Topic
Injective object
About: Injective object is a research topic. Over the lifetime, 44 publications have been published within this topic receiving 306 citations.
Papers
TL;DR: In this article, it was shown that assigning to a field its algebraic closure, to a poset or Boolean algebra its Mac-Neille completion, and to an R-module its injective envelope is not functorial, if one wants the respective embeddings to form a natural transformation.
Abstract: In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless all objects are injective. In particular, assigning to a field its algebraic closure, to a poset or Boolean algebra its Mac-Neille completion, and to an R-module its injective envelope is not functorial, if one wants the respective embeddings to form a natural transformation.
31 citations
Posted Content•
TL;DR: The following result was stated as a "pleasant exercise in homological algebra" in this paper, and its proof was communicated to us privately by M. Van den Bergh many years ago, and we referred to it as quite involved.
Abstract: is an injective object I inthe abelian category GrModA.The following result is stated as a “pleasant exercise in homological algebra” in[VdB]. Its proof was communicated to us privately by M. Van den Bergh manyyears ago, and we referred to it as “quite involved” in [YZ1, Remark 4.9]. Thepurpose of this note is to give a modified proof of this result.
20 citations
TL;DR: Sikorski's extension theorem for finite Boolean algebras is described and turned into a syntactical conservation result, which can facilitate proofs of several related classical principles.
Abstract: Constructive meaning is given to the assertion that every finite Boolean algebra is an injective object in the category of distributive lattices. To this end, we employ Scott's notion of entailment relation, in which context we describe Sikorski's extension theorem for finite Boolean algebras and turn it into a syntactical conservation result. As a by-product, we can facilitate proofs of several related classical principles.
11 citations
10 citations
TL;DR: In this article, the notion of entailment relation is employed to define Sikorski's extension theorem for finite Boolean algebras and turn it into a syntactical conservation result.
Abstract: Constructive meaning is given to the assertion that every finite Boolean
algebra is an injective object in the category of distributive lattices. To
this end, we employ Scott's notion of entailment relation, in which context we
describe Sikorski's extension theorem for finite Boolean algebras and turn it
into a syntactical conservation result. As a by-product, we can facilitate
proofs of several related classical principles.
10 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 4 |
| 2020 | 1 |
| 2019 | 3 |
| 2018 | 1 |
| 2017 | 4 |
| 2016 | 1 |