# A very vague question about the cartesian product in mathematics.

Motivated by this question, I am wondering about Cartesian product analogs in various subfields of mathematics.

The set theoretic Cartesian product creates an "output" set from a set of "input" sets, so that each member of the output set corresponds to the selection of one element each in every input set. There is also a standard Cartesian product of graphs and a standard Cartesian product of functions.

My question is, if I am learning a new subfield of mathematics and I see a "Cartesian product" analog, then what properties should I expect it to have? My naive intuition would be that it should change a list of objects that have sizes $n_1, n_2, \dots , n_k$ into a new object that has size $n_1 \cdot n_2 \cdots n_k$ and that to learn exactly the definition I would have to carefully read how it is defined. Is this intuition misleading or incomplete? Does the concept of "Cartesian product" have a technical meaning that is as wide in scope as it is used in various subfields, or would something like category theory be required for such a meaningful generalization?

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Take a look at the notion of "categorical product". Of course, it doesn't always work: e.g., the cartesian product of graphs is not the categorical product of graphs; as for your link to "cartesian product of functions", I don't know how standard that is. That paper does not even appear on MathReviews (via MathSciNet). – Arturo Magidin Jan 30 '12 at 18:01
If you recast category theory using a model of set theory that uses proper classes, like a conservative NBG set theory model,then the Cartesian product generalizes very straightforwardly to classes AND you get the universal categorical properties as well. So that's how I'd approach it if I had to. And I'm not allergic to proper classes as some mathematicians inexplicably are. I think if you're serious about category theory, you really have to see the inadequacy of ZFC models. – Mathemagician1234 Jan 30 '12 at 18:26
@Mathemagician1234 : I don't think that the issues that trouble the OP have anything to do with set theory. Also, almost everyone I know who uses any category theory (including myself) is perfectly content with ZFC. – Adam Smith Jan 31 '12 at 16:56

If you are still on board, think that in calculus you can think of a function of two variable as a function of the first variable with values in a 'space' of functions in the second variable (this is what is 'going on' in partial differentiation at least up to a point). This trick says $f:X\times Y\to Z$ gives a function $X\to \mathrm{Functions}(Y,Z)$ and vice versa. and this is another neat property of the product of sets. This one generalises to give a whole set of other types of product, whenever there is a reasonable notion the maps between 'things' give a 'thing' of maps. For 'thing = vector space' you find that the corresponding idea on the left is a tensor product. Now look at graphs, and loads of other settings and you will get a very full answer to your query.