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Preimage

Given sets \(X,Y\), a function \(f:X\to Y\) and a subset \(U\subseteq Y\), the preimage of \(U\) under \(f\), written \(f^{-1}(U)\), is the subset of all elements of \(X\) whose image under \(f\) lies in \(U\), i.e. \(f^{-1}(U)=\{x\in X|f(x)\in U\}\). If \(U=\{y\}\) is a singleton, and there is no risk of confusion with the inverse to \(f\), one writes \(f^{-1}(y)\) for \(f^{-1}(\{y\})\). The preimage is an example of a pullback, corresponding to the limit of the diagram \(f:X\rightarrow Y\leftarrow U:\iota\), where \(\iota:U\to Y\) is simply its inclusion, \(y\mapsto y\).