Relative closed shifted symplectic form

Assume the Axiom of Choice (AC): every set-indexed family of nonempty sets has a choice function. This is the premise inherited from the cotangent-complex suppliers for the relative cotangent, derived de Rham, and dual tangent constructions below.

Let $R$ be a discrete commutative $\mathbb{C}$-algebra and let $M$ be a derived Artin stack over $\mathrm{Spec}(R)$ with perfect relative cotangent complex $L_{M\mathrm{/}R}$. On a derived affine $\mathrm{Spec}(A)$ over $R$, form $\mathrm{D}\mathrm{R}(A\mathrm{/}R)={\mathrm{Sym}}_A^{\mathbf{L}}(L_{A\mathrm{/}R}[1])\mathrm{.}$ Its weight-$p$ component is ${\bigwedge }_A^p{L}_{A\mathrm{/}R}[p]$, with the internal cochain differential and the relative mixed de Rham differential of degree $-1$ and weight $+1$, as in Closed n-shifted symplectic form. Define $\mathrm{D}\mathrm{R}(M\mathrm{/}R)$ by derived descent over affine maps to $M$. For a graded mixed complex $E$, use the Hodge-completed weighted negative-cyclic complex ${\mathrm{N}\mathrm{C}}^w(E)(p{)}^j={\prod }_{r\ge 0}{E}^{j-2r}(p+r),$ with differential $(Dm{)}_r=d{m}_r+ϵm_{r-1}$ and $m_{-1}=0$. Both summands have cochain degree $j+1-2r$ and weight $p+r$, and the mixed-complex identities give $D^2=0$. Thus ${\mathcal{A}}^{2,\mathrm{c}\mathrm{l}}(M\mathrm{/}R,n)=\mid {\mathrm{N}\mathrm{C}}^w(\mathrm{D}\mathrm{R}(M\mathrm{/}R))(2)[n-2]\mid$ is the space of closed relative two-forms of cohomological degree $n$; its projection to the weight-two de Rham term is the underlying two-form. Since $L_{M\mathrm{/}R}$ is perfect, the underlying form has an adjoint ${\mathrm{\Theta }}_{\omega }:T_{M\mathrm{/}R}=R{\mathrm{Hom}}_{{\mathcal{O}}_M}(L_{M\mathrm{/}R},{\mathcal{O}}_M)⟶L_{M\mathrm{/}R}[n]\mathrm{.}$ It is nondegenerate exactly when ${\mathrm{\Theta }}_{\omega }$ is a quasi-isomorphism. A relative $n$-shifted symplectic structure is a closed relative two-form of degree $n$ that is nondegenerate.

This is the base-relative analogue of the graded-mixed convention for absolute shifted forms. It fixes the relative cotangent complex, degree, Hodge completion, and nondegeneracy map; it does not assert that an arbitrary relative pairing has a closed lift.

The Axiom of Choice

Derived Artin stack and cotangent complex

Closed n-shifted symplectic form

The boundary-relative mapping-stack construction in Calaque–Ronchi §5.3 is a different interface: it produces a Lagrangian map from an oriented manifold with boundary. It is not used as a supplier for the algebraic family theorem on this page.