Abelian categories are the most general category in which one can develop homological algebra. The idea and the name “abelian category” were first introduced by MacLane [Mac50], but the modern axiomitisation and first substantial applications were given by Grothendieck in his famous Tohoku paper [Gro57]. This paper was motivated by the needs of algebraic geometry, where the category of sheaves ...

Differential categories are now an established abstract setting for differentiation. The paper presents the parallel development for integration by axiomatizing an integral transformation, sA : !A→ !A⊗A, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense...

We introduce a method to lift monads on the base category of a fibration to its total category using codensity monads. This method, called codensity lifting, is applicable to various fibrations which were not supported by the categorical >>-lifting. After introducing the codensity lifting, we illustrate some examples of codensity liftings of monads along the fibrations from the category of preo...

‎in this paper first we define the morphism between geometric spaces in two different types‎. ‎we construct two categories of $uu$ and $l$ from geometric spaces then investigate some properties of the two categories‎, ‎for instance $uu$ is topological‎. ‎the relation between hypergroups and geometric spaces is studied‎. ‎by constructing the category $qh$ of $h_{v}$-groups we answer the question...

For a symmetric monoidal-closed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T,V)-algebra and show that various old and new structures are instances of such algebras. Lawvere’s presentation of a metric space as a V-category is included in our setting, via the Betti-Carboni-Street-Walters interpretation of a V-category as a monad...

• For square contingency tables with ordered categories, this paper proposes some distance subsymmetry models. The one model indicates that the cumulative probability that an observation will fall in row category i or below and column category i + k (k ≥ 2) or above, is equal to the probability that it falls in column category i or below and row category i+k or above. This paper also gives the ...

We describe a baseline system for the VideoCLEF Vid2RSS task. The system uses an unaltered off-the-shelf Information Retrieval system. ASR content is indexed using default stemming and stopping methods. The subject categories are populated by using the category label as a query on the collection, and assigning the retrieved items to that particular category. We describe the results of the syste...

The four-category concept identification task is analyzed as involving two single-cue subproblems. It is supposed that the two concurrent subproblems are worked on independently by the subject, and further, that subproblem learning is a probabilistic, all-or-nothing event. Various lines of evidence are drawn from the data to document these assertions. In particular, the theory successfully pred...

We describe a category of Feynman graphs and show how it relates to compact symmetric multicategories (coloured modular operads) just as linear orders relate to categories and rooted trees relate to multicategories. More specifically we obtain the following nerve theorem: compact symmetric multicategories can be characterised as presheaves on the category of Feynman graphs subject to a Segal co...

In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number ζ an abelian artinian category FS. We call its objects finite factorizable sheaves. They are certain infinite collections of perverse sheaves on configuration spaces, subject to a compatibility (”factorization”) and finiteness conditions. In Chapter 2 the tensor structure on FS is defined using funct...