NDLI: Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings

Content Provider	Springer Nature Link
Author	Vandersmissen, E. Olmen, C. Gerlo, A.
Copyright Year	2006
Abstract	In [2] the subconstruct ${\mathbf{Sob}}$ of sober approach spaces was introduced and it was shown to be a reflective subconstruct of the category ${\mathbf{Ap}}_{0}$ of $T_0$ approach spaces. The main result of this paper states that moreover ${\mathbf{Sob}}$ is firmly ${\mathcal{U}}$ -reflective in ${\mathbf{Ap}}_{0}$ for the class ${\mathcal{U}}$ of epimorphic embeddings. ‘Firm ${\mathcal{U}}$ -reflective’ is a notion introduced in [3] by G.C.L. Brümmer and E. Giuli and is inspired by the exemplary behaviour of the usual completion in the category ${\mathbf{Unif}}_{0}$ of Hausdorff uniform spaces with uniformly continuous maps. It means that ${\mathbf{Sob}}$ is ${\mathcal{U}}$ -reflective in ${\mathbf{Ap}}_{0}$ and that the reflector $\epsilon$ is such that $f:X \rightarrow Y$ belongs to ${\mathcal{U}}$ if and only if $\epsilon(f)$ is an isomorphism. Firm ${\mathcal{U}}$ -reflectiveness implies uniqueness of completion in the sense that whenever $f:X \rightarrow Y$ is a map with $f \in {\mathcal{U}}$ and $Y$ sober, the associated $f^*: \epsilon (X) \rightarrow Y$ is an isomorphism. Our result generalizes the fact that in the category ${\mathbf{Top}}_{0}$ the subconstruct of sober topological spaces is firmly reflective for the class ${\mathcal{U}}_b$ of b-dense embeddings in ${\mathbf{Top}}_{0}$ . Also firmness in some other subconstructs of ${\mathbf{Ap}}_{0}$ will be easily obtained.
Ending Page	258
Page Count	8
Starting Page	251
File Format	PDF
ISSN	09272852
e-ISSN	15729095
Journal	Applied Categorical Structures
Issue Number	3
Volume Number	14
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2006-05-31
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry Convex and Discrete Geometry sober approach spaces Preorders, orders and lattices (viewed as categories) firm reflections Theory of Computation Categorical methods approach frames Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) Mathematical Logic and Foundations
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Algebra and Number Theory Computer Science

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Regularity vs. constructive complete (co)distributivity

Fuzzy Galois connections on fuzzy sets

Algebraic Kan extensions in double categories

Reflective and coreflective subcategories

Adjunction in Monoids

Admissibility, stable units and connected components

Tight limits and completions from Dedekind-MacNeille to Lambek-Isbell

Adjunctions of quasi-functors between dg-categories

$t$-structures for hereditary categories

Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings

Similar Documents

Regularity vs. constructive complete (co)distributivity

Fuzzy Galois connections on fuzzy sets

Algebraic Kan extensions in double categories

Reflective and coreflective subcategories

Adjunction in Monoids

Admissibility, stable units and connected components

Tight limits and completions from Dedekind-MacNeille to Lambek-Isbell

Adjunctions of quasi-functors between dg-categories

$t$-structures for hereditary categories

Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings