Sheaf Theory through Examples (Abridged Version)

A Compositional Sheaf-Theoretic Framework for Event-Based Systems (Extended Version)

A compositional sheaf-theoretic framework for the modeling of complex event-based systems is presented. We show that event-based systems are machines, with inputs and outputs, and that they can be composed with machines of different types, all within a unified, sheaf-theoretic formalism. We take robotic systems as an exemplar of complex systems and rigorously describe actuators, sensors, and algorithms using this framework.

Sheaf representation of monoidal categories

Every small monoidal category with universal finite joins of central idempotents is monoidally equivalent to the category of global sections of a sheaf of local monoidal categories on a topological space. Every small stiff monoidal category monoidally embeds into such a category of global sections. An infinitary version of these theorems also holds in the spatial case. These representation results are functorial and subsume the Lambek-Moerdijk-Awodey sheaf representation for toposes, the Stone representation of Boolean algebras, and the Takahashi representation of Hilbert modules as continuous…

The quot functor of a quasi-coherent sheaf

We build an infinite dimensional scheme parametrizing isomorphism classes of coherent quotients of a quasi-coherent sheaf on a projective scheme. The main tool to achieve the construction is a version of Grothendieck's Grassmannian embedding combined with a result of Deligne, realizing quasi-coherent sheaves as ind-objects in the category of quasi-coherent sheaves of finite presentation. We end our treatment with the discussion of a special case in which we can retain an analog of the Grassmannian embedding.

Higher sheaf theory I: Correspondences

We prove a universal property for the $(\infty, n)$-category of correspondences, generalizing and providing a new proof for the case $n = 2$ from [GR17]. We also provide conditions under which a functor out of a higher category of correspondences of $\mathcal{C}$ can be extended to a higher category of correspondences of the free cocompletion of $\mathcal{C}$. These results will be used in the sequels to this paper to construct $(\infty, n)$-categorical versions of the theories of quasicoherent and ind-coherent sheaves in derived algebraic geometry.

Barr-正合范畴与软层表示

Barr-Exact Categories and Soft Sheaf Representations

本文用 Barr-正合范畴取代代数簇，基于 K-层概念在完全格上建立无点层表示，扩展了 Gehrke 和 van Gool 的软层理论，并应用于紧有序空间对偶及交换 Gelfand 环的表示。