Colimit-dense subcategories

Jiří Adámek; Andrew D. Brooke-Taylor; Tim Campion; Leonid Positselski; Jiří Rosický

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 4, page 447-462
  • ISSN: 0010-2628

Abstract

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Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vopěnka’s Principle, we prove that a cocomplete category is locally presentable if and only if it has a colimit dense subcategory and a generator consisting of presentable objects. We further show that a 3 -element set is colimit-dense in 𝐒𝐞𝐭 op , and spaces of countable dimension are colimit-dense in 𝐕𝐞𝐜 op .

How to cite

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Adámek, Jiří, et al. "Colimit-dense subcategories." Commentationes Mathematicae Universitatis Carolinae 60.4 (2019): 447-462. <http://eudml.org/doc/295075>.

@article{Adámek2019,
abstract = {Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vopěnka’s Principle, we prove that a cocomplete category is locally presentable if and only if it has a colimit dense subcategory and a generator consisting of presentable objects. We further show that a $3$-element set is colimit-dense in $\{\mathbf \{Set\}\}^\{\rm op\}$, and spaces of countable dimension are colimit-dense in $\{\mathbf \{Vec\}\}^\{\rm op\}$.},
author = {Adámek, Jiří, Brooke-Taylor, Andrew D., Campion, Tim, Positselski, Leonid, Rosický, Jiří},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {locally presentable category; colimit-dense subcategory; Vopěnka's Principle},
language = {eng},
number = {4},
pages = {447-462},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Colimit-dense subcategories},
url = {http://eudml.org/doc/295075},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Adámek, Jiří
AU - Brooke-Taylor, Andrew D.
AU - Campion, Tim
AU - Positselski, Leonid
AU - Rosický, Jiří
TI - Colimit-dense subcategories
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 4
SP - 447
EP - 462
AB - Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vopěnka’s Principle, we prove that a cocomplete category is locally presentable if and only if it has a colimit dense subcategory and a generator consisting of presentable objects. We further show that a $3$-element set is colimit-dense in ${\mathbf {Set}}^{\rm op}$, and spaces of countable dimension are colimit-dense in ${\mathbf {Vec}}^{\rm op}$.
LA - eng
KW - locally presentable category; colimit-dense subcategory; Vopěnka's Principle
UR - http://eudml.org/doc/295075
ER -

References

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