Localization, germs, and dense subcategories
Quotient morphisms are identified with germs near the complement of an open set, giving the Witt localization sequence. Comparing the quotients from an open cover then yields an integral density bound: kernel and cokernel annihilated by \(2\). This bound controls the denominators in the sphere calculation.
Throughout, \(\Bcat\) is \(\T\) or \(\Pcat\) and \(\Bzero\) its subcategory before summands. \(\Bcat(U)\) is identified with its image under extension by zero when \(U \subset X\) is open.
Cutoff and germs
Let \(K \subset O \subset X\) with \(K\) closed and \(O\) open. For every \(A \in \Bzero(X)\) there is a morphism \(s : A' \to A\) such that \(A'\) is extended from \(\Bzero(O)\), its cohomology vanishes outside a compact subset of \(O\), and \(s\) restricts to an isomorphism on some open neighborhood of \(K\).
Proof sketch. For a simplex generator, subdivide so the closed stars of simplices meeting \(f^{-1}K\) lie in \(f^{-1}O\); a vertex-valued function gives a polyhedral neighborhood \(E\) with polyhedral closure inside \(f^{-1}O\), and the extension-by-zero map is an isomorphism over \(X \setminus f(\Delta \setminus E)\). Shifts and sums are immediate; cones are cut off compatibly by lifting through open-extension adjunction over the intersection of the two invertibility neighborhoods.
Sources
Dependencies
For an open inclusion \(j : U \hookrightarrow X\), the essential image of \(j_! : \Bcat(U) \to \Bcat(X)\) consists exactly of those objects whose cohomology vanishes outside some compact subset of \(U\). This image is a thick, duality-stable subcategory.
Proof sketch. Extended objects have such a support bound. For the converse, write the object as a summand of \(A \in \Bzero(X)\), cut off over \(U\) invertibly near the support, and lift the splitting through open-extension adjunction; restricting exhibits the object as extended from \(U\). Thickness follows from the cohomology long exact sequence and stalkwise splitting; duality stability from open restriction of Verdier duality.
Restriction induces natural isomorphisms \(\Hom_{\Bcat(X)/\Bcat(U)}(A,B) \simeq \colim_{N \supset X \setminus U} \Hom_{\mathsf D(N)}(A|_N, B|_N)\) for every \(A, B \in \Bcat(X)\), where the colimit runs over open neighborhoods of the complement in the ambient derived categories. These respect composition, open restriction of duality, and restriction of distinguished triangles. The quotient is the Verdier quotient itself.
Proof sketch. Morphisms with cone in \(\Bcat(U)\) become invertible in germs, so restriction factors through the quotient. Fullness: lift a germ representative through a cutoff supported inside its neighborhood, giving a roof with denominator invertible near the complement; summand sources reduce to the finite case. Faithfulness: a morphism with zero germ vanishes in the quotient by cutting off inside a zero neighborhood; roofs reduce to numerators. Restriction was the functor throughout, giving the compatibilities.
Sources
Dependencies
For every open \(U \subset X\) and every \(r \in \Z\), there is a natural long exact sequence \(\cdots \to W_r(\Bcat(U)) \to W_r(\Bcat(X)) \to W_r(\Bcat(X)/\Bcat(U)) \xrightarrow{\partial} W_{r-1}(\Bcat(U)) \to \cdots\). The sequence is natural for \(\Pcat \to \T\) and for pushforwards \(g : X \to X'\) with \(g(U) \subset U'\).
Proof sketch. Apply Balmer--Walter Theorem 2.1 with \(\mathcal J = \Bcat(U)\) and \(\mathcal K = \Bcat(X)\): the support characterization makes \(\mathcal J\) a saturated duality-stable thick subcategory; the categories are essentially small; morphism groups are real vector spaces so \(2\) is bijective; and (TR4\(^+\)) passes to full triangulated subcategories and localizations. The Verdier quotient has full kernel and duality-stable denominators. Naturality follows because the functors preserve support subcategories, and the boundary map commutes since both routes use the same transported symmetric-cone diagram.
Dependencies
Density bounds
If \(X = U \cup V\) with \(U, V\) open, extension by zero induces a fully faithful exact duality-preserving functor \(\Bcat(V)/\Bcat(U \cap V) \to \Bcat(X)/\Bcat(U)\). Its replete image is dense: every target object is an actual summand of an object in the image. These functors commute with \(\Pcat \to \T\).
Proof sketch. Extension sends \(\Bcat(U \cap V)\) into \(\Bcat(U)\), giving the exact functor. With \(K = X \setminus U \subset V\), neighborhoods of \(K\) inside \(V\) are cofinal, so the germ description identifies both morphism groups: full faithfulness. Density: cut off a finite object over \(V\) with cone in $\Bcat(U), so it becomes an isomorphism in the quotient with source from \(V\); summands inherit. Fullness plus exactness makes the replete image triangulated and duality stable.
Dependencies
Let \(\mathcal A \subset \mathcal C\) be a full replete triangulated subcategory stable under duality, dense in the sense that each object of \(\mathcal C\) is a summand of an object of \(\mathcal A\). In every degree, the inclusion \(\iota : W_r(\mathcal A) \to W_r(\mathcal C)\) has kernel and cokernel annihilated by \(2\): there is a homomorphism \(d : W_r(\mathcal C) \to W_r(\mathcal A)\) with \(\iota d = 2\,\mathrm{id}\) and \(d\iota = 2\,\mathrm{id}\). No closure of \(\mathcal A\) under arbitrary summands is required.
Proof sketch. Density gives \(y \oplus y[1] \in \mathcal A\) for all \(y\) (cone on \(0_y \oplus \mathrm{id}_z\)). Double a form as \(D(q) = q \perp q \perp H(x[1])\), whose underlying object lies in \(\mathcal A\) by the shift observation. Neutrality descends: given a lagrangian for \(q\), an explicit lagrangian for \(D(q) \perp H(z)\) is built and verified to lie in \(\mathcal A\) using only fullness and triangles. The hyperbolic summands are neutral on both sides, giving the two doubling identities.
Dependencies
The map on Witt groups induced by the open-cover quotient functor has kernel and cokernel annihilated by \(2\), in every degree and for either test category.
Proof sketch. Apply dense doubling to the replete image of the open-cover dense inclusion.