Talk:Product (category theory)

Definition
In the definition we say that space is a product if there exists projections (morphisms onto the factors) that... I'm not sure that is entirely the right way to phrase it. I'm quite sure that the morphisms onto the factors are part of the data. The correct definition of a product would be a space equipped with morphisms onto the factors such that... — Preceding unsigned comment added by 66.90.153.47 (talk) 14:27, 19 July 2013 (UTC)

Maybe the 2-element case should go into Discussion? Which should be a bit more articulated, I think. Adandrews 21 Apr 2005

Examples
I think the "examples" section is misleading, since it lists things like "in Category C the product is...". It gives the impression that the product is unique, when it should be explicit that it's unique up to isomorphism. Also, I think a negative example should be provided; for example, in the category of fields there ain't no such thing as a product, for example, there's no product of GF(2) and GF(3). Albmont (talk) 19:26, 30 September 2009 (UTC)


 * I agree that uniqueness seems to be handled incorrectly. Though I could be wrong. See below. 178.38.119.178 (talk) 20:39, 4 May 2015 (UTC)


 * A product isn't just the object, but also the two projections. — Preceding unsigned comment added by 190.2.103.165 (talk) 12:32, 9 November 2019 (UTC)

Product functor
The paragraph in "Discussion" about product functor is obscure. The gentle example of a product bifunctor $$C\times C \to C$$ is omitted. No description how a product functor acts on morphisms. The statement that hom-functor is continuous IMHO should go to the article Limit (category_theory). --Beroal (talk) 03:09, 13 October 2010 (UTC)

Merge with Pullback (category theory)?
Shouldn't this be somehow merged with Pullback (category theory)? --Cokaban (talk) 16:46, 7 March 2011 (UTC)
 * No. You can use a pullback as the diagram in a limit, but for the product, here, the diagram has no morpisms. That is, the pullback has a morphism between the objects in the index set.  The product has no morphisms between the objects in the index. If you applied the notion of forgetfull-ness to the pullback, you'd get the product.  Subtle but important difference. linas (talk) 14:06, 13 August 2012 (UTC)

Finite product diagram
Is there any particular reason that the diagram for the finite product does not follow the same layout as the binary product? I think the way it is now makes it more confusing at first glance. It would be relatively simple to fix: swap the X and the Y and then rotate the graph around the vertical axis. —Preceding unsigned comment added by 86.173.27.82 (talk) 18:43, 1 April 2011 (UTC)

Associativity
Why is nothing said about associativity of direct product? Please add it to the article. --VictorPorton (talk) 17:47, 8 May 2012 (UTC) (Oh, I see this at the end of the "Discussion" section. I take my words back. Well, we should have the word associativ* here, at least for search engines.
 * Done linas (talk) 14:13, 13 August 2012 (UTC)

Uniqueness of products — does the theory really develop in the sequence given in the article?
Towards the end of the article:


 * ''If $$I$$ is a set such that all products for families indexed with $$I$$ exist, then it is possible to choose the products in a compatible fashion so that the product turns into a functor $${\mathcal C}^I \to \mathcal C$$.

When can we speak of "the" product? Isn't it only after we define this product functor? Before we select a specific product functor, saying that a category has "products" does not yet give us the right to use the notation A × B, correct?

Yet earlier in the article we find the sentence:


 * ''An object $$X$$ is a product of $$X_1$$ and $$X_2$$, denoted $$X_1 \times X_2$$, iff it satisfies this universal property:

But the universal property only determines the product up to unique isomorphism, correct? So the use of the notation X1 × X2 is premature here, isn't it?


 * ''Above we defined the binary product.

This also seems to be jumping the gun. We've only defined what a binary product is, not picked out a specific one for each pair of factors.

The distinction between "having products" and "equipped with a product" is like the distinction between "orientable" and "oriented". (Though it may make less difference. But it would take a theorem to establish this.) — Preceding unsigned comment added by 178.38.119.178 (talk) 20:54, 4 May 2015 (UTC)

Here is why I think this is important: category theory is not engineering mathematics! (With all due respect to makers and doers.)

We come to category theory to get exact answers to questions about the nature of the information we have at our disposal (or permit ourselves to have). This includes crisp answers to such questions as "the identity of indiscernables" (or non-identity, depending on the precise formulation of the problem at hand).

178.38.119.178 (talk) 20:44, 4 May 2015 (UTC)

Equational definition
The "Equational definition" subsection seems to be giving the same definition of product in a more confusing language, so I propose removing it. Ebony Jackson (talk) 23:24, 6 December 2020 (UTC)