nLab Ab



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 chosen 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 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 \,.

Symmetric 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).

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 σ A,B(a,b)=(b,a)\sigma_{A, B}(a, b) = (b, a).


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).

Pointed objects

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

Closed monoidal structure

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

Natural numbers object

The natural numbers object in AbAb is the free abelian group []= n\mathbb{Z}[\mathbb{N}] = \bigoplus_{n \in \mathbb{N}} \mathbb{Z} on the natural numbers, and comes with group homomorphisms z 0:[]z_0:\mathbb{Z} \to \mathbb{Z}[\mathbb{N}] and z s:[][]z_s:\mathbb{Z}[\mathbb{N}] \to \mathbb{Z}[\mathbb{N}] such that for all abelian groups AA and group homomorphisms f:Af:\mathbb{Z} \to A, g:AAg: A \to A, there is a unique group homomorphism ϕ f,g:[]A\phi_{f, g}:\mathbb{Z}[\mathbb{N}] \to A making the following diagram commute:

z 0 [] z s [] f ϕ f,g ϕ f,g A g A\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:[]v:\mathbb{N} \to \mathbb{Z}[\mathbb{N}] representing the basis of []\mathbb{Z}[\mathbb{N}]; it has the property that for all integers mm \in \mathbb{Z}, mv(0)=z 0(m)m \cdot v(0) = z_0(m) and for all nn \in \mathbb{N}, mv(s(n))=z s(mv(n))m \cdot v(s(n)) = z_s(m \cdot v(n)), where mvm \cdot v is the scalar multiplication of an element vv by an integer mm 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}]


z 0(m)z 0(n)=z 0(mn)z s(v)z 0(n)=z s(vz 0(n))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)z s(v)z s(w)=z s(z s(vw))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,nm, n \in \mathbb{Z} and v,w[]v, w \in \mathbb{Z}[\mathbb{N}] (recall the definition of addition in the natural numbers, inductively defined by 0(p)+0(q)=0(pq)0(p) + 0(q) = 0(p \cdot q), s(m)+0(p)=s(m+0(p))s(m) + 0(p) = s(m + 0(p)), 0(p)+s(n)=s(0(p)+n)0(p) + s(n) = s(0(p) + n), and s(m)+s(n)=s(s(m+n))s(m) + s(n) = s(s(m + n)) for all p,q𝟙p, q \in \mathbb{1} and m,nm, n \in \mathbb{N}). It is a commutative ring and represents multiplication in the polynomial ring [X]\mathbb{Z}[X]; the group homomorphism z 0z_0 represents the function which takes integers to constant polynomials, and z sz_s represents the function which takes a polynomial and multiplies it by the indeterminate XX.

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

Last revised on August 29, 2023 at 00:43:51. See the history of this page for a list of all contributions to it.