$Ab$ 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 chose 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 1 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 produc 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).
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).
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.