Definition talk:Pullback (Category Theory)

$\begin{xy}\xymatrix{

P
 \ar[r]^*+{p_1}
 \ar[d]_*+{p_2}

&

A
 \ar[d]^*+{f}

\\

B
 \ar[r]_*+{g}

&

C

}\end{xy}$

In this diagram:

• Let $A$ be the set of open intervals in $\R$
• Let $B$ be the set of open balls in $\R^2$
• Let $f$ return the size of an open interval.
• Let $g$ return the size of an open ball.

Given a choice of $P$ can this diagram commute in an interesting way? --Jshflynn (talk) 22:40, 6 November 2012 (UTC)

The pullback $P$ in $\mathbf{Set}$ of $A,B,C$ as indicated is given by $\{(x,y) \in A \times B: fx = gy\}$ with $p_1,p_2$ projections. This is the origin of the notation $A \times_C B$. In the present case, it comes down to a ball of radius $r$ conjoined with an interval of length $\pi r^2$. You can thus restrict attention to arrows into $P$, with $P$ so defined. --Lord_Farin (talk) 22:45, 6 November 2012 (UTC)

@Linus44: the indication in the corner of the diagram is one I recently saw for the first time. I'll try to find a way some day. --Lord_Farin (talk) 22:20, 8 November 2012 (UTC)