nLab
cartesian product

Idea

In the strict sense of the word, a cartesian product is a product in Set, the category of sets.

More generally, one says cartesian product to mean the product in any cartesian monoidal category and to distinguish it from other tensor products that this may carry.

For instance one speaks of the cartesian product on Cat and on 2Cat in contrast to the Gray tensor product.

Definition

Given any family $(A_i{)}_{i:I}$ of sets, the cartesian product ${\prod }_i{A}_i$ of the family is the set of all functions $f$ from the index set $I$ with $f_j$ in $A_j$ for each $j$ in $I$.

As stated, the target of such a function depends on the argument, which is natural in dependent type theory; but if you don’t like this, then define ${\prod }_i{A}_i$ to be the set of those functions $f$ from $I$ to the disjoint union ${⨄}_i{A}_i$ such that $f_j\in A_j$ (treating $A_j$ as a subset of ${⨄}_i{A}_i$ as usual) for each $j$ in $I$.

In traditional forms of set theory, one can also take the target of $f$ to be the union ${\bigcup }_i{A}_i$ or even the class of all objects (equivalently, leave it unspecified).

Special cases

Given sets $A$ and $B$, the cartesian product of the binary family $(A,B)$ is written $A\times B$; its elements $(a,b)$ are called ordered pairs. (In set theory, one often makes a special definition for this case, defining

$$(a,b)=\{\{a\},\{a,b\}\}$$(a,b) = \{\{a\},\{a,b\}\}

rather than as a function so that ordered pairs can then be used in the definition of function. From a structural perspective, however, this is unnecessary.)

Given sets $A_1$ through $A_n$, the cartesian product of the $n$-ary family $(A_1,\dots ,A_n)$ is written ${\prod }_{i=1}^n{A}_i$; its elements $(a_1,\dots ,a_n)$ are called ordered $n$-tuples.

Given sets $A_1$, $A_2$, etc, the cartesian product of the countably infinitary family $(A_1,A_2,\dots )$ is written ${\prod }_{i=1}^{\mathrm{\infty }}{A}_i$; its elements $(a_1,a_2,\dots ,)$ are called infinite sequences.

Given a set $A$, the cartesian product of the unary family $(A)$ may be identified with $A$ itself; that is, we identify the ordered singleton $(a)$ with $a$.

The cartesian product of the empty family $()$ is the point, a set whose only element is the empty list $()$; we often call this set $1$ (or $pt$, when we're Urs) and write its element as $*$.

Foundational status

In material set theory, the existence of binary cartesian products follows from the axiom of pairing and the axiom of weak replacement? (which is very weak). In structural set theory, their existence generally must be stated as an axiom: the axiom of products. See ordered pair for more details.

category: foundational axiom