Arrow Categories Research Articles

A relation algebraic approach to universal integrals

In a series of papers Klement et al. investigated discrete integrals such as the Choquet and Sugeno integral and their axiomatization. As part of their study they showed that universal integrals are based on semicopulas, and they provided lower and upper bounds of the integral operations based on a given semicopula. These real-valued resp. unit interval valued integrals can be considered as proper aggregation tool in the context of fuzzy sets. The aim of the current paper is to generalize this approach to so-called L-fuzzy sets and relations, i.e., fuzzy sets and relations that use an arbitrary Heyting algebra L as membership degree instead of the unit interval. Furthermore, we present the theory within arrow categories, i.e., we abstract from concrete sets and relations and work within a suitable algebraic framework. The current paper also shows that the results of the previous work can be proven without referring to the real numbers and specific measures such as the Lebesque measure and the induced measurable spaces.

Non-abelian and ε-curved homological algebra with arrow categories

Grandis's non-abelian homological algebra generalizes standard homological algebra in abelian categories to homological categories, which are a broader class of categories including for example the category of lattices and Galois connections. Here, we prove that if C is any category with an ideal of null morphisms with respect to which (co)kernels exist, then the arrow category of C is a homological category. This broadens the applicability of Grandis's framework substantially. In particular, one can form the homology of chain complexes in C by taking the homology objects to be morphisms of C, which one may think of as maps from an object of cycles to an object of chains modulo boundaries.One situation to which Grandis's original framework does not apply directly is ε-curved homological algebra. This refers to chain complexes of normed spaces whose differential squares to zero only approximately, in the sense that ‖d2‖≤ε for some ε>0. This is relevant for example in the theory of approximate representations of groups, where Kazhdan has successfully employed ε-curved homological techniques in an ad-hoc manner. We develop some basics of ε-curved homological algebra and note that our result on arrow categories facilitates the application of Grandis's theory.

Realisability problem in arrow categories

In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category $\mathcal{C}$ and for arbitrary groups $H\le G_1\times G_2$, is there an object $\phi \colon A_1 \rightarrow A_2$ in $\operatorname{Arr}(\mathcal{C})$ such that $\operatorname{Aut}_{\operatorname{Arr}(\mathcal{C})}(\phi) = H$, $\operatorname{Aut}_{\mathcal{C}}(A_1) = G_1$ and $\operatorname{Aut}_{\mathcal{C}}(A_2) = G_2$? We are interested in solving this problem when $\mathcal C =\mathcal{H}oTop_*$, the homotopy category of pointed topological spaces. To that purpose, we first settle that question in the positive when $\mathcal C = \mathcal{G}raphs$. Then, we construct an almost fully faithful functor from $\mathcal{G}raphs$ to $\operatorname{CDGA}$, the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when $\mathcal C = \operatorname{CDGA}$ and, as long as we work with finite groups, when $\mathcal C =\mathcal{H}oTop_*$. Some results on representability of concrete categories are also obtained.

Equipping weak equivalences with algebraic structure

We investigate the extent to which the weak equivalences in a model category can be equipped with algebraic structure. We prove, for instance, that there exists a monad T such that a morphism of topological spaces admits T-algebra structure if and only it is a weak homotopy equivalence. Likewise for quasi-isomorphisms and many other examples. The basic trick is to consider injectivity in arrow categories. Using algebraic injectivity and cone injectivity we obtain general results about the extent to which the weak equivalences in a combinatorial model category can be equipped with algebraic structure.

Chu connections and back diagonals between [formula omitted]-distributors

Chu connections and back diagonals are introduced as morphisms for distributors between categories enriched in a small quantaloid Q. These notions, meaningful for closed bicategories, dualize the constructions of arrow categories and the Freyd completion of categories. It is shown that, for a small quantaloid Q, the category of complete Q-categories and left adjoints is a retract of the dual of the category of Q-distributors and Chu connections, and it is dually equivalent to the category of Q-distributors and back diagonals. As an application of Chu connections, a postulation of the intuitive idea of reduction of formal contexts in the theory of formal concept analysis is presented, and a characterization of reducts of formal contexts is obtained.

Categories of relations for variable-basis fuzziness

Arrow categories establish a categorical and algebraic description of L-fuzzy relations, i.e., relations that use membership values from an arbitrary but fixed complete Heyting algebra L. With other words arrow categories describe the fixed-basis case. In this paper we are interested in the variable-basis case, i.e., the case where relations between different objects may use different membership values. We will investigate the structure of the collection of lattices of membership values within a given Dedekind category. This will lead to a complete characterization of the variable-basis case in this context.

Membership values in arrow categories

In this paper we consider an internal representation of the lattice of membership values used by the relations in an arbitrary arrow category. This is achieved by defining a more general construction that pairs regular with membership values in one object. We will show basic properties of this construction including its relationship to the lattice of membership values. This characterization of the lattice of membership values allows to specify and reason about the lattice elements as elements of an object of the category. Furthermore, this representation also leads to an abstract approach to higher-order fuzziness based on arrow categories.

Hochschild Cohomology with Support

In this paper, we investigate the functoriality properties of map-graded Hochschild complexes. We show that the category Map of map-graded categories is naturally a stack over the category of small categories endowed with a certain Grothendieck topology of 3-covers. For a related topology of |$\infty $|-covers on the Cartesian morphisms in Map, we prove that taking map-graded Hochschild complexes defines a sheaf. From the functoriality related to “injections” between map-graded categories, we obtain Hochschild complexes “with support”. We revisit Keller's arrow category argument from this perspective, and introduce and investigate a general Grothendieck construction which encompasses both the map-graded categories associated to presheaves of algebras and certain generalized arrow categories, which together constitute a pair of complementary tools for deconstructing Hochschild complexes.

A categorical framework for the transformation of object-oriented systems: Models and data

Refactoring of information systems is hard, for two reasons. On the one hand, large databases exist which have to be adjusted. On the other hand, many programs access those data. Data and programs all have to be migrated in a consistent manner such that their semantics does not change. This paper addresses the data part of the problem and introduces a model for object-oriented structures, describing the schema level with classes, associations, and inheritance as well as the instance level with objects and links. Positive Horn formulas based on predicates are used to formulate constraints to be obeyed by the schema and instance level, in order to reflect object-oriented structures. Homomorphisms are used for the typing of the instance level as well as for the description of refactorings which specify the addition, folding, and unfolding of schema elements. A categorial framework is presented which allows us to derive instance migrations from schema transformations in such a way that instances of the old schema are automatically migrated into instances of the new schema. The natural use of the pullback functor for unfolding is followed by an initial semantics approach: Instance migration is completed with the help of a co-adjoint functor on arrow categories.