Weak tensor products of complete atomistic lattices

Abstract.: Given two complete atomistic lattices $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ , we define a set $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ of complete atomistic lattices by means of three axioms (natural regarding the description of separated quantum compound systems), or in terms of a universal property with respect to a given class of bimorphisms. We call the elements of $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ weak tensor products of $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ . We prove that $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ is a complete lattice. We compare the bottom element $${\mathcal{L}}_{1}$$ $${\mathcal{L}}_{2}$$ with the separated product of Aerts and with the box product of Grätzer and Wehrung. Similarly, we compare the top element $${\mathcal{L}}_{1}$$ $${\mathcal{L}}_{2}$$ with the tensor products of Fraser, Chu and Shmuely. With some additional hypotheses on $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ (true for instance if $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ are moreover irreducible, orthocomplemented and with the covering property), we characterize the automorphisms of weak tensor products in terms of those of $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$

Tensor products of topological abelian groups and Pontryagin duality

Let $G$ be the group of all \ZZ-valued homomorphisms of the Baer-Specker group \ZZ^\NN. The group $G$ is algebraically isomorphic to \ZZ^{(\NN)}, the infinite direct sum of the group of integers, and equipped with the topology of pointwise convergence on \ZZ^\NN, becomes a non reflexive prodiscrete group. It was an open question to find its dual group $\widehat{G}$. Here, we answer this question by proving that $\widehat{G}$ is topologically isomorphic to \ZZ^\NN\otimes_\mathcal{Q}\TT, the (locally quasi-convex) tensor product of \ZZ^\NN and \TT. Furthermore, we investigate the reflexivity properties of the groups of C_p(X,\ZZ), the group of all \ZZ-valued continuous functions on $X$ equipped with the pointwise convergence topology, and $A_p(X)$, the free abelian group on a $0$-dimensional space $X$ equipped with the topology t_p(C(X,\ZZ)) of pointwise convergence topology on C(X,\ZZ). In particular, we prove that \widehat{A_p(X)}\simeq C_p(X,\ZZ)\otimes_\mathcal{Q}\TT and we establish the existence of $0$-dimensional spaces $X$ such that C_p(X,\ZZ) is Pontryagin reflexive

Topological tensor product of bimodules, complete Hopf Algebroids and convolution algebras

Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the Appendix we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.Comment: Minor changes, 33 pages. To appear in CC

Deformation Theory and Partition Lie Algebras

A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and differential graded Lie algebras in characteristic zero, thereby formalising a well-known principle in deformation theory. We introduce a variant of differential graded Lie algebras, called partition Lie algebras, in arbitrary characteristic. We then explicitly compute the homotopy groups of free algebras, which parametrise operations. Finally, we prove generalisations of the Lurie-Pridham correspondence classifying formal moduli problems via partition Lie algebras over an arbitrary field, as well as over a complete local base.Comment: 89 page

Hopf Structures and Duality

info:eu-repo/semantics/nonPublishe

Categorical Foundations for K-theory

K-Theory was originally defined by Grothendieck as a contravariant functor from a subcategory of schemes to abelian groups, known today as K0. The same kind of construction was then applied to other fields of mathematics, like spaces and (not necessarily commutative) rings. In all these cases, it consists of some process applied, not directly to the object one wants to study, but to some category related to it: the category of vector bundles over a space, of finitely generated projective modules over a ring, of locally free modules over a scheme, for instance. Later, Quillen extracted axioms that all these categories satisfy and that allow the Grothendieck construction of K0. The categorical structure he discovered is called today a Quillen-exact category. It led him not only to broaden the domain of application of K-theory, but also to define a whole K-theory spectrum associated to such a category. Waldhausen next generalized Quillen's notion of an exact category by introducing categories with weak equivalences and cofibrations, which one nowadays calls Waldhausen categories. K-theory has since been studied as a functor from the category of suitably structured (Quillen-exact, Waldhausen, symmetric monoidal) small categories to some category of spectra1. This has given rise to a huge field of research, so much so that there is a whole journal devoted to the subject. In this thesis, we want to take advantage of these tools to begin studying K-theory from another perspective. Indeed, we have the impression that, in the generalization of topological and algebraic K-theory that has been started by Quillen, something important has been left aside. K-theory was initiated as a (contravariant) functor from the various categories of spaces, rings, schemes, …, not from the category of Waldhausen small categories. Of course, one obtains information about a ring by studying its Quillen-exact category of (finitely generated projective) modules, but still, the final goal is the study of the ring, and, more globally, of the category of rings. Thus, in a general theory, one should describe a way to associate not only a spectrum to a structured category, but also a structured category to an object. Moreover, this process should take the morphisms of these objects into account. This gives rise to two fundamental questions. What kind of mathematical objects should K-theory be applied to? Given such an object, what category "over it" should one consider and how does it vary over morphisms? Considering examples, we have made the following observations. Suppose C is the category that is to be investigated by means of K-theory, like the category of topological spaces or of schemes, for instance. The category associated to an object of C is a sub-category of the category of modules over some monoid in a monoidal category with additional structure (topological, symmetric, abelian, model). The situation is highly "fibred": not only morphisms of C induce (structured) functors between these sub-categories of modules, but the monoidal category in which theses modules take place might vary from one object of C to another. In important cases, the sub-categories of modules considered are full sub-categories of "locally trivial" modules with respect to some (possibly weakened notion of) Grothendieck topology on C . That is, there are some specific modules that are considered sufficiently simple to be called trivial and locally trivial modules are those that are, locally over a covering of the Grothendieck topology, isomorphic to these. In this thesis, we explore, with K-theory in view, a categorical framework that encodes these kind of data. We also study these structures for their own sake, and give examples in other fields. We do not mention in this abstract set-theoretical issues, but they are handled with care in the discussion. Moreover, an appendix is devoted to the subject. After recalling classical facts of Grothendieck fibrations (and their associated indexed categories), we provide new insights into the concept of a bifibration. We prove that there is a 2-equivalence between the 2-category of bifibrations over a category ℬ and a 2-category of pseudo double functors from ℬ into the double category of adjunctions in CAT. We next turn our attention to composable pairs of fibrations , as they happen to be fundamental objects of the theory. We give a characterization of these objects in terms of pseudo-functors ℬop → FIBc into the 2-category of fibrations and Cartesian functors. We next turn to a short survey about Grothendieck (pre-)topologies. We start with the basic notion of covering function, that associate to each object of a category a family of coverings of the object. We study separately the saturation of a covering function with respect to sieves and to refinements. The Grothendieck topology generated by a pretopology is shown to be the result of these two steps. We define then, inspired by Street [89], the notion of (locally) trivial objects in a fibred category P : ℰ → ℬ equipped with some notion of covering of objects of the base ℬ. The trivial objects are objects chosen in some fibres. An object E in the fibre over B ∈ ℬ is locally trivial if there exists a covering {fi : Bi → B}i ∈ I such the inverse image of E along fi is isomorphic to a trivial object. Among examples are torsors, principal bundles, vector bundles, schemes, locally constant sheaves, quasi-coherent and locally free sheaves of modules, finitely generated projective modules over commutative rings, topological manifolds, … We give conditions under which locally trivial objects form a subfibration of P and describe the relationship between locally trivial objects with respect to subordinated covering functions. We then go into the algebraic part of the theory. We give a definition of monoidal fibred categories and show a 2-equivalence with monoidal indexed categories. We develop algebra (monoids and modules) in these two settings. Modules and monoids in a monoidal fibred category ℰ → ℬ happen to form a pair of fibrations . We end this thesis by explaining how to apply this categorical framework to K-theory and by proposing some prospects of research. ______________________________ 1 Works of Lurie, Toën and Vezzosi have shown that K-theory really depends on the (∞, 1)-category associated to a Waldhausen category [94]. Moreover, topological K-theory of spaces and Banach algebras takes the fact that the Waldhausen category is topological in account [62, 70]