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$Ab$ 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 $\mathbb{Z}$
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 $\mathbb{S}$.
The category $Ab$ is a concrete category, the forgetful functor
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
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 $\mathbb{Z}$.
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 $A, B \in Ab$ 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
This is at the same time the direct sum $A \oplus B$.
Similarly for $I \in$FinSet$\hookrightarrow$ Set a finite set, we have
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
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
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 $I$:
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.
The unit for the tensor product of abelian groups is the additive group of integers:
The tensor product of abelian groups distributes over arbitrary direct sums:
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 $A$:
With the definitions and properties discussed above in Direct sum, etc. we have the following
The category $Ab$ becomes a monoidal category
under direct sum $(Ab, \oplus, 0)$;
under tensor product of abelian groups $(Ab, \otimes, \mathbb{Z})$.
Indeed with both structures combined we have
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 $\sigma_{A, B}(a, b) = (b, a)$.
A monoid internal to $(Ab, \otimes, \mathbb{Z})$ is equivalently a ring.
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).
$Ab$ is a monoidal category with tensor unit $\mathbb{Z}$, so the pointed objects in $Ab$ are the objects $A$ with a group homomorphism $\mathbb{Z} \to A$.
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.
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:
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
by
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 $X$.
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.
Last revised on August 29, 2023 at 00:43:51. See the history of this page for a list of all contributions to it.