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. 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).
In complete analogy to the equivalence of categories between sheaves of sets on a locale $X$ and etale maps $L\to X$ there is an equivalence of categories between cosheaves of sets on a locale $X$ and complete spreads $L\to X$. The analog of the etale space functor is the display locale functor. The analog of the sheaf of sections? functor is the cosheaf of connected components functor.
A decategorified version of this statement was obtained by Marta Bunge and Jonathon Funk in [PLoc]: join-preserving maps $X\to\Omega$ are in bijection with overt weakly closed sublocales of $X$. (Here $\Omega$ is the poset of truth values.)
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.
Glen 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)
Justin M. Curry?, Abstract existence of cosheafification (pdf)
Jonathon Funk, The Display Locale of a Cosheaf , Cah. Top. Géom. Diff. Cat. 36 (1995) pp.53-93.
Andrei V. Prasolov, Cosheafification (arXiv:1605.01555)
Andrei V. Prasolov, Cosheaves (arXiv:1804.07988)
Steve Vickers, Cosheaves and Connectedness in Formal Topology , Ann. Pure Appl. Logic 163 no.2 (2012) pp.157-174. (preprint)
\bibitem{PLoc} Marta Bunge, Jonathon Funk, Constructive theory of the lower power locale, Mathematical Structures in Computer Science, vol. 6, no. 1, pp. 69-83: doi.
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