locally presentable categories - table

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

(n,r)-categories | satisfying Giraud's axioms | inclusion of left exaxt 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 |

Revised on January 5, 2014 21:44:42
by Andrew Stacey
(78.91.23.3)