Cosheafification

It is proved that for any Grothendieck site \(X\), there exists a coreflection (called \(\mathbf{cosheafification}\)) from the category of precosheaves on \(X\) with values in a category \(\mathbf{K}\), to the full subcategory of cosheaves, provided either \(\mathbf{K}\) or \(\mathbf{K}^{op}\) is locally presentable. If \(\mathbf{K}\) is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category \(\mathbf{Pro}% \left( \mathbf{K}\right) \) of pro-objects in \(\mathbf{K}\). In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in \(\mathbf{Pro}\left( \mathbf{K}\right) \) is \(\mathbf{smooth}\), i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.