Ab in nLab

Contents

Contents

Idea

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 ℤ\mathbb{Z} of integers, and for many purposes it is useful to think of AbAb equivalently as the category of modules over ℤ\mathbb{Z}

Ab≃ℤMod. 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 ℤ\mathbb{Z} is the played by the sphere spectrum 𝕊\mathbb{S}.

Properties

Free abelian groups

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.

See at tensor product of abelian groups for details.

Symmetric monoidal and bimonoidal structure

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

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).

Pointed objects

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

Closed monoidal structure

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

Natural numbers object

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

ℤ →z 0 ℤ[ℕ] ←z s ℤ[ℕ] 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 ℤ\mathbb{Z}-modules, so the free ℤ\mathbb{Z}-module ℤ[ℕ]\mathbb{Z}[\mathbb{N}] has a function v:ℕ→ℤ[ℕ]v:\mathbb{N} \to \mathbb{Z}[\mathbb{N}] representing the basis of ℤ[ℕ]\mathbb{Z}[\mathbb{N}]; it has the property that for all integers m∈ℤm \in \mathbb{Z}, m⋅v(0)=z 0(m)m \cdot v(0) = z_0(m) and for all n∈ℕn \in \mathbb{N}, m⋅v(s(n))=z s(m⋅v(n))m \cdot v(s(n)) = z_s(m \cdot v(n)), where m⋅vm \cdot v is the scalar multiplication of an element vv by an integer mm in a ℤ\mathbb{Z}-module.

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

(−)(−):ℤ[ℕ]×ℤ[ℕ]→ℤ[ℕ]⊗ℤ[ℕ]→ℤ[ℕ](-)(-):\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⋅n)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,n∈ℤm, n \in \mathbb{Z} and v,w∈ℤ[ℕ]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⋅q)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,q∈𝟙p, q \in \mathbb{1} and m,n∈ℕm, n \in \mathbb{N}). It is a commutative ring and represents multiplication in the polynomial ring ℤ[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.

• CMon

• Mod

• sAb