A presheaf F on a category C assigns data to objects and restriction maps to morphisms. What additional condition must F satisfy to be a sheaf with respect to a Grothendieck topology J?
AF must be representable — it must equal hom(–, c) for some object c ∈ C
BLocal sections on a covering family that agree on all pairwise overlaps must glue uniquely to a global section on the covered object
CF must preserve all limits and colimits in C, making it a full functor
DThe sets F(c) must be abelian groups, not arbitrary sets, to allow gluing
The gluing (or descent) condition is what distinguishes a sheaf from a mere presheaf. Given a covering family {f_i : c_i → c} and local sections s_i ∈ F(c_i) that agree on all pairwise overlaps (F(c_i ×_c c_j) maps both s_i and s_j to the same element), there must exist a unique global section s ∈ F(c) that restricts to each s_i. Option A is false: not every sheaf is representable, and non-representable sheaves are essential to the theory. Option C confuses functoriality conditions with the sheaf condition. Option D is false: sheaves in the Grothendieck sense take values in any category, including Set.
Question 2 Multiple Choice
Why is the presheaf category [C^op, Set] much 'larger' than the original category C, and what role do non-representable presheaves play?
AIt is larger only in a set-theoretic sense; representable presheaves capture all the relevant structure and non-representables are redundant
B[C^op, Set] is the free cocompletion of C — it freely adds all colimits, including objects that represent 'idealized' or 'generalized' elements that C itself lacks
CThe size difference arises from the contravariance; covariant functors from C to Set would give a category the same size as C
DNon-representable presheaves are artifacts of set-theoretic foundations and have no mathematical content
[C^op, Set] is the free cocompletion of C: it adds all small colimits, creating a much richer universe. Non-representable presheaves represent genuinely new objects — for example, in algebraic geometry, the functor of points of a scheme may be a sheaf that is not representable by any scheme in a naive sense, but is representable in the enlarged category. The Yoneda embedding embeds C fully faithfully into [C^op, Set], but the latter is vastly larger. Option A fundamentally misunderstands the purpose of the presheaf construction: non-representable presheaves are not redundant — they are the objects you freely add when completing C.
Question 3 True / False
Every sheaf is a presheaf, but not every presheaf is a sheaf.
TTrue
FFalse
Answer: True
A sheaf is a presheaf with an additional condition (the gluing axiom). Since sheaves satisfy all presheaf axioms plus the gluing condition, every sheaf is automatically a presheaf. The reverse fails: a presheaf may fail the gluing condition. For example, the presheaf that assigns to each open set U the set of *bounded* continuous functions fails to be a sheaf on R because local sections that agree on overlaps may glue to an unbounded global function. The category of sheaves Sh(C, J) is a full subcategory of the presheaf category [C^op, Set], and the inclusion has a left adjoint (sheafification) that converts arbitrary presheaves into sheaves.
Question 4 True / False
The internal logic of any Grothendieck topos is classical — it satisfies the law of excluded middle and the axiom of choice.
TTrue
FFalse
Answer: False
This is a key misconception about topos theory. The internal logic of a Grothendieck topos is intuitionistic by default — it satisfies intuitionistic higher-order logic but not necessarily classical logic. The law of excluded middle (P ∨ ¬P for all propositions) and the axiom of choice fail in many topoi. For example, the topos of sheaves on a topological space satisfies classical logic only if the space is discrete. Different Grothendieck topologies on the same category C produce different internal logics, some classical and some not. This logical richness is precisely why topos theory is foundational to categorical logic and constructive mathematics.
Question 5 Short Answer
What is the gluing condition for a sheaf, and why is it described as the formal mathematical expression of 'local-to-global' reasoning?
Think about your answer, then reveal below.
Model answer: Given a covering family {c_i → c} and local sections s_i ∈ F(c_i) that pairwise agree on overlaps, the gluing condition requires a unique global section s ∈ F(c) restricting to each s_i. This captures 'local-to-global' reasoning because it says: if you have consistent local data (data on each patch that agrees where patches overlap), you can always reconstruct a unique global datum. The word 'unique' is crucial — not only does a global section exist, but it is the only one compatible with the local data.
The topological motivation is transparent: a continuous function on a space is completely determined by its values on any open cover, and locally defined functions that agree on overlaps patch together uniquely. The sheaf condition formalizes this. When the gluing condition fails, you have a presheaf: local data exists but cannot always be assembled globally. Sheafification forces this assembly to work by identifying presheaf sections that 'want to be' the same global section. In logic, the gluing condition corresponds to the idea that truth is a local-global property: a statement holds globally if and only if it holds locally on every cover.