nLab
cosheaf

Idea

For $C$ a small category, and $\mathrm{PSh}(C)$ its presheaf topos, we have that a colimit-preserving functor $\mathrm{PSh}(C)\to \mathrm{Set}$ is equivalently itself a copresheaf.

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

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

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

Definition

Definition

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

$$F:C\to \mathrm{Set}$$F : C \to Set

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

$$F(U)\simeq \underrightarrow{\lim }(\coprod _{ij}F(U_i{\times }_U{U}_j)\overrightarrow{\to }\coprod _{i}F(U_i))$$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 $\mathrm{CoSh}(C)\subset \mathrm{CoPSh}(C)$ for the full subcategory of cosheaves.

Proposition

Proposition

There is a natural equivalence of categories

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

where on the right we have the category of colimit-preserving functors.

Equivalently: a copresheaf is a cosheaf precisely if its Yoneda extension $\mathrm{PSh}(C)\to \mathrm{Set}$ factors through sheafification $\mathrm{PSh}(C)\to \mathrm{Sh}(C)$.

This is (BungeFunk, prop. 1.4.3).

Related concepts

• presheaf / sheaf / cosheaf

• 2-sheaf / stack

• (∞,1)-sheaf / ∞-stack

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, topological chiral homology.

References

section 1.4 of

• Marta Bunge and Jonathan Funk, Singular coverings of toposes Lecture Notes in Mathematics, (2006) Volume 1890/2006

  chapter 1 Lawvere Distributions on Toposes