Abstract We prove that if the Auslander–Reiten triangles generate relations for Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with (co)generator, then has only finitely many isomorphism classes indecomposable objects up to translation. This gives converse theorem Butler and on groups. Our approach applications in context Frobenius categories.

Abstract Given a compact Kähler manifold X , it is shown that pairs of the form $(E,\, D)$ where E trivial holomorphic vector bundle on and D an integrable connection produce neutral Tannakian category. The corresponding pro-algebraic affine group scheme studied. In particular, this for Riemann surface determines uniquely isomorphism class surface.

Given an algebraic theory which can be described by a (possibly symmetric) operad P , we propose a definition of the weakening (or categorification) of the theory, in which equations that hold strictly for P -algebras hold only up to coherent isomorphism. This generalizes the theories of monoidal categories and symmetric monoidal categories, and several related notions defined in the literature...

This paper is a collection of remarks on control categories, including answers to some frequently asked questions. The paper is not self-contained and must be read in conjunction with [3]. We clarify the definition of response categories, and show that most of the conditions can be dropped. In particular, the requirement of having finite sums can be dropped, leading to an interesting new CPS tr...

A flow in a category X is an object X of X provided with an endomorphism T : X → X in X. In most of the cases X will be the category of (topological) groups and (continuous) group homomorphisms, the category of right modules over a ring R and the R-module homomorphisms, or just the category of topological (resp., measure) spaces and continuous (resp., measure preserving) maps. An isomorphism be...

An improved method to retrieve a library function via its Hindley/Milner type is described. Previous retrieval systems have identiied types that are isomorphic in any Cartesian closed category (CCC), and have retrieved library functions of types that are either isomorphic to the query, or have instances that are. Sometimes it is useful to instantiate the query too, which requires uniication mod...

An orbifold datum is a collection A of algebraic data in modular fusion category C. It allows one to define new CA construction that generalisation taking the Drinfeld centre category. Under certain simplifying assumptions we characterise terms scalars satisfying polynomial equations and give an explicit expression which computes number isomorphism classes simple objects CA. In Ising-type categ...

The m-periodic derived category of an abelian is a natural Z/mZ-periodic analogue the usual category. We determine Grothendieck group periodic skeletally small with enough projectives. In particular, we prove that finitely generated modules over Artin algebra free Z-module if m even, and F2-vector space odd. Moreover, in both cases parity m, rank equal to number isomorphism classes simple cases...

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.