Sheaf categories and homotopy invariance
The invariant lives in Witt groups of sheaf categories generated by proper direct images of constant sheaves on finite polyhedra. This chapter defines the continuous and semialgebraic test categories, establishes Verdier duality on exactly these categories, and proves the cylinder relation that makes their Witt groups homotopy invariant.
Restriction does not preserve the test categories in general (a constant sheaf on \([0,1]\) restricts to a non-compactly-supported sheaf on \((0,1)\)), so later germ arguments take restrictions in the ambient derived categories. Verdier reflexivity here is not stalkwise algebraic reflexivity, which is what allows infinite-dimensional stalks.
Categories and duality
A test space is a locally compact, finite-dimensional, separable metrizable space built from finite polyhedra: a finite polyhedron, a space homeomorphic to one, an open subset of such, or a finite product of these; points and the empty space are included. Write \(\mathsf D(X) = D^b(\mathrm{Sh}_\R(X))\) for the bounded derived category of real sheaves.
\(\Tzero(X)\) is the smallest full replete triangulated subcategory of \(\mathsf D(X)\) containing the proper direct images \(Rf_*\underline\R_B\) for every continuous \(f : B \to X\) with \(B\) a compact finite polyhedron; \(\T(X)\) is its closure under direct summands. \(\Pzero(X)\) and \(\Pcat(X)\) are defined the same way with \(B\) a compact polyhedron in Euclidean space and \(f\) continuous semialgebraic in specified models. An allowed map for \(\T\) is a continuous map of test spaces; for \(\Pcat\), a continuous semialgebraic map (or restriction, or open inclusion) carrying allowed generators to allowed generators.
Well-definedness. Every test space embeds as a locally closed subset of some \(\R^N\), so every allowed map has finite \(c\)-soft dimension; the composition formula used is established before that hypothesis is imposed, so the verification is not circular. The generating maps are proper with bounded images, and the categories are essentially small.
Dependencies
In either test-category definition, closed simplices suffice in place of all compact finite polyhedra. Every object of \(\T(X)\), respectively \(\Pcat(X)\), is a direct summand of an object of \(\Tzero(X)\), respectively \(\Pzero(X)\). In particular these summand closures are triangulated and thick.
Proof sketch. Skeleta filtration plus the open-complement triangle builds \(\underline\R_B\) from constants on closed simplices; pushing along \(f\) proves the first assertion (all maps stay semialgebraic in the \(\Pcat\) case). Summands of a full triangulated subcategory form a triangulated subcategory: for \(u : A \to B\) between summands, a cone on the block map \(u \oplus 0\) exhibits \(\Cone(u)\) as a summand.
Dependencies
The Verdier operation \(\D_X A = R\mathcal Hom(A, \omega_X)\) restricts to a triangulated duality on each of \(\Tzero(X)\), \(\T(X)\), \(\Pzero(X)\), and \(\Pcat(X)\). For every object \(A\) in one of these categories, \(\D_X A\) belongs to the same category and the canonical biduality map \(A \to \D_X^2 A\) is an isomorphism. There is a compact \(K \subset X\) outside which both \(A\) and \(\D_X A\) vanish.
Proof sketch. On a closed \(n\)-simplex, a half-ball local calculation gives \(\D_\Delta\underline\R_\Delta \simeq j_!\underline\R_{\Delta^\circ}[n]\) and biduality is the unit isomorphism, including at the boundary. For a simplex generator \(Rf_*\underline\R_\Delta\), properness plus the internal duality formula gives \(\D_X A \simeq Rf_*\omega_\Delta\), which lies in the same category by the boundary triangle; a tensor--Hom diagram shows proper duality transports the canonical biduality map. The duality class is closed under shifts, triangles, and summands. Compact support passes to duals by open restriction of duality.
For an allowed map \(g : X \to Y\), direct image induces exact functors \(g_* : \Tzero(X) \to \Tzero(Y)\) and \(g_* : \T(X) \to \T(Y)\), and the analogues for \(\Pzero, \Pcat\). They are given by \(Rg_! \simeq Rg_*\) on these categories, commute coherently with Verdier duality and its shifts, and are compatible with composition. For an open inclusion \(j : U \hookrightarrow X\), this functor is \(j_!\) and is fully faithful.
Proof sketch. An object vanishing outside compact \(K\) factors through \(K\), where \(gi\) is proper, identifying \(Rg_! \to Rg_*\) with an invertible comparison. Generators map to generators by composition. Compact support of both \(A\) and \(\D_X A\) gives the duality comparison via evaluation, cup map, and trace; traces compose. For open inclusions, \(Rj_! = j_!\) and the adjunction \((j_!, j^{-1})\) gives full faithfulness.
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The cylinder relation
Let \((\#, \eta)\) be a triangulated duality with exactness sign \(\delta \in \{1,-1\}\) on a category in which multiplication by \(2\) is invertible on every morphism group. Let \(u : L \to \#L\) be symmetric, with no invertibility assumption. In a triangle \(L \to \#L \to C \to L[1]\), the cone carries a nonsingular symmetric form \(\psi : C \to [1]\#C\) for the translated duality. It is neutral and can be chosen to satisfy the cone-compatibility identities \(\psi v = -[1](\#w)\) and \([1](\#v)\psi = \delta[1](\eta_L)w\). For \(\# = \D_{X,r+1}\) this is a neutral form of degree \(r\).
Proof sketch. Dualize and rotate the triangle, complete to a morphism of triangles, and average the third map with its signed transpose (using invertibility of \(2\)) to impose symmetry for the translated biduality. Two-out-of-three gives the isomorphism; the construction is the symmetric boundary of \(u\), hence neutral.
Dependencies
For \(A \in \T(X)\) and \(B \in \mathcal C_I\) (the full thick subcategory of \(\T(I)\) generated by the interval constant sheaf and the two endpoint skyscrapers), the external product \(B \boxtimes A\) belongs to \(\T(I \times X)\); the same holds for \(\Pcat\). For all integers \(s, r\), the natural external-duality map \(\mu_{s,r} : \D_{I,s}B \boxtimes \D_{X,r}A \to \D_{I\times X,s+r}(B \boxtimes A)\) is an isomorphism, adjoint to the product of evaluations. It commutes with proper direct image, duality, and traces; in particular the product of symmetric pairings is symmetric for the sum of the degrees.
Proof sketch. On generators the external-image formula gives a compact polyhedral source; the interval triangle extends membership to \(J\), then exactness and summands to all objects. The map \(\mu_{s,r}\) is defined via the product trace and tensor--Hom adjunction; compatibility with proper pushforward is checked by reducing both routes to the tensor of traces. Invertibility is verified on \(H, E_0, E_1\) over \(I \times \Delta\) by the simplex duality calculation, then propagated by proper duality (even for arbitrary continuous \(f\)), triangles, and summands. Symmetry follows from naturality of the Koszul symmetry.
Dependencies
Let \((A, \alpha)\) be a symmetric degree-\(r\) form in \(\T(X)\) or \(\Pcat(X)\), and \(i_t : X \to I \times X\) the endpoint inclusions. There is a neutral degree-\(r\) form on \(E \boxtimes A \simeq (i_0)_*A \oplus (i_1)_*A\) whose two diagonal forms are \(c(i_0)_*\alpha\) and \(-c(i_1)_*\alpha\) for a common nonzero real scalar \(c\). There are no off-diagonal terms.
Proof sketch. Compact-support cohomology of the interval triangle is the difference exact sequence, so the shifted dual of the boundary has opposite nonzero endpoint coefficients. The product morphism \(u \boxtimes \alpha\) followed by the external-duality comparison is symmetric; its symmetric cone is the required neutral form. Disjoint closed slices force diagonality; precomposition with \(v \boxtimes 1\) determines each diagonal, and the suspension sign is endpoint-independent.
Dependencies
If \(h : I \times X \to Y\) is a continuous homotopy of maps of test spaces, its endpoint maps induce equal homomorphisms \(h_{0*} = h_{1*} : E_r(X) \to E_r(Y)\) for every \(r \in \Z\). The same holds for \(W_r(\Pcat(-))\) under allowed semialgebraic homotopies. In particular, homotopy equivalences induce isomorphisms, and a nonempty contractible test space has the Witt groups of a point.
Proof sketch. The endpoint relation plus rescaling by an inverse square root (either sign of \(c\)) gives \((i_0)_* = (i_1)_*\) on the cylinder; pushing along \(h\) identifies \(h_{0*}\) and \(h_{1*}\). Properness of the homotopy is unnecessary since all objects have compact support. The \(\Pcat\) case uses the same relation with semialgebraic data.
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Appendix signs
With cochain shifts and the translated-duality convention, a symmetric form of degree \(r\) is equivalently a Koszul-symmetric pairing \(\beta : A \otimes A \to \omega_X[-r]\) with invertible adjoint; the exactness sign of \(\D_{X,r}\) is \((-1)^r\), and \(W_r = W^{-r}\). On an oriented \(d\)-manifold the orientation pairing \(1 \otimes 1 \mapsto 1\) is symmetric and positive with no extra parity sign. The interval symmetric cone is the diagonal matrix \(\mathrm{diag}(1,-1)\), and the product cone is \((-1)^r\) times the diagonal of pushed endpoint forms; reversing conventions may negate both entries but keeps their relative sign opposite, so the endpoint Witt-class equality is unaffected.
Proof sketch. Chain-level verification: the Hom shift comparison is a chain map with the stated exponents; unshifted biduality is tensor-symmetry evaluation; shifted biduality acquires \((-1)^{j(j+1)/2}\); tensor--Hom adjunction identifies symmetry with Koszul invariance. The cone identities plus the difference-map normalization determine the endpoint matrices by precomposition.
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For \(B \in \mathcal C_I\), \(A \in \T(X)\) (or the semialgebraic category), and \(s, r \in \Z\), suspending the first product input satisfies \(\gamma_{s+r}(B \boxtimes A)\,\widetilde\mu_{s,r} = (-1)^r\mu_{s,r}^{B,A}[-1]\,\lambda\) with the paper's shift comparisons. The scalar \((-1)^r\) is independent of \(A\), its cohomological degrees, and the point of \(I\).
Proof sketch. Direct exponent comparison from the product evaluation formula: the first-factor Hom comparison contributes \((-1)^{l-s}\), the total comparison \((-1)^{l+n-s-r}\), and suspending changes the braiding exponent by \(-n\); all individual degrees cancel leaving \(r \bmod 2\).