Algebraic structures and combinatorial models

We construct the derived Legendrian category $\mathcal{F}_{c}(X)$ for an $n$-shifted contact derived Artin stack $X$ and the $(\infty,2)$-category $Leg_n$ of Legendrian correspondences in the context of derived algebraic geometry, with several applications to moduli theory. In brief, the objects of the category $\math…

We construct the derived Legendrian category $\mathcal{F}_{c}(X)$ for an $n$-shifted contact derived Artin stack $X$ and the $(\infty,2)$-category $Leg_n$ of Legendrian correspondences in the context of derived algebraic geometry, with several applications to moduli theory. In brief, the objects of the category $\math…

We construct a model categorical equivalence between the category of simplicial vector spaces and the category of representations of a crossed simplicial group \Delta G when each G_{n} is finite and the characteristic of the ground field is 0 .

The matrix Hilbert series of a locally finite elementary twisted Calabi--Yau algebra is the inverse of a matrix polynomial. The Smith normal form of this polynomial over the power series ring at $x=1$ produces a finite list of local exponents refining the Gelfand--Kirillov dimension, which records only the largest of…

For a field $k$ of characteristic $0$, we compare $k$-linear chain complexes, semisimplicial vector spaces, augmented semisimplicial vector spaces, semicubical vector spaces, and arboreal vector spaces through small differential categorical algebras. We prove that semisimplicial modules and augmented semisimplicial mo…

For a field $k$ of characteristic $0$, we compare $k$-linear chain complexes, semisimplicial vector spaces, augmented semisimplicial vector spaces, semicubical vector spaces, and arboreal vector spaces through small differential categorical algebras. We prove that semisimplicial modules and augmented semisimplicial mo…

We construct a category $\OrdFor$ as an arboreal extension of $Δ_{\mathrm{epi}}\subseteqΔ$, whose morphisms are ordered forests composed by grafting. We define a full functor $π\colon \OrdFor\toΔ_{\mathrm{epi}}^{op}$ extracting the semisimplicial shadow. For every complete category $\mathcal C$, this induces a fully f…

We construct a category $\OrdFor$ as an arboreal extension of $Δ_{\mathrm{epi}}\subseteqΔ$, whose morphisms are ordered forests composed by grafting. We define a full functor $π\colon \OrdFor\toΔ_{\mathrm{epi}}^{op}$ extracting the semisimplicial shadow. For every complete category $\mathcal C$, this induces a fully f…

We study quadratic moduli schemes $X$ of algebra laws on a fixed vector space $W$ under the transport-of-structure action of $GL(W)$ on $Hom(W^{\otimes 2},W)$. We construct an intrinsic three-term deformation complex on $X$ whose fibers encode transverse first-order classes and primary obstructions, and whose cohomolo…

We study quadratic moduli schemes $X$ of algebra laws on a fixed vector space $W$ under the transport-of-structure action of $GL(W)$ on $Hom(W^{\otimes 2},W)$. We construct an intrinsic three-term deformation complex on $X$ whose fibers encode transverse first-order classes and primary obstructions, and whose cohomolo…

Abstract Let p be a prime number. We consider diagonal p -permutation functors over a (commutative, unital) ring R $\mathsf {R}$ sans serif upper R in which all prime numbers different from p are invertible. We first determine the finite groups G for which the associated essential algebra E R ( G ) $\mathcal {E}_{\mat…

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