scholarly journals On Flat Objects of Finitely Accessible Categories

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Septimiu Crivei
Keyword(s):  
Abelian Category ◽  
Closure Properties ◽  
Functor Category ◽  
Additive Category ◽  
Accessible Categories ◽  
Flat Cover ◽  
Strongly Flat

Flat objects of a finitely accessible additive category are described in terms of some objects of the associated functor category of , called strongly flat functors. We study closure properties of the class of strongly flat functors, and we use them to deduce the known result that every object of a finitely accessible abelian category has a flat cover.

scholarly journals FLAT RING EPIMORPHISMS OF COUNTABLE TYPE

2019 ◽  
Vol 62 (2) ◽  
pp. 383-439 ◽  
Author(s):  
LEONID POSITSELSKI
Keyword(s):  
Associative Ring ◽  
Abelian Category ◽  
Projective Dimension ◽  
Countable Base ◽  
Linear Topology ◽  
Countable Type ◽  
Flat Ring ◽  
Abelian Categories ◽  
Induced Topology ◽  
Strongly Flat

AbstractLet R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at $\mathbb{G}$ fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology $\mathbb{G}$, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all $\mathbb{G}$-separated $\mathbb{G}$-complete left R-modules.

scholarly journals Embedding categories with exactness into their abelianization

1977 ◽  
Vol 23 (3) ◽  
pp. 275-283
Author(s):  
Murray Adelman
Keyword(s):  
Abelian Groups ◽  
Abelian Category ◽  
Partial Answer ◽  
Additive Category ◽  
Small Set ◽  
Exact Sequences

AbstractWe give a partial answer to the following questions: Given an additive category and a class of sequences, under what conditions is the universal functor to the abelian category faithful and what other sequences are taken to exact sequences?The “answers” to these questions appear as theorems 2.2 and 3.2 respectively and amount to a weakening of the condition that there be enough relative projectives.We then characterize those additive categories with exactness which have the property that all relative exact sequences are determined by a small set of functors into the category of abelian groups.

scholarly journals Tameness, powerful images, and large cardinals

2020 ◽  
Vol 21 (01) ◽  
pp. 2050024
Author(s):  
Will Boney ◽  
Michael Lieberman
Keyword(s):  
Large Cardinals ◽  
Classification Theory ◽  
Closure Properties ◽  
Weakly Compact ◽  
Accessible Categories ◽  
Elementary Classes ◽  
Comprehensive Level ◽  
Theoretic Characterization ◽  
Measurable Cardinals

We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Proc. Amer. Math. Soc. 145(10) (2017) 4517–4532; A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. AMS 145(3) (2016) 1317–1327; M. Lieberman, A category-theoretic characterization of almost measurable cardinals (Submitted, 2018), http://arxiv.org/abs/1809.06963; M. Lieberman and J. Rosický, Classification theory for accessible categories. J. Symbolic Logic 81(1) (2016) 1647–1648].

ON KRULL–SCHMIDT FINITELY ACCESSIBLE CATEGORIES

2011 ◽  
Vol 84 (1) ◽  
pp. 90-97
Author(s):  
SEPTIMIU CRIVEI
Keyword(s):  
Additive Category ◽  
Epimorphic Image ◽  
Accessible Categories ◽  
Finitely Presented

AbstractLet 𝒞 be a finitely accessible additive category with products, and let (Ui)i∈Ibe a family of representative classes of finitely presented objects in 𝒞 such that each objectUiis pure-injective. We show that 𝒞 is a Krull–Schmidt category if and only if every pure epimorphic image of the objectsUiis pure-injective.

On Auslander’s formula and cohereditary torsion pairs

2018 ◽  
Vol 20 (06) ◽  
pp. 1750071 ◽  
Author(s):  
Abhishek Banerjee
Keyword(s):  
Regular Cardinal ◽  
Abelian Category ◽  
Triangulated Category ◽  
Functor Category ◽  
Torsion Pair ◽  
Torsion Pairs ◽  
Torsion Free ◽  
Coherent Functors

For a small abelian category [Formula: see text], Auslander’s formula allows us to express [Formula: see text] as a quotient of the category [Formula: see text] of coherent functors on [Formula: see text]. We consider an abelian category with the added structure of a cohereditary torsion pair [Formula: see text]. We prove versions of Auslander’s formula for the torsion-free class [Formula: see text] of [Formula: see text], for the derived torsion-free class [Formula: see text] of the triangulated category [Formula: see text] as well as the induced torsion-free class in the ind-category [Formula: see text] of [Formula: see text]. Further, for a given regular cardinal [Formula: see text], we also consider the category [Formula: see text] of [Formula: see text]-presentable objects in the functor category [Formula: see text]. Then, under certain conditions, we show that the torsion-free class [Formula: see text] can be recovered as a subquotient of [Formula: see text].

scholarly journals On the Category of Weakly U-Complexes

2020 ◽  
Vol 13 (2) ◽  
pp. 323-345
Author(s):  
Gustina Elfiyanti ◽  
Intan Muchtadi-Alamsyah ◽  
Fajar Yuliawan ◽  
Dellavitha Nasution
Keyword(s):  
Homological Algebra ◽  
Abelian Category ◽  
Universal Property ◽  
Homotopy Category ◽  
Triangulated Category ◽  
Additive Category

Motivated by a study of Davvaz and Shabbani which introduced the concept of U-complexes and proposed a generalization on some results in homological algebra, we study thecategory of U-complexes and the homotopy category of U-complexes. In [8] we said that the category of U-complexes is an abelian category. Here, we show that the object that we claimed to be the kernel of a morphism of U-omplexes does not satisfy the universal property of the kernel, hence wecan not conclude that the category of U-complexes is an abelian category. The homotopy category of U-complexes is an additive category. In this paper, we propose a weakly chain U-complex by changing the second condition of the chain U-complex. We prove that the homotopy category ofweakly U-complexes is a triangulated category.

scholarly journals GORENSTEIN PROJECTIVE OBJECTS IN FUNCTOR CATEGORIES

2018 ◽  
Vol 240 ◽  
pp. 1-41 ◽  
Author(s):  
SONDRE KVAMME
Keyword(s):  
Commutative Ring ◽  
Abelian Category ◽  
Functor Category ◽  
Functor Categories

Let $k$ be a commutative ring, let ${\mathcal{C}}$ be a small, $k$-linear, Hom-finite, locally bounded category, and let ${\mathcal{B}}$ be a $k$-linear abelian category. We construct a Frobenius exact subcategory ${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))$ of the functor category ${\mathcal{B}}^{{\mathcal{C}}}$, and we show that it is a subcategory of the Gorenstein projective objects ${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ in ${\mathcal{B}}^{{\mathcal{C}}}$. Furthermore, we obtain criteria for when ${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))={\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$. We show in examples that this can be used to compute ${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ explicitly.

scholarly journals Covers of Acts Over Monoids and Pure Epimorphisms

2014 ◽  
Vol 57 (3) ◽  
pp. 589-617 ◽  
Author(s):  
Alex Bailey ◽  
James H. Renshaw
Keyword(s):  
The Other ◽  
The Subject ◽  
Definition Of ◽  
Flat Cover ◽  
Strongly Flat ◽  
Torsion Free

AbstractIn 2001, Enochs's celebrated flat cover conjecture was finally proven, and the proofs (two different proofs were presented in the same paper) have since generated a great deal of interest among researchers. The results have been recast in a number of other categories and, in particular, for additive categories. In 2008, Mahmoudi and Renshaw considered a similar problem for acts over monoids but used a slightly different definition of cover. They proved that, in general, their definition was not equivalent to that of Enochs, except in the projective case, and left open a number of questions regarding the ‘other’ definition. This ‘other’ definition is the subject of the present paper and we attempt to emulate some of Enochs's work for the category of acts over monoids, and concentrate, in the main, on strongly flat acts. We hope to extend this work to other classes of acts, such as injective, torsion free, divisible and free, in a future report.

A Relationship between Left Exact and Representable Functors

1971 ◽  
Vol 23 (2) ◽  
pp. 374-380 ◽  
Author(s):  
H. B. Stauffer
Keyword(s):  
Abelian Groups ◽  
Direct Limit ◽  
Abelian Category ◽  
Functor Category ◽  
Natural Transformations ◽  
Comma Category

Our aim in this paper is to demonstrate a relationship between left exact and representable functors. More precisely, in the functor category whose objects are the additive functors from the dual of an abelian category 𝔄 to the category of abelian groups and whose morphisms are the natural transformations between them, the left exact functors can be characterized as those equivalent to a direct limit of representable functors taken over a directed class. The proof will proceed in the following manner. Lambek [3] and Ulmer [7] have shown that any functor T in can be expressed as a direct limit of representable functors taken over a comma category. When T is left exact, it is easily observed that this comma category is a filtered category. When T is left exact, it is easily observed that this comma category is a filtered category.

scholarly journals Failure Analysis on a Collapsed Flat Cover of an Adjustable Ballast Tank Used in Deep-Sea Submersibles

2019 ◽  
Vol 9 (23) ◽  
pp. 5258
Author(s):  
Fang Wang ◽  
Mian Wu ◽  
Genqi Tian ◽  
Zhe Jiang ◽  
Shun Zhang ◽  
...  
Keyword(s):  
Deep Sea ◽  
Material Selection ◽  
High Strength ◽  
External Pressure ◽  
Element Analysis ◽  
Ballast Tank ◽  
Material Inhomogeneity ◽  
Comprehensive Properties ◽  
Ultimate Cause ◽  
Flat Cover

A flat cover of an adjustable ballast tank made of high-strength maraging steel used in deep-sea submersibles collapsed during the loading process of external pressure in the high-pressure chamber. The pressure was high, which was the trigger of the collapse, but still considerably below the design limit of the adjustable ballast tank. The failure may have been caused by material properties that may be defective, the possible stress concentration resulting from design/processing, or inappropriate installation method. The present paper focuses on the visual inspections of the material inhomogeneity, ultimate cause of the collapse of the flat cover in pressure testing, and finite element analysis. Special attention is paid to the toughness characteristics of the material. The present study demonstrates the importance of material selection for engineering components based on the comprehensive properties of the materials.

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