For $C$ a small category, and $PSh(C)$ its presheaf topos, we have (by the discussion at Profunctor – In terms of colimit preserving functors on presheaf categories) that a cocontinuous, i.e., colimit-preserving, functor $PSh(C) \to Set$ is equivalently itself a copresheaf on $C$:
If we replace in this statement presheaves with sheaves, we obtain the notion of cosheaf on $C$:
Let $C$ be a site. A cosheaf on $C$ is a copresheaf
such that it takes covers to colimits: for each covering family $\{U_i \to U\}$ in $C$ we have
Write $CoSh(C) \subset CoPSh(C)$ for the full subcategory of cosheaves.
There is a natural equivalence of categories
where on the left we have the category of cosheaves from def. 1 and on the right we have the category of colimit-preserving functors on the sheaf topos of $C$.
Equivalently: a copresheaf is a cosheaf precisely if its Yoneda extension $PSh(C) \to Set$ factors through the sheafification functor $PSh(C) \to Sh(C)$.
This is (Bunge-Funk 06, prop. 1.4.3).
Cosheaves of algebras, or notions similar to this, appear in AQFT as local nets of observables. Similar structures in higher category theory are factorization algebras, factorization homology, and topological chiral homology. Notably the definition of factorization algebra typically explicitly involves the notion of cosheaf.
G. E. Bredon, Sheaf Theory , Springer Heidelberg 1997. (chap. V-VI)
Marta Bunge and Jonathon Funk, Singular coverings of toposes , Lecture Notes in Mathematics vol. 1890 Springer Heidelberg (2006). (sec. 1.4)
Jonathon Funk, The Display Locale of a Cosheaf , Cah. Top. Géom. Diff. Cat. 36 (1995) pp.53-93.
Justin M. Curry, Abstract existence of cosheafification (pdf)
Andrei V. Prasolov, Cosheafification (arXiv:1605.01555)