The category of motives over the algebraic closure of a finite field is known to be a semisimple Q-linear Tannakian category, but unless one assumes the Tate conjecture there is little further one can say about it. However, once this conjecture is assumed, it is possible to give an almost entirely satisfactory description of the category together with its standard fibre functors. In particular ...

In this paper we establish Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for algebras conilpotent coalgebras. We show that normalized chain complex functor transforms Quillen equivalence quasicategories simplicial into duality. This allows us to give conceptual interpretation nerve category its adjoint. As an application, prove repr...

This is the notes for the author’s presentation in the student seminar. We follow the paper [BMR1] to talk about a derived equivalence between the representations of a semisimple Lie algebra in positive characteristic and the category of twisted D-modules supported generalized Springer fibers. We work over a field k, and consider a semisimple Lie algebra g. Let U = U(g) be the enveloping algebr...

let $r$ be a commutative ring. we write $mbox{hom}(mu_a, nu_b)$ for the set of all fuzzy $r$-morphisms from $mu_a$ to $nu_b$, where $mu_a$ and $nu_b$ are two fuzzy $r$-modules. we make$mbox{hom}(mu_a, nu_b)$ into fuzzy $r$-module by redefining a function $alpha:mbox{hom}(mu_a, nu_b)longrightarrow [0,1]$. we study the properties of the functor $mbox{hom}(mu_a,-):frmbox{-mod}rightarrow frmbox{-mo...

We construct a functor from the partition complex of finite set A A to category trees with leaves labelled by and prove that it is homo...

Let G be a group of automorphisms of a category C. We give a definition for a functor F : C → C to be a G-covering and three constructions of the orbit category C/G, which generalizes the notion of a Galois covering of locally finitedimensional categories with group G whose action on C is free and locally bonded. Here C/G is defined for any category C and does not require that the action of G i...

the ordinary tensor product of modules is defined using bilinear maps (bimorphisms), that are linear in eachcomponent. keeping this in mind, linton and banaschewski with nelson defined and studied the tensor product in an equational category and in a general (concrete) category k, respectively, using bimorphisms, that is, defined via the hom-functor on k. also, the so-called sesquilinear, or on...

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices a class of semirings called semirings. The semiring perspective leads necessary sufficient condition for interval [Formula: see text] be subalgebra an lattice, where is dualizing element. also import some results techniques semimodule theory in study this semirings, generalizing about ...

Mal'tsev categories turned out to be a central concept in categorical algebra. On one hand, the simplicity and beauty of notion is revealed through lot characterizations different flavour. Depending on context, can define as those for which `any reflexive relation an equivalence'; difunctional'; `the composition equivalence relations same object commutative'; `each fibre fibration points unital...

We investigate when an exact functor F ∼= − ⊗Λ MΓ : mod-Λ → mod-Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard’s for self-injective algebras. As a corol...