Authors:Chris Heunen; Vaia Patta Pages: 205 - 237 Abstract: The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful and having a nearly full Kan extension; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial. Finally, a characterization of matroids in terms of optimality of the greedy algorithm can be rephrased in terms of limits. PubDate: 2018-04-01 DOI: 10.1007/s10485-017-9490-2 Issue No:Vol. 26, No. 2 (2018)

Authors:Martín Ortiz-Morales Pages: 239 - 285 Abstract: Quasi-hereditary algebras were introduced by E. Cline, B. Parshall and L. Scott in order to deal with highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups. These categories have been a very important tool in the study of finite-dimensional algebras. On the other hand, functor categories were introduced in representation theory by M. Auslander, and used in his proof of the first Brauer–Thrall conjecture and later used systematically in his joint work with I. Reiten on stable equivalence, as well as many other applications. Recently, functor categories were used by Martínez-Villa and Solberg to study the Auslander–Reiten components of finite-dimensional algebras. The aim of the paper is to introduce the concept of quasi-hereditary category. We can think of the Auslander–Reiten components as quasi-hereditary categories. In this way, we have applications to the functor category \(\mathrm {Mod}(\mathcal {C} )\) , with \(\mathcal C\) a component of the Auslander–Reiten quiver. PubDate: 2018-04-01 DOI: 10.1007/s10485-017-9493-z Issue No:Vol. 26, No. 2 (2018)

Authors:Martin Brandenburg Pages: 287 - 308 Abstract: Let k be a commutative \(\mathbb {Q}\) -algebra. We study families of functors between categories of finitely generated modules which are defined for all commutative k-algebras simultaneously and are compatible with base changes. These operations turn out to be Schur functors associated to k-linear representations of symmetric groups. This result is closely related to Macdonald’s classification of polynomial functors. PubDate: 2018-04-01 DOI: 10.1007/s10485-017-9494-y Issue No:Vol. 26, No. 2 (2018)

Authors:Pedro Nicolás; Manuel Saorín Pages: 309 - 368 Abstract: Given small dg categories A and B and a B-A-bimodule T, we give necessary and sufficient conditions for the associated derived functors of Hom and the tensor product to be fully faithful. Special emphasis is put on the case when RHom \(_\mathrm{A}\) (T,') is fully faithful and preserves compact objects, in which case nice recollements situations appear. It is also shown that, given an algebraic compactly generated triangulated category D, all compactly generated co-smashing triangulated subcategories which contain the compact objects appear as the image of such a RHom \(_\mathrm{A}\) (T,'). The results are then applied to the case when A and B are ordinary algebras, comparing the situation with the well-stablished tilting theory of modules. In this way we recover and extend recent results by Bazzoni–Mantese–Tonolo, Chen-Xi and D. Yang. PubDate: 2018-04-01 DOI: 10.1007/s10485-017-9495-x Issue No:Vol. 26, No. 2 (2018)

Authors:Rory B. B. Lucyshyn-Wright Pages: 369 - 400 Abstract: Certain axiomatic notions of affine space over a ring and convex space over a preordered ring are examples of the notion of \(\mathscr {T}\) -algebra for an algebraic theory \(\mathscr {T}\) in the sense of Lawvere. Herein we study the notion of commutant for Lawvere theories that was defined by Wraith and generalizes the notion of centralizer clone. We focus on the Lawvere theory of left R -affine spaces for a ring or rig R, proving that this theory can be described as a commutant of the theory of pointed right R-modules. Further, we show that for a wide class of rigs R that includes all rings, these theories are commutants of one another in the full finitary theory of R in the category of sets. We define left R -convex spaces for a preordered ring R as left affine spaces over the positive part \(R_+\) of R. We show that for any firmly archimedean preordered algebra R over the dyadic rationals, the theories of left R-convex spaces and pointed right \(R_+\) -modules are commutants of one another within the full finitary theory of \(R_+\) in the category of sets. Applied to the ring of real numbers \(\mathbb {R}\) , this result shows that the connection between convex spaces and pointed \(\mathbb {R}_+\) -modules that is implicit in the integral representation of probability measures is a perfect ‘duality’ of algebraic theories. PubDate: 2018-04-01 DOI: 10.1007/s10485-017-9496-9 Issue No:Vol. 26, No. 2 (2018)

Authors:Dragan Mašulović Pages: 401 - 412 Abstract: In this paper we give a new proof of the Nešetřil–Rödl Theorem, a deep result of discrete mathematics which is one of the cornerstones of the structural Ramsey theory. In contrast to the well-known proofs which employ intricate combinatorial strategies, this proof is spelled out in the language of category theory and the main result follows by applying several simple categorical constructions. The gain from the approach we present here is that, instead of giving the proof in the form of a large combinatorial construction, we can start from a few building blocks and then combine them into the final proof using general principles. PubDate: 2018-04-01 DOI: 10.1007/s10485-017-9500-4 Issue No:Vol. 26, No. 2 (2018)

Authors:Jan Grebík Pages: 1 - 6 Abstract: We disprove a conjecture from Kubiś and Mašulović [2] by showing the existence of a Fraïssé class \(\mathcal {C}\) which does not admit a Katětov functor. On the other hand, we show that the automorphism group of the Fraïssé limit of \(\mathcal {C}\) is universal, as it happens in the presence of a Katětov functor. PubDate: 2018-02-01 DOI: 10.1007/s10485-016-9469-4 Issue No:Vol. 26, No. 1 (2018)

Authors:George Janelidze; Ross Street Abstract: We make various observations on infinitary addition in the context of the series monoids introduced in our previous paper on real sets. In particular, we explore additional conditions on such monoids suggested by Tarski’s Arithmetic of Cardinal Algebras, and present a monad-theoretic construction that generalizes our construction of paradoxical real numbers. PubDate: 2018-04-18 DOI: 10.1007/s10485-018-9524-4

Authors:John Frith; Anneliese Schauerte Abstract: The congruence lattice of a frame has long been an object of considerable interest, not least because it turns out to be a frame itself. Perhaps more surprisingly congruence lattices of, for instance, \(\sigma \) -frames, \(\kappa \) -frames and some partial frames also turn out to be frames. The situation for congruences of a meet-semilattice is notably different. In this paper we analyze the meet-semilattice congruence lattices of arbitrary frames and compare them with the corresponding lattices of frame congruences. In the course of this, we provide a structure theorem as well as many examples and counter-examples. PubDate: 2018-04-13 DOI: 10.1007/s10485-018-9521-7

Authors:Dikran Dikranjan; Hans-Peter A. Künzi Abstract: We study cowellpoweredness in the category \(\mathbf{QUnif}\) of quasi-uniform spaces and uniformly continuous maps. A full subcategory \(\mathcal{A}\) of \(\mathbf{QUnif}\) is cowellpowered when the cardinality of the codomains of any class of epimorphisms in \(\mathcal{A}\) , with a fixed common domain, is bounded. We use closure operators in the sense of Dikranjan–Giuli–Tholen which provide a convenient tool for describing the subcategories \(\mathcal{A}\) of \(\mathbf{QUnif}\) and their epimorphisms. Some of the results are obtained by using the knowledge of closure operators, epimorphisms and cowellpoweredness of subcategories of the category \(\mathbf{Top}\) of topological spaces and continuous maps. The transfer is realized by lifting these subcategories along the forgetful functor \(T{:}\mathbf{QUnif}\rightarrow \mathbf{Top}\) and studying when epimorphisms and cowellpoweredness are preserved by the lifting. In other cases closure operators of \(\mathbf{QUnif}\) are used to provide specific results for \(\mathbf{QUnif}\) that have no counterpart in \(\mathbf{Top}\) . This leads to a wealth of cowellpowered categories and a wealth of non-cowellpowered categories of quasi-uniform spaces, in contrast with the current situation in the case of the smaller category \(\mathbf{Unif}\) of uniform spaces, where no example of a non-cowellpowered subcategory is known so far. Finally, we present our main example: a non-cowellpowered full subcategory of \(\mathbf{QUnif}\) which is the intersection of two “symmetric” cowellpowered full subcategories of \(\mathbf{QUnif}\) . PubDate: 2018-04-07 DOI: 10.1007/s10485-018-9523-5

Authors:Arthur J. Parzygnat Abstract: Given a representation of a \(C^*\) -algebra, thought of as an abstract collection of physical observables, together with a unit vector, one obtains a state on the algebra via restriction. We show that the Gelfand–Naimark–Segal (GNS) construction furnishes a left adjoint of this restriction. To properly formulate this adjoint, it must be viewed as a weak natural transformation, a 1-morphism in a suitable 2-category, rather than as a functor between categories. Weak naturality encodes the functoriality and the universal property of adjunctions encodes the characterizing features of the GNS construction. Mathematical definitions and results are accompanied by physical interpretations. PubDate: 2018-03-23 DOI: 10.1007/s10485-018-9522-6

Authors:B. Banaschewski Abstract: For the \(\ell \) -ring F(L) introduced in Karimi Feizabadi et al. (Categ Gen Algebr Struct Appl 5:85–102, 2016) and then shown to have an embedding into the familiar \(\ell \) -ring \({\mathfrak R}L\) of all real-valued continuous function on a frame L, the resulting image \({\mathfrak S}L\) in \({\mathfrak R}L\) is characterized here by internal properties within \({\mathfrak R}L\) . Further, a number of results concerning the \({\mathfrak S}L\) are obtained on the basis of this characterization. PubDate: 2018-03-19 DOI: 10.1007/s10485-018-9514-6

Authors:Stefan Schröer Abstract: We show that each sheaf of modules admits a flasque hull, such that homomorphisms into flasque sheaves factor over the flasque hull. On the other hand, we give examples of modules over non-noetherian rings that do not inject into flasque modules. This reveals the impossibility to extend the proof of Serre’s vanishing result for affine schemes with flasque quasicoherent resolutions to the non-noetherian setting. However, we outline how hypercoverings can be used for a reduction to the noetherian case. PubDate: 2018-03-08 DOI: 10.1007/s10485-018-9520-8

Authors:Christian Pech; Maja Pech Abstract: Fraïssé’s theorem characterizing the existence of universal homogeneous structures is a cornerstone of model theory. A categorical version of these results was developed by Droste and Göbel. Such an abstract version of Fraïssé theory allows to construct unusual objects that are far away from the usual structures. In this paper we are going to derive sufficient conditions for a comma category to contain universal homogeneous objects. Using this criterion, we characterize homogeneous structures that possess universal homogeneous endomorphisms. The existence of such endomorphisms helps to reduce questions about the full endomorphism monoid to the self-embedding monoid of the structure. As another application we characterize the retracts of homogeneous structures that are induced by universal homogeneous retractions. This extends previous results by Bonato, Delić, Mudrinski, Dolinka, and Kubiś. PubDate: 2018-02-28 DOI: 10.1007/s10485-018-9519-1

Authors:Matthew Burke Abstract: We formulate and prove a generalisation of Lie’s second theorem that integrates homomorphisms between formal group laws to homomorphisms between Lie groups. Firstly we generalise classical Lie theory by replacing groups with groupoids. Secondly we include groupoids whose underlying spaces are not smooth manifolds. The main intended application is when we replace the category of smooth manifolds with a well-adapted model of synthetic differential geometry. In addition we provide an axiomatic system that provides all the abstract structure that is required to prove Lie’s second theorem. PubDate: 2018-02-27 DOI: 10.1007/s10485-018-9518-2

Authors:Maria Manuel Clementino; Jorge Picado; Aleš Pultr Abstract: Sublocales of a locale (frame, generalized space) can be equivalently represented by frame congruences. In this paper we discuss, a.o., the sublocales corresponding to complete congruences, that is, to frame congruences which are closed under arbitrary meets, and present a “geometric” condition for a sublocale to be complete. To this end we make use of a certain closure operator on the coframe of sublocales that allows not only to formulate the condition but also to analyze certain weak separation properties akin to subfitness or \(T_1\) . Trivially, every open sublocale is complete. We specify a very wide class of frames, containing all the subfit ones, where there are no others. In consequence, e.g., in this class of frames, complete homomorphisms are automatically Heyting. PubDate: 2018-02-13 DOI: 10.1007/s10485-018-9516-4

Authors:Shilong Zhang; Li Guo; William Keigher Abstract: In a previous study, the algebraic formulation of the First Fundamental Theorem of Calculus (FFTC) is shown to allow extensions of differential and Rota–Baxter operators on the one hand, and to give rise to categorical explanations using the ideas of liftings of monads and comonads, and mixed distributive laws on the other. Generalizing the FFTC, we consider in this paper a class of constraints between a differential operator and a Rota–Baxter operator. For a given constraint, we show that the existences of extensions of differential and Rota–Baxter operators, of liftings of monads and comonads, and of mixed distributive laws are equivalent. PubDate: 2018-02-13 DOI: 10.1007/s10485-018-9517-3

Authors:Alan S. Cigoli; Tomas Everaert; Marino Gran Abstract: Given an exact category \({\mathcal {C}}\) , it is well known that the connected component reflector \( \pi _0 :\mathsf {Gpd}(\mathcal {C}) \rightarrow \mathcal {C}\) from the category \(\mathsf {Gpd}(\mathcal {C})\) of internal groupoids in \(\mathcal {C}\) to the base category \(\mathcal {C}\) is semi-left-exact. In this article we investigate the existence of a monotone-light factorization system associated with this reflector. We show that, in general, there is no monotone-light factorization system \((\mathcal {E}',\mathcal {M}^*)\) in \(\mathsf {Gpd}\) ( \(\mathcal {C}\) ), where \(\mathcal {M}^*\) is the class of coverings in the sense of the corresponding Galois theory. However, when restricting to the case where \(\mathcal {C}\) is an exact Mal’tsev category, we show that the so-called comprehensive factorization of regular epimorphisms in \(\mathsf {Gpd}\) ( \(\mathcal {C}\) ) is the relative monotone-light factorization system (in the sense of Chikhladze) in the category \(\mathsf {Gpd}\) ( \(\mathcal {C}\) ) corresponding to the connected component reflector, where \(\mathcal {E}'\) is the class of final functors and \( \mathcal {M}^*\) the class of regular epimorphic discrete fibrations. PubDate: 2018-02-12 DOI: 10.1007/s10485-018-9515-5

Authors:Walter Tholen Abstract: When replacing the non-negative real numbers with their addition by a commutative quantale \(\mathsf{V}\) , under a metric lens one may then view small \(\mathsf{V}\) -categories as sets that come with a \(\mathsf{V}\) -valued distance function. The ensuing category \(\mathsf{V}\text {-}\mathbf{Cat}\) is well known to be a concrete topological category that is symmetric monoidal closed. In this paper we show which concrete symmetric monoidal-closed topological categories may be fully and bireflectively embedded into \(\mathsf{V}\text {-}\mathbf{Cat}\) , for some \(\mathsf{V}\) . PubDate: 2018-02-12 DOI: 10.1007/s10485-018-9513-7

Authors:Ben Berckmoes Abstract: We establish an approach theoretic version of Anscombe’s theorem, which we apply to justify the use of confidence intervals based on the sample mean after a group sequential trial. PubDate: 2018-02-09 DOI: 10.1007/s10485-018-9512-8