nLab
cartesian product

Context

Category theory

Monoidal categories

Contents

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(A_i)_{i:I} of sets, the cartesian product iA i\prod_i A_i of the family is the set of all functions ff from the index set II with f jf_j in A jA_j for each jj in II.

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 iA i\prod_i A_i to be the set of those functions ff from II to the disjoint union iA i\biguplus_i A_i such that f jA jf_j \in A_j (treating A jA_j as a subset of iA i\biguplus_i A_i as usual) for each jj in II.

In traditional forms of set theory, one can also take the target of ff to be the union iA i\bigcup_i A_i or even the class of all objects (equivalently, leave it unspecified).

Special cases

Given sets AA and BB, the cartesian product of the binary family (A,B)(A,B) is written A×BA \times B; its elements (a,b)(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 1A_1 through A nA_n, the cartesian product of the nn-ary family (A 1,,A n)(A_1,\ldots,A_n) is written i=1 nA i\prod_{i=1}^n A_i; its elements (a 1,,a n)(a_1,\ldots,a_n) are called ordered nn-tuples.

Given sets A 1A_1, A 2A_2, etc, the cartesian product of the countably infinitary family (A 1,A 2,)(A_1,A_2,\ldots) is written i=1 A i\prod_{i=1}^\infty A_i; its elements (a 1,a 2,,)(a_1,a_2,\ldots,) are called infinite sequences.

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

The cartesian product of the empty family ()() is the point, a set whose only element is the empty list ()(); we often call this set 11 (or pt\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.

Revised on March 26, 2014 21:00:59 by Urs Schreiber (89.204.137.178)