# Pullbacks as manifolds versus ones as topological spaces

Let $Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2$ be smooth maps with a common target. Suppose that we have a pullback $Z$ of the diagram in (Mfd).

Questions:

1. Suppose that we have a manifold $Z$ satisfying the universal property of pullbacks. Is the subset $\{(y_1,y_2) \in Y_1 \times Y_2 \mid f_1(y_1)=f_2(y_2)\}$ a submanifold of $Y_1 \times Y_2$ diffeomorphic to $Z$?
2. Is a pullback in (Mfd) always a pullback in (Top)?
3. Does the forgetful functor (Mfd)$\to$(Top) preserve all small (or finite) limits?
4. If you know a good reference to categorical properties of manifolds, please let me know.

Note:

• (Mfd) is the category of smooth manifolds and smooth maps.
• (Top) is the category of topological spaces and continuous maps.
• In my question, a pullback is a limit in the sense of the theory of categories, i.e. a pullback is a manifold/topological space satisfying the universal property of pullback.
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I slightly modified Q1. – H. Shindoh Mar 11 '13 at 18:56
Related (but not identical): math.stackexchange.com/questions/322485/… – Martin Brandenburg Mar 11 '13 at 23:24
I have asked the same question at mathoverflow.net/questions/124311/… – H. Shindoh Mar 12 '13 at 12:32