nLab
Locally Presentable and Accessible Categories

Locally Presentable and Accessible Categories is a book by Jiří Adámek and Jiří Rosický about, unsurprisingly, locally presentable and accessible categories. It was published by Cambridge University Press in the London Mathematical Society Lecture Note Series, number 189, in 1994.

Content

1. locally presentable categories

2. accessible categories: see also sketch

3. algebraic categories: see algebraic category, equationally presentable category, essentially algebraic theory, and variety of algebras

4. injectivity class?es: see also weak factorization system

5. categories of models: see internal logic, theory, essentially algebraic theory, model

6. Vopěnka's principle

• Appendix: large cardinals

Related concepts

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categories…	satisfying Giraud's axioms	inclusion of left exact localizations	generated under colimits from small objects		localization of free cocompletion		generated under filtered colimits from small objects
(0,1)-category theory	(0,1)-toposes	$\hookrightarrow$	algebraic lattices	$\simeq$ Porst’s theorem	subobject lattices in accessible reflective subcategories of presheaf categories
category theory	toposes	$\hookrightarrow$	locally presentable categories	$\simeq$ Adámek-Rosický’s theorem	accessible reflective subcategories of presheaf categories	$\hookrightarrow$	accessible categories
model category theory	model toposes	$\hookrightarrow$	combinatorial model categories	$\simeq$ Dugger’s theorem	left Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory	(∞,1)-toposes	$\hookrightarrow$	locally presentable (∞,1)-categories	$\simeq$ Simpson’s theorem	accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories	$\hookrightarrow$	accessible (∞,1)-categories

category: reference