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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
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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
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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
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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
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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?
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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.)
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Category C the product is...". It gives the impression that the product is
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The "Equational definition" subsection seems to be giving the same definition of product in a more confusing language, so I propose removing it.
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is omitted. No description how a product functor acts on morphisms. The statement that hom-functor is continuous IMHO should go to the article
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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.
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Here is why I think this is important: category theory is not engineering mathematics! (With all due respect to makers and doers.)
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Maybe the 2-element case should go into
Discussion? Which should be a bit more articulated, I think. Adandrews 21 Apr 2005
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But the universal property only determines the product up to unique isomorphism, correct? So the use of the notation X
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The paragraph in "Discussion" about product functor is obscure. The gentle example of a product bifunctor
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Uniqueness of products — does the theory really develop in the sequence given in the article?
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I agree that uniqueness seems to be handled incorrectly. Though I could be wrong. See below.
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Why is nothing said about associativity of direct product? Please add it to the article. --
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pullback, you'd get the product. Subtle but important difference.
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A product isn't just the object, but also the two projections.
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