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In two recent papers, the author has developed a theory of graded annihilators of left modules over the Frobenius skew polynomial ring over a commutative Noetherian ring $R$ of prime characteristic $p$, and has shown that this theory is…
The aim of this note is threefold. The first is to obtain a simple characterization of relative constructible sheaves when the parameter space is projective. The second is to study the relative Fourier-Mukai for relative constructible…
Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category $\mathcal{C}$ and $\mathcal{C}$-module category $\mathcal{M}$, we introduce a version…
This is the second in a series of papers on the relation between algebraic set theory and predicative formal systems. In part I, we introduced the notion of a predicative category of small maps and obtained the result that such categories…
Recent advances in foundation models have led to a promising trend of developing large recommendation models to leverage vast amounts of available data. Still, mainstream models remain embarrassingly small in size and na\"ive enlarging does…
We review briefly the existing vertex-operator-algebraic constructions of various tensor category structures on module categories for affine Lie algebras. We discuss the results first conjectured in the work of Moore and Seiberg that led us…
This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in…
In Part 1, we describe six projective-type model structures on the category of differential graded modules over a differential graded algebra A over a commutative ring R. When R is a field, the six collapse to three and are well-known, at…
Given a torsion theory (Y,X) in an abelian category C, the reflector I from C to the torsion-free subcategory X induces a reflective factorisation system (E, M) on C. It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Par\'e that…
Jinwu Ye has shown that two flux quanta are attached in one layer while the electron is in the other layer to form a mutually composite fermion (MCF). This idea is based on an earlier idea that CF are formed by attaching two flux quanta to…
We continue the theory of $\tT$-systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at…
The goal of this paper is to construct a category of motivic "sheaves" on an algebraic variety defined over a subfield of C, using Nori's method. This categoryis abelian and it possesses faithful exact realization functors to the…
We show that in $K$-theory-like categories many corner embeddings into a discrete algebra of compact operators are invertible, and consequently functors on splitexact algebraic $KK$-theory are faithful if and only if they are faithful on…
Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to…
Let $A$ be an algebra in a monoidal category $\Cc$, and let $X$ be an object in $\Cc$. We study $A$-(co)ring structures on the left $A$-module $A\ot X$. These correspond to (co)algebra structures in $EM(\Cc)(A)$, the Eilenberg-Moore…
We study the moduli space $\mathbf{M}_X(\Lambda, n)$ of semistable $\Lambda$-modules of vanishing Chern classes over an abelian variety $X$, where $\Lambda$ belongs to a certain subclass of $D$-algebras. In particular, for $\Lambda =…
In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version of model structures, for operads and…
Assume $k$ is a field and let $F:C\rightarrow Vect_{k}$ be a small $k$-linear functor from a $k$-linear abelian category $C$ to the category of vector spaces over the field $k$, the purpose of this note is to use a little knowledge of…
We describe the proof that the period map from the Torelli space of Calabi-Yau manifolds to the classifying space of polarized Hodge structures is an embedding. The proof is based on the constructions of holomorphic affine structure on the…
We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartments. Assuming the generalized Riemann hypothesis, this shows that the Steinberg…