Group Theory

Category theory



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

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

The category AbAb 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 AbAb 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\mathbb{S}


Free abelian groups


The category AbAb is a concrete category, the forgetful functor

U:AbSet U : Ab \to Set

to Set sends a group, regarded as a set AA equipped with the structure (+,0)(+,0) of a chose element 0A0 \in A and a binary, associative and 0-unital operation ++ to its underlying set

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

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


For A,BAbA, B \in Ab two abelian groups, their direct product A×BA \times B is the abelian group whose elements are pairs (a,b)(a, b) with aAa \in A and bBb \in B, whose 0-element is (0,0)(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 ABA \oplus B.

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

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

But for ISetI \in Set a set which is not finite, there is a difference: the direct sum iIA i\oplus_{i \in I} A_i of an II-indexed family A i iI{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

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

The trivial group 0Ab0 \in Ab (the group with a single element) is a unit for the direct sum: for every abelian group we have

A00AA. A \oplus 0 \simeq 0 \oplus A \simeq A \,.

In view of remark 1 this means that the direct sum of |I|{\vert I \vert} copies of the additive group of integers with themselves is equivalently the free abelian group on II:

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

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

See at tensor product of abelian groups for details.


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

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

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

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

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

iIA( iI)A([I])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


The category AbAb becomes a monoidal category

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

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

Indeed with both structures combined we have

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

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


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


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


Enrichment over AbAb

Categories enriched over AbAb 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: category

Revised on August 21, 2015 02:59:22 by Urs Schreiber (