nLab
cosheaf

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topos theory

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Extra stuff, structure, properties

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In higher category theory

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Contents

Idea

For CC a small category, and PSh(C)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)SetPSh(C) \to Set is equivalently itself a copresheaf on CC:

[PSh(C),Set] cocCoPSh(C).[PSh(C), Set]_{coc} \simeq CoPSh(C).

If we replace in this statement presheaves with sheaves, we obtain the notion of cosheaf on CC:

[Sh(C),Set] cocCoSh(C).[Sh(C), Set]_{coc} \simeq CoSh(C).

Definition

Definition

Let CC be a site. A cosheaf on CC is a copresheaf

F:CSetF : C \to Set

such that it takes covers to colimits: for each covering family {U iU}\{U_i \to U\} in CC we have

F(U)lim ( ijF(U i× UU j) iF(U i)) F(U) \simeq \lim_{\to} \left( \coprod_{i j} F(U_i \times_{U} U_j) \stackrel{\to}{\to} \coprod_i F(U_i) \right)

Write CoSh(C)CoPSh(C)CoSh(C) \subset CoPSh(C) for the full subcategory of cosheaves.

Proposition

Proposition

There is a natural equivalence of categories

CoSh(C)Func coc(Sh(C),Set), CoSh(C) \simeq Func_{coc}(Sh(C), Set) \,,

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 CC.

Equivalently: a copresheaf is a cosheaf precisely if its Yoneda extension PSh(C)SetPSh(C) \to Set factors through the sheafification functor PSh(C)Sh(C)PSh(C) \to Sh(C).

This is (Bunge-Funk 06, prop. 1.4.3).

Equivalence between cosheaves of sets and complete spreads

In complete analogy to the equivalence of categories between sheaves of sets on a locale XX and etale maps LXL\to X there is an equivalence of categories between cosheaves of sets on a locale XX and complete spreads LXL\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 XΩX\to\Omega are in bijection with overt weakly closed sublocales of XX. (Here Ω\Omega is the poset of truth values.)

Examples

In AQFT and higher AQFT

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.

References

\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.

Last revised on April 19, 2020 at 14:53:25. See the history of this page for a list of all contributions to it.