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Given a regular cardinal
$\kappa $
such that
$\kappa ^{<\kappa }=\kappa $
(or any regular
$\kappa $
if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the
$\kappa $
-separable toposes. These are equivalent to sheaf toposes over a site with
$\kappa $
-small limits that has at most
$\kappa $
many objects and morphisms, the (basis for the) topology being generated by at most
$\kappa $
many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough
$\kappa $
-points, that is, points whose inverse image preserve all
$\kappa $
-small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when
$\kappa =\omega $
, when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call
$\kappa $
-geometric, where conjunctions of less than
$\kappa $
formulas and existential quantification on less than
$\kappa $
many variables is allowed. We prove that
$\kappa $
-geometric theories have a
$\kappa $
-classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to
$\kappa $
-geometric morphisms (geometric morphisms the inverse image of which preserves all
$\kappa $
-small limits) into that topos. Moreover, we prove that
$\kappa $
-separable toposes occur as the
$\kappa $
-classifying toposes of
$\kappa $
-geometric theories of at most
$\kappa $
many axioms in canonical form, and that every such
$\kappa $
-classifying topos is
$\kappa $
-separable. Finally, we consider the case when
$\kappa $
is weakly compact and study the
$\kappa $
-classifying topos of a
$\kappa $
-coherent theory (with at most
$\kappa $
many axioms), that is, a theory where only disjunction of less than
$\kappa $
formulas are allowed, obtaining a version of Deligne’s theorem for
$\kappa $
-coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic.
Using the category of metric spaces as a template, we develop a metric analogue of the categorical semantics of classical/intuitionistic logic, and show that the natural notion of predicate in this “continuous semantics” is equivalent to the a priori separate notion of predicate in continuous logic, a logic which is independently well-studied by model theorists and which finds various applications. We show this equivalence by exhibiting the real interval
$[0,1]$
in the category of metric spaces as a “continuous subobject classifier” giving a correspondence not only between the two notions of predicate, but also between the natural notion of quantification in the continuous semantics and the existing notion of quantification in continuous logic.
Along the way, we formulate what it means for a given category to behave like the category of metric spaces, and afterwards show that any such category supports the aforementioned continuous semantics. As an application, we show that categories of presheaves of metric spaces are examples of such, and in fact even possess continuous subobject classifiers.
In this short paper, we combine the representability theorem introduced in [Porta and Yu, Representability theorem in derived analytic geometry, preprint, 2017, arXiv:1704.01683; Porta and Yu, Derived Hom spaces in rigid analytic geometry, preprint, 2018, arXiv:1801.07730] with the theory of derived formal models introduced in [António,
$p$
-adic derived formal geometry and derived Raynaud localization theorem, preprint, 2018, arXiv:1805.03302] to prove the existence representability of the derived Hilbert space
$\mathbf{R}\text{Hilb}(X)$
for a separated
$k$
-analytic space
$X$
. Such representability results rely on a localization theorem stating that if
$\mathfrak{X}$
is a quasi-compact and quasi-separated formal scheme, then the
$\infty$
-category
$\text{Coh}^{-}(\mathfrak{X}^{\text{rig}})$
of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the
$\infty$
-category
$\text{Coh}^{-}(\mathfrak{X})$
. Along the way, we prove several results concerning the
$\infty$
-categories of formal models for almost perfect modules on derived
$k$
-analytic spaces.
Given an action
${\varphi }$
of inverse semigroup S on a ring A (with domain of
${\varphi }(s)$
denoted by
$D_{s^*}$
), we show that if the ideals
$D_e$
, with e an idempotent, are unital, then the skew inverse semigroup ring
$A\rtimes S$
can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.
Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics (QM) that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, ‘Q-worlds’. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.
In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects (i.e., objects satisfying the axiom of choice). Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS.
Kaplansky introduced the notions of CCR and GCR
$C^{\ast }$
-algebras, because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR
$C^{\ast }$
-algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.
Let X be a monoid scheme. We will show that the stalk at any point of X defines a point of the topos of quasi-coherent sheaves over X. As it turns out, every topos point of is of this form if X satisfies some finiteness conditions. In particular, it suffices for M/M× to be finitely generated when X is affine, where M× is the group of invertible elements.
This allows us to prove that two quasi-projective monoid schemes X and Y are isomorphic if and only if and are equivalent.
The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani [3]. We will study the topos points of free commutative monoids and show that already for ℕ∞, there are ‘hidden’ points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting ‘geometry of monoids’.
In this paper, we introduce the notion of the equivalence relation, called n-isoclinism, between crossed modules of groups, and give some basic properties of this notion. In particular, we obtain some criteria under which crossed modules are n-isoclinic. Also, we present the notion of n-stem crossed module and, under some conditions, determine them within an n-isoclinism class.
For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.
Let $S|_{R}$ be a groupoid Galois extension with Galois groupoid $G$ such that $E_{g}^{G_{r(g)}}\subseteq C1_{g}$, for all $g\in G$, where $C$ is the centre of $S$, $G_{r(g)}$ is the principal group associated to $r(g)$ and $\{E_{g}\}_{g\in G}$ are the ideals of $S$. We give a complete characterisation in terms of a partial isomorphism groupoid for such extensions, showing that $G=\dot{\bigcup }_{g\in G}\text{Isom}_{R}(E_{g^{-1}},E_{g})$ if and only if $E_{g}$ is a connected commutative algebra or $E_{g}=E_{g}^{G_{r(g)}}\oplus E_{g}^{G_{r(g)}}$, where $E_{g}^{G_{r(g)}}$ is connected, for all $g\in G$.
The aim of the present paper is to extend the dualizing object approach to Stone duality to the noncommutative setting of skew Boolean algebras. This continues the study of noncommutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In this paper we construct a series of dual adjunctions between the categories of left-handed skew Boolean algebras and Boolean spaces, the unital versions of which are induced by dualizing objects $\{ 0, 1, \ldots , n+ 1\} $, $n\geq 0$. We describe the categories of Eilenberg-Moore algebras of the monads of the adjunctions and construct easily understood noncommutative reflections of left-handed skew Boolean algebras, where the latter can be faithfully embedded (if $n\geq 1$) in a canonical way. As an application, we answer the question that arose in a recent paper by Leech and Spinks to describe the left adjoint to their ‘twisted product’ functor $\omega $.
Paterson showed how to construct an étale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz’s construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear representations of inverse semigroups can be constructed from the groups occurring in the associated topological groupoid.
It follows from methods of B. Steinberg, extended to inverse categories, that finite inverse category algebras are isomorphic to their associated groupoid algebras; in particular, they are symmetric algebras with canonical symmetrizing forms.We deduce the existence of transfer maps in cohomology and Hochschild cohomology from certain inverse subcategories. This is in part motivated by the observation that, for certain categories , being a Mackey functor on is equivalent to being extendible to a suitable inverse category containing . We further show that extensions of inverse categories by abelian groups are again inverse categories.
The main purpose of this paper is to develop a point-free notion of topological transitivity. First, we define transitive frame maps and transitive completely prime filters in Frm, the category of frames and frame maps. Then we discuss the relationship between these notions in Frm and the notions of topological transitive and transitive points in Top. Finally, we investigate the relationship between transitive frame maps and the existence of transitive completely prime filters.
We describe measurable Hilbert sheaves as Hilbert space objects in a sheaf category constructed from a measure space. These are quite useful for the interpretation of the direct integral of Hilbert spaces as an indexed functor. We set up a framework to put this and similar constructions of operator theory on an indexed categorical footing.
Let C be a small category with weak finite limits, and let Flat(C) be the category of flat functors from C to the category of small sets. We prove that the free exact completion of C is the category of set-valued functors of Flat (C) which preserve small products and filtered colimits. In case C has finite limits, this gives A. Carboni and R. C. Magno's result on the free exact completion of a small category with finite limits.
Let Cb be the admissible functorial quasi-uniformity on the completely regular bispaces which is spanned by the upper quasi-uniformity on the real line. Answering a question posed by B. Banaschewski and G. C. L. Brümmer in the affirmative we show that CbX is transitive for every strongly zero-dimensional bispace X.