1. under direct sum $(\mathrm{Ab},\oplus ,0)$;

2. under tensor product of abelian groups $(\mathrm{Ab},\otimes ,\mathbb{Z})$.

Indeed with both structures combined we have

• $(\mathrm{Ab},\oplus ,\otimes ,0,\mathbb{Z})$

is a bimonoidal category (and can be made a bipermutative category).

Remark

A monoid internal to $(\mathrm{Ab},\otimes ,\mathbb{Z})$ is equivalrntly a ring.

Remark

A monoid in $(\mathrm{Ab},\oplus ,0)$ is equivalently just an abelian group again (since $\oplus$ is the coproduct in $\mathrm{Ab}$, so every object has a unique monoid structure with respect to it).

Enrichment over $\mathrm{Ab}$

Categories enriched over $\mathrm{Ab}$ are called pre-additive categories or sometimes just additive categories. If they satisfy an extra exactness condition they are called abelian categories. See at additive and abelian categories.

Related concepts

• Mod

• sAb

category: categories