Definition 2.2.1. A presheaf (resp. copresheaf) on a metric site C with values in a category D is a contravariant (resp. covariant) functor A: C^{op} ® D (resp. C ® D) from C to D. We say A is a sheaf (resp. cosheaf) if for any object X of C, and any open effective cover {U_{i}} of X, A(X) is the limit (resp. colimit) of the system A(U_{ij}) for all i, j, where U_{ij} = U_{i} Ç U_{j} Í X, via the morphisms induced by the inclusions among U_{ij}. Denote by [C, D] (resp. {C, D}) the category of copresheaves (resp. cosheaves) on C with values in D. We denote by [C_D] the subcategory of [C, D] consisting of functors preserve colimits. If colimits exist D, then colimits exist in [C, D] which are computed pointwise. Suppose E is a category. Any functor j: C ® E determines a functor j*: [E, D] ® [C, D] sending each copresheaf B Î [C, D] to j*(A) = Aj Î [E, D]. Suppose colimits exist in D. Then j* has a left adjoint j!: [C, D] ® [E, D]. If A Î [C, D] we say that j!(A) is the Kan extension of A (along j). If D is the category of sets (resp. abelian groups, resp. rings), a presheaf or sheaf on C with values in D is called a presheaf (or sheaf) of sets (resp. abelian groups, resp. rings) on C. Remark 2.2.2. Suppose C is a metric site and A is a presheaf (resp. copresheaf) on C with values in a category D. Then A is a sheaf (resp. cosheaf) if and only if for any Z Î D, the presheaf of sets on C given by X ® hom (Z, A(X)) (resp. hom (A(X), Z)) is a sheaf. Remark 2.2.3.
A presheaf A of sets on a metric site C is a sheaf if and
only if the following condition is satisfied:
Denote by C^{^} and C^{~} the categories of presheaves and sheaves of sets on a metric site C respectively. We may identify C with a subcategory of C^{^} via Yoneda embedding; the inclusion functor C ® C^{^} preserves limits. Remark 2.2.4.
(a) Suppose j: C
® D is a functor.
Then (2.2.2) implies that j
is a cosheaf if and only if j*(D)
Í C^{~}.
Example 2.2.5.
(a) A metric site C is strict
if and only if the identity functor C ®
C is a cosheaf (this follows from (2.2.2) and (2.2.3)).
Proposition 2.2.6. If colimits exist in D, then the same is true for {C, D}. Colimits in {C, D} are computed pointwise. The inclusion functor {C, D} ® [C, D] preserves colimits. Proof. It suffices to consider the case that C = W(X) for a topological space X. The assertions then follow from the transitivity of colimits in [W(X), D]. Example 2.2.7. We have C^{~} = {C, Set^{op}}^{op}. Since Set^{op} has colimits, {C, Set^{op}} has colimits and the inclusion {C, Set^{op}} ® [C, Set^{op}] preserves colimits. It follows that limits exist in C^{~} and the inclusion C^{~} ® C^{^} preserves limits. Remark 2.2.8. (2.2.7) also follow from that fact that C^{~} is an isometric reflective subcategory of C^{^} with the reflector (+): C^{^} ® C^{~}: a presheaf O Î C^{^} induces a presheaf O_{X} on the space X of any object X, generated by O(U) for effective open sets U of X. Denote by O_{X}^{+} the associated sheaf to O_{X}. Then X ® O_{X}^{+}(X) defines a sheaf O^{+} of sets on C, which is the associated sheaf to O. It is easy to see that (+) preserves finite limits. Proposition 2.2.9. Suppose C' is a metric site and C a full subsite of C'. Suppose D is a category with colimits and j: C ® D is a cosheaf. Then the Kan extension j' of j on C' is a cosheaf. Proof. Suppose X Î C'. Then j' induces a copresheaf j'_{X} on X. j'_{X} is the colimit of the direct images of j'_{Y} = j_{Y} on Y for all the morphisms f: Y ® X with Y Î C. Since each j_{Y} is a cosheaf, j'_{X} is a cosheaf by (2.2.6). Thus j': C' ® D is a cosheaf. Remark 2.2.10. If C is a Cauchybase (resp. finite Cauchybase, resp. effectivebase) of C', for any cosheaf j: C d D, the Kan extension j': C' d D is called the cosheaf on C' generated by j. The correspondence j d j' is an equivalence from {C, D} to {C', D}. 2.2.11 The Yoneda embedding
Y: C ®
C^{^} is a universal copresheaf with values in a category
with colimits in the following sense:
2.2.12 Suppose C is
strict. The embedding S: C ®
C^{~} is a cosheaf. It is a universal cosheaf on C
with values in a category with colimits in the following sense:
