Categorical pull-backs and pasting laws

Question

Let $\cal C$ be a category with pull-backs. Is it always true that given two objects $X$,$Y\in \cal C$ then $X\times_{Y} Y\cong X$?

My guess is that we have to require at least that $\cal {C}$$(X,Y)$ is non-empty.

The second question is the following: if this holds true, may I always naively insert or cancel the term $\times_Y Y$ in a pull-back in wich X appears or should I take care of the existence of other maps?

Is what I have written so far true also for 2-categorical pull-backs?

Answer 1

Pullbacks aren't an operation on objects; they are an operation on diagrams. While we often write $X \times_Z Y$ for pullbacks of the form $$\begin{CD} X \times_Z Y @>p_1 >> X \\ @VV p_0 V @VVg V \\ Y @> f>> Z \end{CD} $$ strictly speaking it is an abuse of notation to do so, for two reasons:

• The pullback depends on $f$ and $g$, so it's somewhat of a lie to use notation that ignores them
• The pullback isn't just an object; the maps $p_0$ and $p_1$ are part of the data of a pullback as well

There is a third issue, in that we speak of "the" pullback, but in the usual way it's only well-defined up to isomorphism.

Anyways, the pullback of an isomorphism is an isomorphism; i.e. in the diagram above, if $g$ is an isomorphism, then so is $p_0$.