Related papers: Left-exact Localizations of $\infty$-Topoi I: High…

This paper provides an overview of the applications of sheaf theory in deep learning, data science, and computer science in general. The primary text of this work serves as a friendly introduction to applied and computational sheaf theory…

We show that the cohomology table of any coherent sheaf on projective space is a convergent--but possibly infinite--sum of positive real multiples of the cohomology tables of what we call supernatural sheaves.

We use (non-)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Ext's in a suitable bisheaf category. We compare our approach to various definitions present in the literature.

We study Structural Reflection beyond Vop\v{e}nka's Principle, at the level of almost-huge cardinals and higher, up to rank-into-rank embeddings. We identify and classify new large cardinal notions in that region that correspond to some…

Given a smooth projective toric variety $X_\Sigma$ of complex dimension $n$, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $Coh(X_\Sigma)$…

The strong shape category of compact metrizable spaces (compacta) is very well-studied; extending it to noncompact spaces, however, introduces computational complexity that makes it hard to work with. The fine shape category, as defined by…

This article introduces strongly near proximity, which represents a new kind of proximity called \emph{almost proximity}. A main result in this paper is the introduction of a hit-and-miss topology on ${CL}(X)$, the hyperspace of nonempty…

This is a foundation for algebraic geometry, developed internal to the Zariski topos, building on the work of Kock and Blechschmidt. The Zariski topos consists of sheaves on the site opposite to the category of finitely presented algebras…

Let $X$ be a locally Noetherian scheme with a closed subscheme $Z$. Let $\mathcal{X}$ be the completion of $X$ at $Z$, considered as a formal scheme. We show that a coherent sheaf on $X$ is equivalently given by a coherent sheaf on…

We characterise the class of exponentiable $\infty$-toposes: $\mathcal X$ is exponentiable if and only if $\mathcal S\mathrm{h}(\mathcal X)$ is a continuous $\infty$-category. The heart of the proof is the description of the…

This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of $n$-nerve and Pelissier's thesis. If $M$ is a tractable left proper cartesian model category, we construct a…

We introduce a spectral embedding algorithm for finding proximal relationships between nodes in signed graphs, where edges can take either positive or negative weights. Adopting a physical perspective, we construct a Hamiltonian which is…

To any moment graph G we assign a subcategory V of the category of sheaves on G together with an exact structure. We show that in the case that the graph is associated to a non-critical block of the equivariant category O over a…

Let S be a K3 surface and S^[n] the Hilbert scheme of length n subschemes of S. Over the cartesian square of S^[n] there exists a natural reflexive rank 2n-2 coherent sheaf E, which is locally free away from the diagonal. The fiber of E,…

Given a combinatorial (semi-)model category $M$ and a set of morphisms $C$, we establish the existence of a semi-model category $L_C M$ satisfying the universal property of the left Bousfield localization in the category of semi-model…

We arrange morphisms and comorphisms of sites as the horizontal and vertical cells of a double category of sites; using the formalism of extensions and restrictions of presheaves, we explains how one can define a sheafification double…

How discriminative position information is for image classification depends on the data. On the one hand, the camera position is arbitrary and objects can appear anywhere in the image, arguing for translation invariance. At the same time,…

We prove an Atiyah-Segal isomorphism for the higher $K$-theory of coherent sheaves on quotient Deligne-Mumford stacks over $\C$. As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an…

Broadly speaking the present is a homotopy complement to the book of Giraud, albeit in a couple of different ways. In the first place there is a representability theorem for maps to a topological champ (a.k.a. stack) and whence an extremely…

We prove that, for closed exact embedded Lagrangian submanifolds of cotangent bundles, the homomorphism of homotopy groups induced by the stable Lagrangian Gauss map vanishes. In particular, we prove that this map is null-homotopic for all…