The in…nite combinatorics here give statements in which, from some sequence, an in…nite subsequence will satisfy some condition – for example, belong to some speci…ed set. Our results give such statements generically –that is, for ‘nearly all’points, or as we shall say, for quasi all points –all o¤ a null set in the measure case, or all o¤ a meagre set in the category case. The prototypical res...

in this article, we have shown, for the add-point monad t, thepartial morphism category set*is isomorphic to the kleisli category sett. alsowe have proved that the category, sett, of t-algebras is isomorphic to thecategory set of pointed sets. finally we have established commutative squaresinvolving these categories.

Crop seeds are important sources of protein, oil, and carbohydrates for food, animal feeds, and industrial products. Recently, much attention has been paid to quality and functional properties of crop seeds. However, seed traits possess some distinct genetic characteristics in comparison with plant traits, which increase the difficulty of genetically improving these traits. In this study, diall...

In this paper, we classify additive closed symmetric monoidal structures on the category of left R-modules by using Watts’ theorem. An additive closed symmetric monoidal structure is equivalent to an R-module ΛA,B equipped with two commuting right R-module structures represented by the symbols A and B, an R-module K to serve as the unit, and certain isomorphisms. We use this result to look at s...

We prove that any rigid additive symmetric monoidal category can be mapped to a abelian in universal way. This yields novel approach Grothendieck's standard conjecture D and Voevodsky's smash nilpotence conjecture.

It is proved that in any pointed category with pullbacks, coequalizers and regular epi-mono factorizations, the class of regular epimorphisms is stable under pullback along the so-called balanced effective descent morphisms. Here “balanced” can be omitted if the category is additive. A balanced effective descent morphism is defined as an effective descent morphism p : E → B such that any subobj...

In this paper we show that the Farrell-Jones isomorphism conjectures are inherited in group extensions for assembly maps in algebraic K-theory and L-theory. Introduction Under what assumptions are the Farrell-Jones isomorphism conjectures inherited by group extensions or subgroups? We will formulate a version of the standard conjectures (see Farrell-Jones [9]) with twisted coefficients in an ad...

For a given (fixed) category, we consider the category of all 2-cell structures (over it) and study some naturality properties. A category with a 2-cell structure is a sesquicategory; we use additive notation for the vertical composition of 2-cells; instead of a law for horizontal composition we consider a relation saying which pairs of 2-cells can be horizontally composed; for a 2-cell structu...

We define and study the category Cohn(P) of normal coherent sheaves on the monoid scheme P (equivalently, the M0-scheme P/F1 in the sense of ConnesConsani-Marcolli [4] ). This category resembles in most ways a finitary abelian category, but is not additive. As an application, we define and study the Hall algebra of Cohn(P). We show that it is isomorphic as a Hopf algebra to the enveloping algeb...