Left Kan extension of a functor from a poset to a cocomplete category along the down-set embedding yields a cosheaf.
Dihomology III. A generalization of the Poincar´ e duality for manifolds
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Functors on Posets Left Kan Extend to Cosheaves: an Erratum
Left Kan extension of a functor from a poset to a cocomplete category along the down-set embedding yields a cosheaf.