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:
- 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 $Z$?
- Is a pullback in (Mfd) always a pullback in (Top)?
- Does the forgetful functor (Mfd)$\to$(Top) preserve all small (or finite) limits?
- 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.
