Reflective subcategories (original) (raw)

A note on reflective subcategories defined by partial algebras

1984

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE 25.2 (1984) A NOTE ON REFLECTIVE SUBCATEGORIES DEFINED BY PARTIAL ALGEBRAS Jeno SZIGETI Abo tract s By uoing a gemoralized partial F-algebra a full oubcategory of a cert aim comma category will ho defined* Them a ouf fioiont oondition will bo giTen to proTldo the reflecti-Tity of thio oubcategory.

Remarks on equivalences of additive subcategories

We study category equivalences between some additive subcategories of module categories. As its application, we show that the group of aut- ofunctors of the category of reflexive modules over a normal domain is isomorphic to the divisor class group.

Some remarks on homologically finite subcategories

Boletín de la Sociedad Matemática Mexicana, 2017

Let be an Artin algebra and C a full subcategory of-mod closed under direct summands and closed under extensions. It is known that if C is functorially finite, then it has almost split sequences. Here we review an example of a covariantly finite subcategory that has right almost split morphisms except for one isomorphism class, and we compute its almost split sequences.

On Auslander-Reiten translates in functorially finite subcategories and applications

We consider functorially finite subcategories in module categories over Artin algebras. One main result provides a method, in the setup of bounded derived categories, to compute approximations and the end terms of relative Auslander–Reiten sequences. We also prove an Auslander–Reiten formula for the setting of functorially finite subcategories. Furthermore, we study the category of modules filtered by standard modules for certain quasi-hereditary algebras and we classify precisely when this category has finite type. The class of these algebras contains all blocks of Schur algebras S(2,r).

Intrinsic characterizations of some functors that preserve direct limits

arXiv (Cornell University), 2018

Finite modules, finitely presented modules and Mittag-Leffler modules are characterized by their behaviour by tensoring with direct products of modules. In this paper, we study and characterize the functors of modules that preserve direct products and direct limits.

Categories at Thick Subcategories

2016

Given a thick subcategory of a triangulated category, we define a colocalisation and a natural long exact sequence that involves the original category and its localisation and colocalisation at the subcategory. Similarly, we construct a natural long exact sequence containing the canonical map between a homological functor and its total derived functor with respect to a thick subcategory. 2000 Mathematics Subject Classification. 18E30.

On the fullness of certain functors

Journal of Pure and Applied Algebra, 1989

functor from a complete category to a 'weakly bounded' category (a concept including locally presentable categories, Cartesian closed topological categories, generalized varieties and many others). We show that the fullness of K depends only on its behaviour on objects and on isomorphisms. As a consequence, it is an isomorphism if and only if it is bijective on objects and creates isomorphisms. These can be seen as generalizations of recent 'Beth's definability type' results obtained by R. McKenzie, G. Weaver and the author. In what follows, an ordinal will be identified with the set of smaller ordinals, and cardinals with initial ordinals. We give or recall some definitions, not all in [l 11. Definitions. Let a be an infinite regular cardinal, (Y+ be the cardinal following it, and ~8 be a category. An a-nice limit is a finite product, a Pth-power or the equalizer of p (parallel) * This paper has been written while the author was at McGill University (Montreal) under a postdoctoral fellowship of the NSERC (Canada). The author also had the support of a grant from the FCAR (Quebec).

Exactness of limits and colimits in abelian categories revisited

2022

Let Σ be a small category and A be a Σ-co-complete (resp. Σcomplete) abelian category. It is a well-known fact that the category Fun(Σ, A) of functors of Σ in A is an abelian category, and that the functor colim Σ (−) : Fun(Σ, A) → A (resp. lim Σ (−) : Fun(Σ, A) → A) is left (resp. right) adjoint to κ Σ : A → Fun(Σ, A), where κ Σ is the associated constant diagram functor. In this paper we will show that the functor colim Σ (−) (resp. lim Σ (−)) is exact if and only if the pair of functors colim Σ (−), κ Σ (resp. κ Σ , lim Σ (−)) is Ext-adjoint. As an application of our findings, we will give new proofs of known results on the exactness of limits and colimits in abelian categories. Contents 1. Introduction 1 2. Preliminaries 3 3. A one-point extension of a small category 4 4. Main results 6 References 13