Related papers: The Relation between Ann-Categories and Ring Categ…

We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit…

The finitistic dimension of a triangulated category is introduced. For the category of perfect complexes over a ring it is shown that this dimension is finite if and only if the small finitistic dimension of the ring is finite.

A twisted ring is a ring endowed with a family of endomorphisms satisfying certain relations. One may then consider the notions of twisted module and twisted differential module. We study them and show that, under some general hypothesis,…

We introduce the concepts of dependence and independence in a very general framework. We use a concept of rank to study dependence and independence. By means of the rank we identify (total) dependence with inability to create more…

We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over…

We introduce a 3-dimensional categorical structure which we call intercategory. This is a kind of weak triple category with three kinds of arrows, three kinds of 2-dimensional cells and one kind of 3-dimensional cells. In one dimension, the…

We define the Chow ring of the classifying space of a linear algebraic group. In all the examples where we can compute it, such as the symmetric groups and the orthogonal groups, it is isomorphic to a natural quotient of the complex…

In a recent paper, Bruhn, Diestel, Kriesell and Wollan (arXiv:1003.3919) present four systems of axioms for infinite matroids, in terms of independent sets, bases, closure and circuits. No system of rank axioms is given. We give an easy…

In mathematical applications, category theory remains a contentious issue, with enthusiastic fans and a skeptical majority. In a muted form this split applies to the authors of this note. When we learned that the only mathematically sound…

In this paper we introduce and study the class of graded U-nil clean rings, as a generalization of graded nil-good class defined in [3]. We also investigate the transfer of the graded U-nil cleaness to matrix rings, and to graded group…

We consider a convenient category of "quadratic" multirings, that allows simple functorial relations with categories associated with abstract quadratic forms theories and shares many good aspects of the theories of Special Groups and of…

In what follows we generalize the notion of a complemented ring to rings that are not necessarily reduced. We then determine how our concepts fit in with other well-known classes of rings.

We examine the use of classes to formulate several categorical notions. This leads to two proposals: an explicit structure for working with subobjects, and a hierarchy of $k$-classes. We apply the latter to both ordinary and higher…

T*-categories are introduced as a ternary generalization of C*-categories. Their linking C*-categories are constructed and the Gelfand-Naimark representation theorems of Zettl for C*-ternary rings and for W*-ternary rings, are generalized…

We highlight the underlying category-theoretic structure of measures of information flow. We present an axiomatic framework in which communication systems are represented as morphisms, and information flow is characterized by its behavior…

The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…

Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…

Coclass theory can be used to define infinite families of finite p-groups of a fixed coclass. It is conjectured that the groups in one of these infinite families all have isomorphic mod-p cohomology rings. Here we prove that almost all…

Category theory provides an alternative to Hilbert's Formal Axiomatic method and goes beyond Mathematical Structuralism

We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality…