Homotopy *Creatio Ex Nihilo*

We realize the Pareigis Hopf algebra, which encodes the monoidal structure of the category of complexes (via the Pareigis transform which is the identity on objects), as a universal quantum symmetry of the algebra of dual numbers. We show that under the Pareigis transform the category of corresponding equivariant quasicoherent sheaves on the double point isequivalent to the category of complexes with square zero homotopies. In particular, the Pareigis transform of the algebra of dual numbers is the terminal object of the extended Hinich category of local pseudo-compact algebras.

We also introduce the notion of the support of a Frobenius algebra and prove that the Pareigis transform of the Frobenius support of the algebra of dual numbers is a closed graded trace of dimension -1 on the terminal Hinich algebra, being a boundary of the Pareigis transform of the augmentation of dual numbers. This can be understood as a dga model of the empty set with a homologically trivial (-1)-dimensional fundamental cycle. If time permits, a physical application to modeling the creation of a virtual pair of anti-particles and the Boyle-Finn-Turok CPT-Symmetric Universe will be sketched.