nLab Ab

Idea

$Ab$ 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 $Ab$ equivalently as the category of modules over $\mathbb{Z}$

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

The category $Ab$ 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 $Ab$ 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 $\mathbb{S}$.

Properties

Free abelian groups

Remark

The category $Ab$ is a concrete category, the forgetful functor

$$U : Ab \to Set$$

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

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

This functor has a left adjoint $F : Set \to Ab$ 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 $A, B \in Ab$ 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

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

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

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

$$\oplus_{i \in I} A_i \simeq \prod_i A_{i} \,.$$

But for $I \in Set$ 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

$$\oplus_{i \in I} A_i \hookrightarrow \prod_i A_i \,.$$

Example

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

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

Example

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

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

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 product of abelian groups is the additive group of integers:

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

Proposition

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

$$A \otimes (\oplus_{i \in I} B_i) \simeq \oplus_{i \in I} A \otimes B_i \,.$$

Example

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

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

Symmetric monoidal and bimonoidal structure

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

Proposition

The category $Ab$ becomes a monoidal category

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

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

Indeed with both structures combined we have

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

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

It’s also easy to see that under direct sum or tensor product, Ab can be turned into a symmetric monoidal category by equipping it with the appropriate braiding map. For example, under $\oplus$, the braiding is $\sigma_{A, B}(a, b) = (b, a)$.

Remark

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

Remark

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

Pointed objects

$Ab$ is a monoidal category with tensor unit $\mathbb{Z}$, so the pointed objects in $Ab$ are the objects $A$ with a group homomorphism $\mathbb{Z} \to A$.

Closed monoidal structure

Abelian groups are equivalently $\mathbb{Z}$-modules. Because the category of $R$-modules $Mod_R$ is closed monoidal for all commutative rings $R$, $Ab = Mod_{\mathbb{Z}}$ is also closed monoidal.

Natural numbers object

The natural numbers object in $Ab$ is the free abelian group $\mathbb{Z}[\mathbb{N}] = \bigoplus_{n \in \mathbb{N}} \mathbb{Z}$ on the natural numbers, and comes with group homomorphisms $z_0:\mathbb{Z} \to \mathbb{Z}[\mathbb{N}]$ and $z_s:\mathbb{Z}[\mathbb{N}] \to \mathbb{Z}[\mathbb{N}]$ such that for all abelian groups $A$ and group homomorphisms $f:\mathbb{Z} \to A$, $g: A \to A$, there is a unique group homomorphism $\phi_{f, g}:\mathbb{Z}[\mathbb{N}] \to A$ making the following diagram commute:

$$\array{ \mathbb{Z} & \stackrel{z_0}{\to} & \mathbb{Z}[\mathbb{N}] & \stackrel{z_s}{\leftarrow} & \mathbb{Z}[\mathbb{N}] \\ & \mathllap{f} \searrow & \downarrow \mathrlap{\phi_{f, g}} & & \downarrow \mathrlap{\phi_{f, g}} \\ & & A & \underset{g}{\leftarrow} & A }$$

Abelian groups are $\mathbb{Z}$-modules, so the free $\mathbb{Z}$-module $\mathbb{Z}[\mathbb{N}]$ has a function $v:\mathbb{N} \to \mathbb{Z}[\mathbb{N}]$ representing the basis of $\mathbb{Z}[\mathbb{N}]$; it has the property that for all integers $m \in \mathbb{Z}$, $m \cdot v(0) = z_0(m)$ and for all $n \in \mathbb{N}$, $m \cdot v(s(n)) = z_s(m \cdot v(n))$, where $m \cdot v$ is the scalar multiplication of an element $v$ by an integer $m$ in a $\mathbb{Z}$-module.

The ring structure on $\mathbb{Z}[\mathbb{N}]$ is defined by double induction on $\mathbb{Z}[\mathbb{N}]$, we define

$$(-)(-):\mathbb{Z}[\mathbb{N}] \times \mathbb{Z}[\mathbb{N}] \to \mathbb{Z}[\mathbb{N}] \otimes \mathbb{Z}[\mathbb{N}] \to \mathbb{Z}[\mathbb{N}]$$

by

$$z_0(m)z_0(n) = z_0(m \cdot n) \qquad z_s(v)z_0(n) = z_s(v z_0(n))$$

$$z_0(m)z_s(w) = z_s(z_0(m) w) \qquad z_s(v)z_s(w) = z_s(z_s(vw))$$

for all $m, n \in \mathbb{Z}$ and $v, w \in \mathbb{Z}[\mathbb{N}]$ (recall the definition of addition in the natural numbers, inductively defined by $0(p) + 0(q) = 0(p \cdot q)$, $s(m) + 0(p) = s(m + 0(p))$, $0(p) + s(n) = s(0(p) + n)$, and $s(m) + s(n) = s(s(m + n))$ for all $p, q \in \mathbb{1}$ and $m, n \in \mathbb{N}$). It is a commutative ring and represents multiplication in the polynomial ring $\mathbb{Z}[X]$; the group homomorphism $z_0$ represents the function which takes integers to constant polynomials, and $z_s$ represents the function which takes a polynomial and multiplies it by the indeterminate $X$.

Enrichment over $Ab$

Categories enriched over $Ab$ 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

• CMon

• Mod

• sAb

category: category