Smallness in topology

J 2023

Smallness in topology

ADÁMEK, Jiří, Miroslav HUŠEK, Jiří ROSICKÝ and Walter THOLEN

Basic information

Original name

Smallness in topology

Authors

ADÁMEK, Jiří (203 Czech Republic), Miroslav HUŠEK, Jiří ROSICKÝ (203 Czech Republic, guarantor, belonging to the institution) and Walter THOLEN

Edition

Quaestiones Mathematicae, TAYLOR & FRANCIS LTD, 2023, 1607-3606

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

United Kingdom of Great Britain and Northern Ireland

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 0.700 in 2022

RIV identification code

RIV/00216224:14310/23:00134276

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.2989/16073606.2023.2247720

UT WoS

001170576200001

Keywords in English

Finitely presentable object; finitely generated object; finitely small object; directed colimit; Hausdorff space; T0-space; T1-space; compact space

Abstract

V originále

Quillen’s notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category of all topological spaces.

Links

GA22-02964S, research and development project

Name: Obohacené kategorie a jejich aplikace (Acronym: ECATA)

Investor: Czech Science Foundation

Citovat

ADÁMEK, Jiří, Miroslav HUŠEK, Jiří ROSICKÝ and Walter THOLEN. Smallness in topology. Quaestiones Mathematicae. TAYLOR & FRANCIS LTD, 2023, vol. 46, S1, p. 13-39. ISSN 1607-3606. Available from: https://dx.doi.org/10.2989/16073606.2023.2247720.

@article{2333977,
   author = {Adámek, Jiří and Hušek, Miroslav and Rosický, Jiří and Tholen, Walter},
   article_number = {S1},
   doi = {http://dx.doi.org/10.2989/16073606.2023.2247720},
   keywords = {Finitely presentable object; finitely generated object; finitely small object; directed colimit; Hausdorff space; T0-space; T1-space; compact space},
   language = {eng},
   issn = {1607-3606},
   journal = {Quaestiones Mathematicae},
   title = {Smallness in topology},
   url = {https://doi.org/10.2989/16073606.2023.2247720},
   volume = {46},
   year = {2023}
}

TY  - JOUR
ID  - 2333977
AU  - Adámek, Jiří - Hušek, Miroslav - Rosický, Jiří - Tholen, Walter
PY  - 2023
TI  - Smallness in topology
JF  - Quaestiones Mathematicae
VL  - 46
IS  - S1
SP  - 13-39
EP  - 13-39
PB  - TAYLOR & FRANCIS LTD
SN  - 16073606
KW  - Finitely presentable object
KW  - finitely generated object
KW  - finitely small object
KW  - directed colimit
KW  - Hausdorff space
KW  - T0-space
KW  - T1-space
KW  - compact space
UR  - https://doi.org/10.2989/16073606.2023.2247720
N2  - Quillen’s notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category of all topological spaces.
ER  -

ADÁMEK, Jiří, Miroslav HUŠEK, Jiří ROSICKÝ and Walter THOLEN. Smallness in topology. \textit{Quaestiones Mathematicae}. TAYLOR \&{} FRANCIS LTD, 2023, vol.~46, S1, p.~13-39. ISSN~1607-3606. Available from: https://dx.doi.org/10.2989/16073606.2023.2247720.

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