Recent zbMATH articles in MSC 18Chttps://www.zbmath.org/atom/cc/18C2021-04-16T16:22:00+00:00WerkzeugForking independence from the categorical point of view.https://www.zbmath.org/1456.030592021-04-16T16:22:00+00:00"Lieberman, Michael"https://www.zbmath.org/authors/?q=ai:lieberman.michael-j"Rosický, Jiří"https://www.zbmath.org/authors/?q=ai:rosicky.jiri"Vasey, Sebastien"https://www.zbmath.org/authors/?q=ai:vasey.sebastienThe model-theoretic notion of forking introduced by Shelah for classes of models axiomatized by stable first-order theories is a generalization of linear independence in vector spaces and algebraic independence in fields. In fact, (non)forking can be seen as a commutative diagram of embeddings also known as an amalgam.
In this paper, the broad model-theoretic framework of abstract elementary classes (AECs) is used. A \(\mu\)-AEC is simply an accessible category with all morphisms monomorphisms. The authors describe when a category has a `stable independence notion' -- a class of distinguished commutative squares that itself forms an accessible category -- and show that this is a purely category-theoretic axiomatization of forking in a \(\mu\)-AEC. This generalizes a result of [\textit{W. Boney} et al., Ann. Pure Appl. Logic 167, No. 7, 590--613 (2016; Zbl 1400.03060)] that characterized stable forking in the framework of AECs but depended on set-representations of their objects. The category \(\mathcal K_{\mathrm{reg}}\) of regular monomorphisms in a locally presentable coregular category \(\mathcal K\) that has effective unions, in the sense of Barr, is shown to have a stable independence notion, thus showing that forking occurs in both Grothendieck toposes and Grothendieck abelian categories.
Assuming a large cardinal axiom the authors also characterize when a stable independence notion exists in a \(\mu\)-AEC. Thus, it is established that model-theoretic stability is invariant under equivalence of categories.
Reviewer: Amit Kuber (Kanpur)Aspects of algebraic algebras.https://www.zbmath.org/1456.180062021-04-16T16:22:00+00:00"Hofmann, Dirk"https://www.zbmath.org/authors/?q=ai:hofmann.dirk"Sousa, Lurdes"https://www.zbmath.org/authors/?q=ai:sousa.lurdesSummary: In this paper we investigate important categories lying strictly between the Kleisli category and the Eilenberg-Moore category, for a Kock-Zöberlein monad on an order-enriched category. Firstly, we give a characterisation of free algebras in the spirit of domain theory. Secondly, we study the existence of weighted (co)limits, both on the abstract level and for specific categories of domain theory like the category of algebraic lattices. Finally, we apply these results to give a description of the idempotent split completion of the Kleisli category of the filter monad on the category of topological spaces.On a generalization of equilogical spaces.https://www.zbmath.org/1456.031102021-04-16T16:22:00+00:00"Pasquali, Fabio"https://www.zbmath.org/authors/?q=ai:pasquali.fabioSummary: We use the theory of triposes to prove that every (non-degenerate) locale \(\mathsf{H}\) is the set of truth values of a complete and co-complete quasi-topos into which the category of topological spaces embeds and the topos of sheaves over \(\mathsf{H}\) reflectively embeds.Algebraic dynamics.https://www.zbmath.org/1456.370132021-04-16T16:22:00+00:00"Manes, Ernie"https://www.zbmath.org/authors/?q=ai:manes.ernie-gSummary: Dynamical notions are introduced in arbitrary tight categories. The enveloping semigroup of \(X\) is the free object on one generator in the variety generated by \(X\). Two new examples are dynamical systems in which all spaces are countably tight and compact spaces which are homeomorphic to their square. All dynamic varieties have a universal minimal object. Comfort types are identified with certain singly-generated submonads of the ultrafilter monad.Purity of the Batanin monad. I.https://www.zbmath.org/1456.180052021-04-16T16:22:00+00:00"Penon, Jacques"https://www.zbmath.org/authors/?q=ai:penon.jacquesAmong the many approaches to defining weak \(\omega\)-categories is Batanin's definition that encodes the definition as a monad \(\mathbb{B}\) whose algebras are the weak \(\omega\)-categories. The work under review is the first of two parts, the second of which intends to produce examples of weak \(\omega\)-categories in Batanin's sense. Toward producing these examples the author requires some syntactic equipment to be in place on the monad \(\mathbb{B}\). In Part I the author considers a property of monads called purity. This is quite a strong property that is intimately linked to monads that arise in a syntactic manner from a language in the usual sense of logic.
While Part I is a prequel to a study of Batanin's monad, which is over globular sets, the material is presented in full generality and can be seen as independent of the particular application the author has in mind. In that sense, the work can be seen to introduce a new kind of monad that abstracts the syntactic structure of term construction over a language.
A concrete monad is a concrete category \(\mathbb{C}\) (over \(\mathbf{Set}\)) by means of a functor \(U\), together with a monad \(M\) on \(\mathbb{C}\). Such a monad is syntactic if it is equipped with functions \(L\colon UM(C)\to\mathbb{N}\), suitably natural, satisfying compatibility (in)equalities with regard to the monad unit and multiplication. Intuitively, \(L\) measures the construction length of an element, so that elements in the image of the unit have length \(1\), etc. There is also a condition that does not involve \(L\), namely that the evident function \(UM^{2}(C)\to UM(1)\times UM(C)\) (where \(1\) is a terminal object in \(\mathbb{C}\)) be injective. This immediately excludes monads whose algebras are strict categories such as the monad for strict \(\omega\)-categories on globular sets (as well as the free monoid monad on \(\mathbf{Set}\)). It appears that the notion of syntactic monad without this condition still makes sense and that this condition is presented early on primarily with a view toward the second part of the work.
A concrete cartesian monad is then defined to be a cartesian monad (i.e., the structure maps are cartesian) while \(\mathbb{C}\) is required to have all finite limits and \(U\) commutes with fibered products. Purity is then a property of a concrete cartesian syntactic monad. The ensuing 20 pages lead to the first example of a pure monad, a certain monad \(\mathbb{A}\) of trees on a language.
Section \(2\) is thus concerned with the beginnings of the construction of a pure monad. The section is quite dense with combinatorial definitions of the more or less standard way of viewing terms in a language of symbols with arities as trees. Most of the proofs are left as easy exercises. Section \(3\) proceeds to define, given a language, the concrete cartesian syntactic monad \(\mathbb{A}\) (via a lengthy process involving an operad of trees based on monoids). The final two sections introduce further combinatorial gadgets to the existing structure.
Reviewer: Ittay Weiss (Portsmouth)