A protolocalisation of a regular category is a full reflective regular subcategory, whose reflection preserves pullbacks of regular epimorphisms along arbitrary morphisms. We devote special attention to the epireflective protolocalisations of an exact Mal’cev category; we characterise them in terms of a corresponding closure operator on equivalence relations. We give some examples in algebra an...

For any free partially commutative monoid M(E, I), we compute the global dimension of the category of M(E, I)-objects in an Abelian category with exact coproducts. As a corollary, we generalize Hilbert’s Syzygy Theorem to polynomial rings in partially commuting variables.

The main purpose of this paper is to introduce the concept of (Γ, n)semihypergroups as a generalization of hypergroups, as a generalization of nary hypergroups and obtain an exact covariant functor between the category (Γ, n)-semihypergrous and the category semigroups. Moreover, we introduce and study complete part. Finally, we obtain some new results and some fundamental theorems in this respect.

We define an exact functor Fn,k from the category of Harish-Chandra modules for GL(n, R) to the category of finite-dimensional representations for the degenerate affine Hecke algebra for gl(k). Under certain natural hypotheses, we prove that the functor maps standard modules to standard modules (or zero) and irreducibles to irreducibles (or zero).

We show that K1(E) of an exact category E agrees with K1(DE) of the associated triangulated derivator DE under the hypothesis of the GilletWaldhausen theorem. More generally we show that K1(C(E)) of the category of bounded complexes in E always coincides with K1(DE).

We define an exact functor Fn,k from the category of Harish-Chandra modules for GL(n, R) to the category of finite-dimensional representations for the degenerate affine Hecke algebra for gl(k). Under certain natural hypotheses, we prove that the functor maps standard modules to standard modules (or zero) and irreducibles to irreducibles (or zero).

We solve a problem proposed by Khovanov by constructing, for any set of primes S, a triangulated category (in fact a stable∞-category) whose Grothendieck group is S−1Z. More generally, for any exact∞-category E, we construct an exact∞-category S−1E of equivariant sheaves on the Cantor space with respect to an action of a dense subgroup of the circle. We show that this∞-category is precisely the...

We introduce and develop the notion of scalar extension for abelian categories. Given a field extension F /F , to every F -linear abelian category A satisfying a suitable finiteness condition we associate an F -linear abelian category A ⊗F F ′ and an exact F -linear functor t : A → A ⊗F F . This functor is universal among F -linear right exact functors with target an F -linear abelian category....

We give a classification of substructures (= closed subbifunctors) given skeletally small extriangulated category by using the defects, in similar way to author’s exact structures additive category. More precisely, for an category, possible are bijection with Serre subcategories abelian consisting defects conflations. As byproduct, we prove that poset on it is isomorphic some

The notion of normal subobject having an intrinsic meaning in any protomodular category, we introduce the notion of normal functor, namely left exact conservative functor which reflects normal subobjects. The point is that for the category Gp of groups the change of base functors, with respect to the fibration of pointed objects, are not only conservative (this is the definition of a protomodul...