# Product (Category Theory)

This si­mul­ta­neously cap­tures the con­cept of a product of sets, posets, groups, topolog­i­cal spaces etc. In ad­di­tion, like any uni­ver­sal con­struc­tion, this char­ac­ter­i­za­tion does not differ­en­ti­ate be­tween iso­mor­phic ver­sions of the product, thus al­low­ing one to ab­stract away from an ar­bi­trary, spe­cific con­struc­tion.

## Definition

Given a pair of ob­jects $$X$$ and $$Y$$ in a cat­e­gory $$\mathbb{C}$$, the product of $$X$$ and $$Y$$ is an ob­ject $$P$$ along with a pair of mor­phisms $$f: P \rightarrow X$$ and $$g: P \rightarrow Y$$ satis­fy­ing the fol­low­ing uni­ver­sal con­di­tion:

Given any other ob­ject $$W$$ and mor­phisms $$u: W \rightarrow X$$ and $$v:W \rightarrow Y$$ there is a unique mor­phism $$h: W \rightarrow P$$ such that $$fh = u$$ and $$gh = v$$.

Children:

• Universal property of the product

The product can be defined in a very gen­eral way, ap­pli­ca­ble to the nat­u­ral num­bers, to sets, to alge­braic struc­tures, and so on.

Parents:

• Category theory

How math­e­mat­i­cal ob­jects are re­lated to oth­ers in the same cat­e­gory.

• Would Product (math­e­mat­ics) be an ap­pro­pri­ate name, or does cat­e­gory the­ory’s use of the term point to only a sub­set of the things product can mean?

• @1yq Whether Product (math­e­mat­ics) is ap­pro­pri­ate re­ally de­pends if you’re ask­ing a cat­e­gory the­o­rist (who would say yes) or not . ;-)

In se­ri­ous­ness, spe­cific kinds of prod­ucts in­clude carte­sian prod­ucts, prod­ucts of alge­braic struc­tures, prod­ucts of topolog­i­cal spaces and the most well known: product of num­bers. All of these are spe­cial cases of the cat­e­gor­i­cal product (if you pick your cat­e­gory right), but I can imag­ine some­one want­ing to look up ‘product’ as in mul­ti­pli­ca­tion and get­ting hit with cat­e­gory the­ory.

I don’t know. It’s a mat­ter of taste I sup­pose. I get the idea that cat­e­gory the­ory is not yet quite widely-known enough for this to be con­sid­ered “the” defi­ni­tion by most math­e­mat­i­ci­ans, but if other con­trib­u­tors feel it should be given that sta­tus I cer­tainly won’t com­plain. I just thought this was the safer ap­proach.

See, for ex­am­ple product on Wikipe­dia.

• Yeah, I think keep­ing it as it is now is prob­a­bly the best way of fol­low­ing the “one idea per page” method­ol­ogy. The page on Prod­ucts (math­e­mat­ics) can have this page as child.