Monoid Properties as Invariants of Toposes of Monoid Actions

Toposes of Discrete Monoid Actions

Properties of toposes of right $M$-sets are studied, and these toposes are characterised up to equivalence by their canonical points. The solution to the corresponding Morita equivalence problem is presented in the form of an equivalence between a 2-category of monoids and the corresponding 2-category of toposes.

Toposes of Monoid Actions

We study toposes of actions of monoids on sets. We begin with ordinary actions, producing a class of presheaf toposes which we characterize. As groundwork for considering topological monoids, we branch out into a study of supercompactly generated toposes (a class strictly larger than presheaf toposes). This enables us to efficiently study and characterize toposes of continuous actions of topological monoids on sets, where the latter are viewed as discrete spaces. Finally, we refine this characterization into necessary and sufficient conditions for a supercompactly generated topos to be equival…

拓扑幺半群作用的拓扑斯

Toposes of Topological Monoid Actions

本文证明拓扑幺半群在离散空间上的连续作用范畴是格罗滕迪克拓扑斯，解决相应莫里塔等价问题，并用典范点刻画这些拓扑斯，识别出粉末幺半群和完备幺半群等自然代表类，进一步证明该构造具有（2-）函子性。

Geometric morphisms between toposes of monoid actions: factorization systems

Let M, N be monoids, and PSh(M), PSh(N) their respective categories of right actions on sets. In this paper, we systematically investigate correspondences between properties of geometric morphisms PSh(M) $\rightarrow$ PSh(N) and properties of the semigroup homomorphisms M $\rightarrow$ N or flat-left-N-right-M-sets inducing them. More specifically, we consider properties of geometric morphisms featuring in factorization systems, namely: surjections, inclusions, localic morphisms, hyperconnected morphisms, terminal-connected morphisms, {é}tale morphisms, pure morphisms and complete spreads. We …

非交换Gelfand对偶：布尔局部分离拓扑与单调完备$C^{*}$ -范畴

Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete $C^{*}$-categories

对于任何布尔拓扑，我们都可以将其内部希尔伯特空间的范畴联系起来，如果拓扑是局部分离的，我们可以考虑平方可积〔的〕的满子范畴希尔伯特空间。在这两种情况下，它都是对称monoidal单调完备$C^{*}$范畴。我们将证明任何布尔局部分离拓扑都可以被重构为“非退化”monoidal正规$*$ -表示的分类拓扑，它既是它的内部Hilbert空间范畴，也是它的平方可积〔的〕希尔伯特空间.这表明一个可能的扩展通常Gelfand对偶之间的一类toposes（或更一般的局部堆栈或局部广群）和一类对称monoidal $C^{*}$ -类别尚未发现。

拓扑的几何不变量的场地刻画

Site characterizations for geometric invariants of toposes

我们讨论的问题的特点的Grothendieck拓扑的属性，以满足一个给定的“几何”不变量作为其网站的定义的属性，并指出了一套通用的技术，建立这样的标准。然后，我们将我们的方法应用于特定的不变量，特别是包括Grothendieck拓扑的属性是本地的（分别）。原子，局部连接，相当于一个预层拓扑），获得明确的网站特征。