Related papers: Algebraic Geometry over model categories (a genera…

We consider K\"ahler toric manifolds $N$ that are torifications of statistical manifolds $\mathcal{E}$ in the sense of [M. Molitor, "K\"ahler toric manifolds from dually flat spaces", arXiv:2109.04839], and prove a geometric analogue of the…

A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…

Let $p$ be a prime. Given a split semisimple group scheme $G$ over a normal integral domain $R$ which is a faithfully flat $\mathbb Z_{(p)}$-algebra, we classify all finite dimensional representations $V$ of the fiber $G_K$ of $G$ over…

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…

Let k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack \hat G over k, the metaplectic extension of the Greenberg realization of Sp_{2n}(R). We also…

In this paper, we study the K-theory on higher modules in spectral algebraic geometry. We relate the K-theory of an $\infty$-category of finitely generated projective modules on certain $\mathbb{E}_{\infty}$-rings with the K-theory of an…

In view of applications to the construction of moduli spaces of objects in algebraic supergeometry, we start a systematic study of stacks in that context. After defining a superstack as a stack over the \'etale site of superschemes, we…

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued'. This paper largely consisits of two…

We show that the higher Grothendieck-Witt groups, a.k.a. algebraic hermitian K-groups, are represented by an infinite orthogonal Grassmannian in the A1-homotopy category of smooth schemes over a regular base for which 2 is a unit in the…

Universal algebraic geometry allows considering of geometric properties of every universal algebra. When two algebras have same algebraic geometry? We must consider the categories of algebraic closed sets of these algebras to answer this…

The goal of this paper is to define a notion of non-commutative Gelfand duality. Using techniques from derived algebraic geometry, we show that the category of rings is anti-equivalent to a subcategory of pre-ringed sites, inspired by…

We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the…

The notion of the \emph{homotopy type} of a topological stack has been around in the literature for some time. The basic idea is that an atlas $X \to \mathfrak{X}$ of a stack determines a topological groupoid $\mathbb{X}$ with object space…

For a commutative ring $A$, we have the category of (bounded-below) chain complexes of $A$-modules $Ch_{+}(A\mymod)$, a closed symmetric monoidal category with a compatible stable Quillen model structure. The associated homotopy category is…

Directed Algebraic Topology is beginning to emerge from various applications. The basic structure we shall use for such a theory, a 'd-space', is a topological space equipped with a family of 'directed paths', closed under some operations.…

Using the machinery of etale homotopy theory a' la Artin-Mazur we determine the etale homotopy types of moduli stacks over $\bar{\Q}$ parametrizing families of algebraic curves of genus g greater than 1 endowed with an action of a finite…

Since the time when the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way to formalize it in mathematics is…

Let $\mathcal{C}$ be a small category, $\mathfrak{A}$ be a precosheaf of unital $k$-algebras on $\mathcal{C}$ and $\mathfrak{M}$ be an $\mathfrak{A}$-bimodule. We introduce two new notions, namely, the Grothendieck construction…

In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…

Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework for studying the homological algebra of…