Talk:Pullback (category theory)

Pullbacks preserve isomorphisms?

In the section on properties it states that pullbacks preserve isomorphisms. However, the supplied references only support that retractions are preserved under pullbacks. The consequence that is stated can be proved directly from the universal property. But I think the claim about isomorphims should be removed, or supplied with a citation or restricted to cases where it holds. 2001:638:208:FD5F:8864:6713:49B1:7E4 (talk) 14:47, 11 April 2016 (UTC)

It turns out that the result is at least true, though I don't have a reference for it. It can be proved straightforwardly. 2A02:810B:8940:CA0:B053:F303:FDDA:2044 (talk) 12:46, 17 April 2016 (UTC)

some weaknesses in this article

"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)

If Z = Y and g = id_Y, then isn't the pullback just X with the identity morphism on it? I don't understand how it could be a "join of X and Y on the foreign key f".138.38.103.61 (talk) 13:16, 23 April 2015 (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 $f:A\to C$ and $g:B\to C$ the categorical product of $f$ and $g$ in the slice category over $C$ ? 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. $X\times Y$ is (isomorphic to) the pullback of the unique maps $X\to 1$ and $Y\to 1$; I have added this to the article. Cyrapas (talk) 16:18, 26 December 2013 (UTC)

That's not a limit!

A limit is a cone. A pullback is not a cone; where is the arrow to $Z$? --Mathmensch (talk) 22:39, 19 December 2016 (UTC)