Ideals of closed categories
type: Item
instance of: scholarly article
title: Ideals of closed categories
zbMATH Open document ID: 0649.18009
DOI: 10.1016/0022-4049(88)90068-0
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: By an ideal of a (symmetric monoidal) closed category, the authors mean a subobject of the unit for the monoidal structure. The main result of this paper is that, for a class of ``suitable'' closed categories, the operation which sends a closed category to its poset of ideals can be made into a functor, which is right lax (or colax) adjoint to the inclusion functor from closed posets to closed categories. This generalizes known results for categories of modules over commutative rings and for Grothendieck toposes.
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 18B35
zbMATH DE Number: 4059658
zbMATH Keywords: poset of ideals
zbMATH Keywords: closed posets
zbMATH Keywords: closed categories
Wikidata QID: Q72421838
author: Susan Niefield
author: K. I. Rosenthal
reviewed by: Peter Johnstone
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90068-0
OpenAlex ID: W2062048624
cites work: Fake rings, fake modules, and duality
cites work: Q5620723
cites work: Q5586461
cites work: Formal category theory: Adjointness for 2-categories
cites work: Q4145861
cites work: Q3965241
cites work: An extension of the Galois theory of Grothendieck
cites work: Q4100657
cites work: Q4100671
cites work: Q5647004
cites work: Metric spaces, generalized logic, and closed categories
cites work: Strong De Morgan's law and the spectrum of a commutative ring
cites work: Rigidity in closed categories and generalized ''sup-inf'' theorems
cites work: Q5665322
Recommended article: Ideals and clots in pointed regular categories
Recommended article: Q4863987
Recommended article: On closed categories of functors
Recommended article: Ideal Banach category theorems
Recommended article: On the structure of closed ideals
Recommended article: INTEGRAL CLOSURE AND IDEAL TOPOLOGIES IN MODULES
Recommended article: Ideals of the second category
Recommended article: A category base for Mycielski's ideals
Recommended article: Cartesian closed algebraic categories
Recommended article: Ideal cotorsion theories in triangulated categories
description: scientific article; zbMATH DE number 4059658