group theory

category theory

# Contents

## 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 $ℤ$ of integers, and for many purposes it is useful to think of $\mathrm{Ab}$ equivalently as the category of modules over $ℤ$

$\mathrm{Ab}\simeq ℤ\mathrm{Mod}\phantom{\rule{thinmathspace}{0ex}}.$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 $ℤ$ 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 $\left(+,0\right)$ of a chose element $0\in A$ and a binary, associative and 0-unital operation $+$ to its underlying set

$\left(A,+,0\right)↦A\phantom{\rule{thinmathspace}{0ex}}.$(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 $ℤ\left[S\right]$ on this set: the group of formal linear combinations of elements in $S$ with coefficients in $ℤ$.

### 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 $A×B$ is the abelian group whose elements are pairs $\left(a,b\right)$ with $a\in A$ and $b\in B$, whose 0-element is $\left(0,0\right)$ and whose addition operation is the componentwise addition

$\left({a}_{1},{b}_{1}\right)+\left({a}_{2},{b}_{2}\right)=\left({a}_{1}+{a}_{2},{b}_{1}+{b}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$(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$↪$ Set a finite set, we have

${\oplus }_{i\in I}{A}_{i}\simeq \prod _{i}{A}_{i}\phantom{\rule{thinmathspace}{0ex}}.$\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

${\oplus }_{i\in I}{A}_{i}↪\prod _{i}{A}_{i}\phantom{\rule{thinmathspace}{0ex}}.$\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

$A\oplus 0\simeq 0\oplus A\simeq A\phantom{\rule{thinmathspace}{0ex}}.$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 $I$:

${\oplus }_{i\in I}ℤ\simeq ℤ\left[I\right]\phantom{\rule{thinmathspace}{0ex}}.$\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 $A×B$.

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:

$A\otimes ℤ\simeq ℤ\otimes A\simeq A\phantom{\rule{thinmathspace}{0ex}}.$A \otimes \mathbb{Z} \simeq \mathbb{Z} \otimes A \simeq A \,.
###### Proposition

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

$A\otimes \left({\oplus }_{i\in I}{B}_{i}\right)\simeq {\oplus }_{i\in I}A\otimes {B}_{o}\phantom{\rule{thinmathspace}{0ex}}.$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 $A$:

${\oplus }_{i\in I}A\simeq \left({\oplus }_{i\in I}ℤ\right)\otimes A\simeq \left(ℤ\left[I\right]\right)\otimes A\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(\mathrm{Ab},\oplus ,0\right)$;

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

Indeed with both structures combined we have

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

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

###### Remark

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

###### Remark

A monoid in $\left(\mathrm{Ab},\oplus ,0\right)$ 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.

category: categories

Revised on May 5, 2013 20:52:49 by Urs Schreiber (150.212.92.41)