As HNuers example shows, the pushforward of a constant sheaf isn't constant anymore. However one also doesn't get arbitrary sheaves this way. The direct image is still a constructible sheaf, which means that your space is a finite disjoint union of locally closed pieces on which the sheaf is locally constant. In Nhuers For example the sheaf in NHuers example is locally constant (but not constant!) on the whole space. The pushforward of a possible choice vector space on a point to some space in the example of such decomposition K.J. Moi would be a constant on the point and its complement.

Edit: Of course one needs assumptions here. For example everything works for pushforward along morphisms of complex algebraic varieties (sheaves considered in analytic topology!).

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As HNuers example shows, the pushforward of a constant sheaf isn't constant anymore. However one also doesn't get arbitrary sheaves this way. The direct image is still a constructible sheaf, which means that your space is a finite disjoint union of locally closed pieces on which the sheaf is constant. In Nhuers example a possible choice of such decomposition would be a point and its complement.

Edit: Of course one needs assumptions here. For example everything works for pushforward along morphisms of complex algebraic varieties (sheaves considered in analytic topology!).

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As HNuers example shows, the pushforward of a constant sheaf isn't constant anymore. However one also doesn't get arbitrary sheaves this way. The direct image is still a constructible sheaf, which means that your space is a finite disjoint union of locally closed pieces on which the sheaf is constant. In Nhuers example a possible choice of such decomposition would be a point and its complement.