Talk:Product (category theory)

message: 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)
thread_title:  Definition 

message: 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)
replies: text: 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)

text: 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)
thread_title:  Examples 

message: 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)#Preservation_of_limits. --Beroal (talk) 03:09, 13 October 2010 (UTC)
thread_title:  Product functor 

message: Shouldn't this be somehow merged with Pullback (category theory)? --Cokaban (talk) 16:46, 7 March 2011 (UTC)
replies: text: 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)
thread_title:  Merge with [[Pullback (category theory)]]? 

message: 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)
thread_title:  Finite product diagram 

message: Why is nothing said about associativity of direct product? Please add it to the article. --VictorPorton (talk) 17:47, 8 May 2012 (UTC)
replies: text: Done linas (talk) 14:13, 13 August 2012 (UTC)
thread_title:  Associativity 

message: 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)
thread_title:  Uniqueness of products — does the theory really develop in the sequence given in the article? 

message: 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)
thread_title:  Equational definition