A category $\mathcal{C}$ consists of:

1. A collection of objects: $A, B, C, \ldots$
2. For each pair of objects $A, B$, a collection of morphisms $\mathrm{Hom}(A, B)$, written $f : A \to B$
3. For each object $A$, an identity morphism $\mathrm{id}_A : A \to A$
4. A composition operation: given $f : A \to B$ and $g : B \to C$, there exists $g \circ f : A \to C$

satisfying associativity $(h \circ g) \circ f = h \circ (g \circ f)$ and identity $f \circ \mathrm{id}_A = f = \mathrm{id}_B \circ f$.

A functor $F : \mathcal{C} \to \mathcal{D}$ consists of a mapping on objects ($A \mapsto F(A)$) and a mapping on morphisms ($f \mapsto F(f)$) satisfying:

• $F(\mathrm{id}_A) = \mathrm{id}_{F(A)}$
• $F(g \circ f) = F(g) \circ F(f)$

Given functors $F, G : \mathcal{C} \to \mathcal{D}$, a natural transformation $\alpha : F \Rightarrow G$ assigns to each object $A$ a morphism $\alpha_A : F(A) \to G(A)$ such that for every $f : A \to B$:

A presheaf on $\mathcal{C}$ is a functor $F : \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$. Concretely: for each object $U$, a set $F(U)$ of "sections over $U$"; for each morphism $f : V \to U$, a restriction map $F(f) : F(U) \to F(V)$, satisfying $F(\mathrm{id}_U) = \mathrm{id}_{F(U)}$ and $F(g \circ f) = F(f) \circ F(g)$.

An adjunction $F \dashv G$ consists of functors $F : \mathcal{C} \to \mathcal{D}$ and $G : \mathcal{D} \to \mathcal{C}$ with a natural bijection:

Equivalently, natural transformations $\eta_A : A \to G(F(A))$ (unit) and $\varepsilon_B : F(G(B)) \to B$ (counit) satisfying the triangle identities.

A morphism $f : A \to B$ is an isomorphism if there exists $g : B \to A$ with $g \circ f = \mathrm{id}_A$ and $f \circ g = \mathrm{id}_B$. The pair $(f, g)$ is the witness. Objects $A$ and $B$ are isomorphic, written $A \cong B$.

An equivalence between $\mathcal{C}$ and $\mathcal{D}$ consists of functors $F : \mathcal{C} \to \mathcal{D}$ and $G : \mathcal{D} \to \mathcal{C}$ with natural isomorphisms $G \circ F \cong \mathrm{Id}_{\mathcal{C}}$ and $F \circ G \cong \mathrm{Id}_{\mathcal{D}}$.

A site is a category $\mathcal{C}$ equipped with a Grothendieck topology $J$: for each object $U$, a specification of which families $\{U_i \to U\}$ count as covers, satisfying stability, transitivity, and identity axioms.

A Context $U = (N, \Sigma, L, \pi)$ where $N$ is a name, $\Sigma = (T, P, I)$ is a signature, $L \in \{\text{CWA}, \text{OWA}, \text{3VL}\}$ is a logic regime, and $\pi$ is provenance metadata.

A Context Morphism $f : V \to U$ consists of a signature inclusion $\Sigma_U \hookrightarrow \Sigma_V$ and a logic compatibility condition ($L_V$ at least as strong as $L_U$). Direction: $V$ refines $U$ (more specific to more general).

Given $f : V \to U$ and $g : W \to U$, the overlap $V \times_U W = (N_{V \cap W}, \Sigma_V \cap \Sigma_W, \max(L_V, L_W), \pi_{\text{derived}})$.

A cover of $U$ is a family $\{U_i \to U\}$ with $\bigcup_i \Sigma_{U_i} \supseteq \Sigma_U$. The Context Site $(\mathbf{Ctx}, J)$ equips the context category with this topology, satisfying stability, transitivity, and identity.

A presheaf $F$ on $(\mathbf{Ctx}, J)$ is a sheaf iff for every cover $\{U_i \to U\}$: (1) Locality — $s|_{U_i} = t|_{U_i}$ for all $i$ implies $s = t$; (2) Gluing — matching sections $s_i|_{U_i \times_U U_j} = s_j|_{U_i \times_U U_j}$ for all $i,j$ yield unique $s \in F(U)$ with $s|_{U_i} = s_i$.