nLab cosheaf

Idea

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 $C$:

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

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

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

Definition

Definition

Let $C$ be a site. A cosheaf on $C$ is a copresheaf

$$F : C \to Set$$

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

$$F(U) \simeq \lim_{\to} \left( \coprod_{i j} F(U_i \times_{U} U_j) \stackrel{\to}{\to} \coprod_i F(U_i) \right)$$

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

Proposition

Proposition

There is a natural equivalence of categories

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

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 $C$.

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).

Equivalence between cosheaves of sets and complete spreads

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 this article: join-preserving maps $X\to\Omega$ are in bijection with overt weakly closed sublocales of $X$. (Here $\Omega$ 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.

Related concepts

• (0,1)-presheaf / (0,1)-sheaf

• presheaf / sheaf / cosheaf

• 2-sheaf / stack

• (∞,1)-sheaf / ∞-stack

• Lawvere distribution

References

• 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)

• Marta Bunge, Jonathon Funk, Constructive theory of the lower power locale, Mathematical Structures in Computer Science, vol. 6, no. 1, pp. 69-83: doi.