For $A$, $B$ and $C$ abelian groups and $A \times B$ the cartesian product group, a bilinear map
from $A$ and $B$ to $C$ is a function of the underlying sets (that is, a binary function from $A$ and $B$ to $C$) which is a linear map – that is a group homomorphism – in each argument separately.
In terms of elements this means that a bilinear map $f : A \times B \to C$ is a function of sets that satisfies for all elements $a_1, a_2 \in A$ and $b_1, b_2 \in B$ the two relations
and
Notice that this is not a group homomorphism out of the direct product group. The product group $A \times B$ is the group whose elements are pairs $(a,b)$ with $a \in A$ and $b \in B$, and whose group operation is
hence satisfies
and hence in particular
which is (in general) different from the behaviour of a bilinear map.
The definition of tensor product of abelian groups is precisely such that the following is an equivalent definition of bilinear map:
For $A, B, C \in Ab$ a function of sets $f : A \times B \to C$ is a bilinear map from $A$ and $B$ to $C$ precisely if it factors through the tensor product of abelian groups $A \otimes B$ as
The analogous defintion for more than two arguments yields multilinear maps. There is a multicategory of abelian groups and multilinear maps between them; the bilinear maps are the binary morphisms, and the multilinear maps are the multimorphisms.
More generally :
For $R$ a ring (or rig) and $A, B, C \in R$Mod being modules (say on the left, but on the right works similarly) over $R$, a bilinear map from $A$ and $B$ to $C$ is a function of the underlying sets
which is a bilinear map of the underlying abelian groups as in def. 1 and in addition such that for all $r \in R$ we have
and
As before, this is equivalent to $f$ factoring through the tensor product of modules
Multilinear maps are again a generalisation.
See at tensor product of ∞-modules
binary function, bilinear map, multilinear map
In the context of higher algebra/(∞,1)-category theory bilinear maps in an (∞,1)-category are discussed in section 4.3.4 of