Given a site $C$, the sheafification functor universally turns presheaves on $C$ into sheaves.
It is characterized as being the left adjoint functor $L : PSh(C) \to Sh(C)$ to the inclusion $Sh(C) \hookrightarrow PSh(C)$ of sheaves into all presheaves, exhibiting this as a reflective subcategory.
Therefore sheafification is a special case of a very general phenonemon of localizations of categories. See category of sheaves for more.
Let $(C,J)$ be a site in the sense of: small category equipped with a coverage. Write $PSh(C)$ for the category of presheaves on $C$ and
for the Yoneda embedding. Write
for the category of sheaves: the full subcategory on those presheaves that are $J$-sheaves.
The inclusion of sheaves into presheaves admits a left adjoint functor, which hence exhibits $Sh(C)$ as a reflective subcategory or reflective localization of $PSh(C)$:
This follows by general properties discussed at reflective subcategory. We spell out the argument using the theory of localization at a set of morphisms satisfying a calculus of fractions.
Recall from the discussion at sheaf that $Sh_J(C)$ is by definition the full subcategory of $PSh(C)$ on the local objects with respect to the morphisms
of sieve inclusions for all covering families of the coverage $J$. (Here the colimit $\lim_\to$ is the coequalizer of the two injection maps, as indicated. This is spelled out in more detail at sheaf and at sieve.)
Now we invoke the following results:
The localization proposition says that every full subcategory of a locally presentable category on the $W$-local objects for a small set $W$ of morphisms is a reflective subcategory, given by the localization at these morphisms;
By Gabriel-Zisman's theorem every reflective subcategory is the localization at the collection of morphisms inverted by the left adjoint (which by the localization proposition is the saturation of the original set of morphisms).
If $W$ satisfies the axioms of a calculus of fractions then, by the discussion there, this localization is equivalently given by the category $PSh(C)[W^{-1}]$ whose objects are those of $PSh(C)$ and whose morphisms are given by $PSh(C)[W^{-1}](X,A) \simeq {\lim_{\to}}_{\hat X \stackrel{w \in W}{\to} X} PSh_C(\hat X,A)$.
Notice that an object is a local object with respect to the above set of morphisms $W$ precisely if it is local with respect to the set of all small colimits (in the arrow category $Arr(PSh(C))$ ) of such morphims (since the hom-functor $PSh_C(-,A)$ sends colimits in the first argument to limits, and a limit of isomorphisms is an isomorphism).
Let hence $\bar W$ be the completion of $A$ under forming small colimits in $Arr(PSh(C))$.
We claim that the morphisms in $\bar W$ form a calculus of fractions. The first condition to check is that for all morphisms of presheaves $X \to j(U)$ and every covering family $\{U_i \to U\}$ there is a morphism $Y \to X$ in $\bar W$ and a commuting diagram
in $PSh(C)$. (It is sufficient to demand this for $s \in W \subset \bar W$ to deduce the stability conditions for all morphisms in $\bar W$, since by universal colimits in the presheaf topos $PSh(C)$ the pullback of a colimit is a colimit of pullbacks.)
Similarly, to see that we can find $Y \to X$ we use the co-Yoneda lemma to decompose $X$ as a colimit of representables $X \simeq {\lim_{\to}_j} K_j$ and then use universal colimits to deduce that we are looking at a diagram of the form
Since $\bar W$ is closed under colimits, it is hence sufficient that we show the stability condition for $X$ any representable. So we need to fill diagrams of the form
For this now use the single condition on a coverage: that for $\{U_i \to U\}$ a covering family in the site $(C,J)$ we can find a covering family $\{K_j \to K\}$ such that every $K_j \to K$ factors through one of the $U_i \to U$. But this means that also the sieves factor, and we have a commuting diagram
This shows that $\bar W$ satisfies the first condition at factorization system.
The second condition at calculus of fractions demands that if two composites of the form
are equal in $PSh(C)$, then there is a morphism $Y \to X$ in $PSh(C)$ such that the two composites
are equal. But the sieve inclusions are monomorphisms, hence this condition is trivially satisyfied (choose $Y \to X$ to be the identity on $X$). Again by decomposing into colimits by the co-Yoneda lemma and using universal colimits and the pasting law for pullbacks, the same follows for general morphisms in $\bar W$
by applying the above on each component $k$ of the colimit.
This gives us the localization $L : PSh(C) \to Sh_J(C)$ as described at calculus of fractions. By the discussion there we have that $Sh_J(C)$ is equivalently given by the category $PSh(C)[W^{-1}]$ with the same objects as $PSh(C)$ and hom-sets given by
So we have that for $X \in PSh(C)$ a presheaf and $A \in Sh_J(C)$ the hom-set $Sh_C(L(X),A)$ is given by
But if $A$ is a sheaf, it is a $\bar W$-local object and hence $PSh_C(\hat X \stackrel{w}{\to} X, A)$ is an isomorphism for all $w \in \bar W$. Hence the above colimit is over a diagram constant on its value at $w = Id : X \to X$ therefore we have a natural isomorphism
This demonstrates the adjunction $(L \dashv i)$.
For $X \in PSh(C)$ a presheaf on the site $(C,J)$, its sheafification $L(X)$ is the presheaf given on any $U \in C$ by
where the colimit on the right is over all $w \in \bar W$.
By the Yoneda lemma we have
By the above proposition this is
By the proof of the above proposition, using the formulas discussed at calculus of fractions, this hom-set is given by
By the definition of $\bar W$, the morphisms $\hat U \to j(U)$ in $\bar W$ are colimits of diagrams of covering sieves
This means (…) that the above coimit may be computed as two consecutive colimits of the form
One such application is called the plus construction.
A morphism $S(\{U_i\}) \to X$ out of a sieve into any presheaf is in components precisely a matching family of the presheaf $X$ on the covering $\{U_i \to U\}$.
Use that the sieve is the coequalizer
and that the hom-functor $PSh_C(-,X)$ sends colimits to limits. More details on this computation are at sheaf.
The unit of the $(L \dashv i)$-adjunction has as components natural morphisms
in $PSh(C)$, from any presheaf into its sheafification. By general properties of reflective subcategories these morphisms are mapped to isomorphisms by $L : PSh(C) \to Sh(C)$. Therefore these are local isomorphisms.
So every presheaf is related by a local isomorphism to its sheafification.
For presheaves with values in categories other than Set, sheafification may be a difficult problem, unless one has some extra assumptions.
Consider a type of structure $T$ defined in terms of an essentially algebraic theory finite limits (such as groups, algebras, modules, etc.), then internal $T$-models are preserved by both direct images and inverse images of geometric morphisms. Therefore, the adjunction $(L \dashv i) : Sh_J(C) \to PSh(C)$ 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-$Set$.
If a category $A$ satisfies the following assumptions, sheafification of presheaves in $[S^{op}, A]$ exists and is constructed analogously as for Set-valued sheaves.
This is true for instance for
the category Set of sets;
the category Grp of groups;
the category $k Alg$ of $k$-algebras;
the category $Mod(R)$ 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-$Set$.
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-$Set$. 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?
sheafification, plus-construction on presheaves
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
The description in terms of dense monomorphisms using Lawvere-Tierney topology is in section V.3 of
Extension of sheafification of presheaves with values in other categories has been advanced in
Alex 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
Alexander 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)