## § Sheaves in geometry and logic 1.2: Pullbacks

• Pullbacks are fiber bundles.
• Pullbacks for presheaves are constructed pointwise.
• The pullback of $f$ along itself in set is going to be the set of $(x, y)$ such that $f(x) = f(y)$.
• The pullback of $f: X \to Y$ along itself in an arbitrary category is an object $P$ together parallel pair of arrows P -k,k'-> X called the kernel pair.
• $f$ is monic iff both arrows in the kernel pair are identity X -> X.
• Thus, any functor preserving pullbacks preserves monics, (because it preserves pullback squares, it sends the kernel pair with both arrows identity to another kernel pair with both arrows identity. This means that the image of the arrow is again a monic).
• The pullback of a monic along any arrow is monic.
• The pullback of an epi along any arrow is epi in set, but not necessarily always!