# Talk:Pullback (category theory)

"Thus, the pullback can be thought of as the ordinary (Cartesian) product, but with additional structure." This sentence seems plain wrong. The pullback is generally not set-isomorphic to the cartesian product, except if f and g are both constant maps that map to the same point in Z. Maybe what's meant is that pullback is isomorphic to some SUBSET of X x Y? The article does not mention important special cases, like Z = Y and g = id_Y, which is what an SQL programmers may call an (inner) "join of X and Y on the foreign key f". — Preceding unsigned comment added by 84.132.133.4 (talk) 22:03, 16 January 2014 (UTC)

## Ahhh

Hah, I wish I still understood this. Ahhh. --SPUI (talk) 03:37, 3 Mar 2005 (UTC)

## Picture relevance

I'm not sure what the picture is supposed to illustrate (and I partially understand pullbacks), and it is not mentioned/explained in the text. I move to have it stricken? 129.107.75.207 (talk) 00:59, 31 March 2008 (UTC)

The picture under the "Universal Property" heading? That picture says it all! In category theory, many concepts are easier to introduce by supplying a commutative diagram rather than a slough of symbols embedded in a cryptic sentence. I do not second the motion. —Preceding unsigned comment added by 68.150.218.1 (talk) 03:45, 4 October 2010 (UTC)

## Merge with Product (category theory)?

Shouldn't this be merged with product (category theory)? --Cokaban (talk) 16:48, 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.
I just added the simplest possible paragraph I could think of, explaining this, to this article. I guess I should also talk about limits. linas (talk) 14:33, 13 August 2012 (UTC)
Isn't the pullback of ${\displaystyle f:A\to C}$ and ${\displaystyle g:B\to C}$ the categorical product of ${\displaystyle f}$ and ${\displaystyle g}$ in the slice category over ${\displaystyle C}$ ? If so, this seems to be a more direct link between the notions. --Pcagne (talk) 17:36, 17 December 2012 (UTC)
I believe you are correct about the slice category connection. Additionally, a product is a pullback of a cospan over a terminal object, i.e. ${\displaystyle X\times Y}$ is (isomorphic to) the pullback of the unique maps ${\displaystyle X\to 1}$ and ${\displaystyle Y\to 1}$; I have added this to the article. Cyrapas (talk) 16:18, 26 December 2013 (UTC)