For a continuous map , the inverse-image functor takes a sheaf on to the sheafification of its local inverse-image presheaf on . It is left adjoint to the direct image functor. For sheaves of modules on ringed spaces, extension of scalars gives the separate pullback of a sheaf of modules.
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The inverse image functor, often denoted by \( f^{-1} \), is a concept from category theory and algebraic topology. It is a construction that relates to how functions (morphisms) between objects (like sets, topological spaces, or algebraic structures) induce relationships between their respective structures.