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category of sheaves

Idea

This entry is about the properties and the characterization of the category $\mathrm{Sh}(S)$ of sheaves on a site $S$ – a Grothendieck topos. Among other things it does give a definition and a characterization of the notion of sheaf itself, but for more on the traditional information on sheaves see there.

Definition and Properties

Consider a site $S$, i.e. a category $S$ equipped with a coverage, a Grothendieck topology. Think of this topology equivalently as encoded in a system of local isomorphisms (see there) on the presheaf category $\mathrm{PSh}(S):=[S^{\mathrm{op}},\mathrm{Set}]$.

Then

• a sheaf on $S$ is precisely a presheaf $F\in [S^{\mathrm{op}},\mathrm{Set}]$ such that ${\mathrm{Hom}}_{\mathrm{PSh}(S)}(-,F)$ sends all local isomorphisms to isomorphisms;

• the category $\mathrm{Sh}(S)$ of sheaves on $S$ is the full subcategory of the category of presheaves $\mathrm{PSh}(S)$ on sheaves;

• the fully faithful forgetful inclusion functor $i:\mathrm{Sh}(S)\hookrightarrow \mathrm{PSh}(S)$ has an left adjoint $\overline{(-)}:\mathrm{PSh}(S)\to \mathrm{Sh}(S)$ – sheafification – which is exact – hence the inclusion is a geometric morphism:

  $$\mathrm{Sh}(C)\stackrel{\stackrel{\mathrm{lex}}{\leftarrow }}{\hookrightarrow }\mathrm{PSh}(C);$$Sh(C) \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} PSh(C) \,;

• the functor $L:=i\circ \overline{(-)}:\mathrm{PSh}(S)\to \mathrm{PSh}(S)$ whose image is $\mathrm{Sh}(S)$ has the property that $L(A_1\to A_2)$ is an isomorphism if and only if $A_1\to A_2$ is a local isomorphism;

• the category of sheaves is equivalent to the homotopy category of the category with weak equivalences $\mathrm{PSh}(C)$ with the weak equivalences given by $W=$local isomorphisms

$$\mathrm{Sh}(S)\simeq {\mathrm{Ho}}_{\mathrm{PSh}(S)}=\mathrm{PSh}(C)[\mathrm{local}\mathrm{isomorphisms}{]}^{-1}.$$Sh(S) \simeq Ho_{PSh(S)} = PSh(C)[local isomorphisms]^{-1} \,.

• also the converse is true: for every left exaxt functor $L:\mathrm{PSh}(S)\to \mathrm{PSh}(S)$ (preserving finite limits) which is left adjoint to the inclusion of its image, there is a Grothendieck topology on $S$ such that the image of $L$ is the category of sheaves on $S$ with respect to that topology

• Categories of sheaves are examples of categories that are toposes: they are the Grothendieck toposes characterized among all toposes as those satisfying Giraud's axioms.

In topos-theoretic language we therefore have that

Sheaf toposes are precisely the subtoposes of presheaf toposes.

In higher category theory

The notion of category of sheaves has analogs in higher category theory.

For (∞,1)-category theory see (∞,1)-category of (∞,1)-sheaves.

References

For some standard references see the list of references at sheaf and topos.

The characterization of sheaf toposes and Grothendieck topologiws in terms of left exact reflective subcategories of a presheaf category is in

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

where it is implied by the combination of Corollary VII 4.7 and theorem V.4.1.

The statement is also theorem A.4.4.8 in

• Peter Johnstone, Sketches of an Elephant .

The characterization of $\mathrm{Sh}(S)$ as the homotopy category of $\mathrm{PSh}(S)$ with respect to local isomorphisms is emphasized at the beginning of the text

• Bertrand Toen, Stacks and non-abelian cohomology (web) .

Details are in

• Kashiwara-Schapira, Categories and Sheaves .

It’s a bit odd that the full final statement does not seem to be stated there explicitly, but all the ingredients are discussed:

• exercise 16.7 shows that sheafification inverts precisely the local isomorphisms, so that in particular every local isomorphism between sheaves is an isomorphism;

• lemma 16.3.2 states that the unit of the adjunction ${\mathrm{Id}}_{\mathrm{PSh}(S)}\to i\circ \overline{(-)}:\mathrm{PSh}(S)\to \mathrm{PSh}(S)$ is componentwise a local isomorphism;

• using this corollary 7.2.2 says that $\mathrm{Sh}(S)\simeq {\mathrm{Ho}}_{\mathrm{PSh}(S)}$ with the homotopy category ${\mathrm{Ho}}_{\mathrm{PSh}(S)}$ formed using local isomorphisms as weak equivalences.

The entirely analogous story in the wider context of (infinity,1)-categories is in

• Charles Rezk, Toposes and homotopy toposes (pdf)

and

• Jacob Lurie, Higher Topos Theory .

For more on this see (∞,1)-category of (∞,1)-sheaves and models for ∞-stack (∞,1)-toposes.