nLab
sheafification

Definition

Given a site $S$, every Set-valued presheaf $F$ in $\mathrm{PSh}(S):=[S^{\mathrm{op}},\mathrm{Set}]$ is locally isomorphic (weakly equivalent) to a sheaf $\overline{F}$. This construction extends to a functor

$$\overline{(-)}:\mathrm{PSh}(S)\to \mathrm{Sh}(S).$$\bar{(-)} : PSh(S) \to Sh(S) \,.

This functor is sheafification.

Sheafification is left exact left adjoint to the fully faithful forgetful subcategory functor $i:\mathrm{Sh}(S)\hookrightarrow \mathrm{PSh}(S)$. Therefore it constitutes a geometric morphism of Grothendieck topoi

$$\mathrm{PSh}(S)\stackrel{\stackrel{\overline{(-)}}{\to }}{\stackrel{i}{\leftarrow }}\mathrm{Sh}(S).$$PSh(S) \stackrel{\stackrel{\bar {(-)}}{\to}}{\stackrel{i}{\leftarrow}} Sh(S) \,.

Sheafification for more general presheaves

For presheaves with values in categories other than Set, sheafification may be a difficult problem, unless one has some extra assumptions.

Sheafification with values in models for finit-limit theories

Consider a type of structure $T$ defined in terms of finite limits (such as groups, algebras, modules, etc.), then internal $T$-models are preserved by both direct and inverse images of geometric morphisms. Therefore, the adjunction between sheaves and presheaves of sets directly induces an adjunction between $T$-models in sheaves and presheaves. And since finite limits of sheaves and presheaves are computed pointwise, $T$-models in the category of (pre)sheaves are the same as (pre)sheaves of $T$-models-in-$\mathrm{Set}$.

Sheafification using IPC-property

If a category $A$ satisfies the following assumptions, sheafification of presheaves in $[S^{\mathrm{op}},A]$ exists and is constructed analogously as for Set-valued sheaves.

• $A$ admits small limits;
• $A$ admits small colimits;
• small filtered limits in $A$ are exact;
• $A$ satisfies the IPC-property .

This is true for instance for

• the category Set of sets;

• the category Grp of groups;

• the category $k\mathrm{Alg}$ of $k$-algebras;

• the category $\mathrm{Mod}(R)$ of modules,

(but all of these are also $T$-models for finite-limit theories $T$).

One should say more: there are so many applications and fairly difficult theorems there; for example van
Osdol’s work.

Mike: Another way to think about this is: if you have a type of structure $T$ defined in terms of finite limits (such as groups, algebras, modules, etc.), then internal $T$-models are preserved by both direct and inverse images of geometric morphisms. Therefore, the adjunction between sheaves and presheaves of sets directly induces an adjunction between $T$-models in sheaves and presheaves. And since finite limits of sheaves and presheaves are computed pointwise, $T$-models in the category of (pre)sheaves are the same as (pre)sheaves of $T$-models-in-$\mathrm{Set}$.

If $T$ is not defined in terms of finite limits, then internal $T$-models in sheaves need not be the same as sheaves of $T$-models-in-$\mathrm{Set}$. My intuition would be that the former, rather than the latter, is the more interesting and important notion. For instance, a local ring in a topos of sheaves is a sheaf of rings whose stalks are local, rather than a sheaf taking values in the category of local rings, and this is usually what people care about. But since people have studied the other version, there must be important examples of it as well?

Urs: okay, I have added this to the above now – so is the IPC-property business really unnecessary for the examples above?

Mike: I’m pretty sure it is not. Does anyone have any examples where the IPC-property business is important?

Construction

In terms of matching families

A concrete component-based construction of sheafification proceeds via matching families. See there for details.

In terms of local isomorphisms

Encoding the Grothendieck topology on $S$ equivalently in a system of local isomorphisms, sheafification can be expressed as follows.

Write ${\mathrm{Ho}}_{\mathrm{PSh}(S)}$ for the homotopy category induced by letting weak equivalences be the local isomorphisms with respect to a Grothendieck topology on $S$ as described at category of sheaves.

Let $F\in \mathrm{PSh}(S)$ be a presheaf on $S$. Its sheafification is the presheaf

$$\overline{F}:U\mapsto {\mathrm{Hom}}_{\mathrm{Ho}(\mathrm{PSh}(S))}(Y(U),F),$$\bar F : U \mapsto Hom_{Ho(PSh(S))}(Y(U), F) \,,

where $Y$ denotes the Yoneda embedding. This can be computed explicitly as

$$\overline{F}(U)={colim}_{\hat{U}\stackrel{\mathrm{local}\mathrm{iso}}{\to }U}{\mathrm{Hom}}_{\mathrm{PSh}(S)}(\hat{U},F),$$\bar F(U) = \colim_{ \hat U \stackrel{local iso}{\to} U } Hom_{PSh(S)}(\hat U, F) \,,

Remarks

• For more on the role of sheafification see category of sheaves.

• The notion of sheafification generalizes from presheaf Grothendieck topoi to arbitrary topoi with Lawvere-Tierney topology. See the last.

• The notion of sheafification also generalizes from the 1-categorical to the (infinity,1)-categorical context. See (infinity,1)-category of (infinity,1)-sheaves.

References

The description of sheafification in terms of local isomorphisms is in section 16.3 (for Set-valued presheaves) and section 17.4 (for more general presheaves) of

• Kashiwara–Schapira, Categories and Sheaves

The description in terms of dense monomorphisms using Lawvere-Tierney topology is in section V.3 of

• Mac Lane, Moerdijk, Sheaves in Geometry and Logic

Extension of sheafification of presheaves with values in other categories has been advanced in

• A. Heller, K. A. Rowe, On the category of sheaves, Amer. J. Math. 84, 1962, 205-216.

• Barr, Grillet and Van Osdol, Exact categories and categories of sheaves, Lecture Notes in Math., Vol. 236, Springer, Berlin, 1971

• A. Rosenberg, Almost quotient categories, sheaves and localizations, 181 p. Seminar on supermanifolds 25, University of Stockholm, D. Leites editor, 1988 (in Russian; partial remake in English exists)