nLab class-locally presentable category

Idea

The notion of a class-locally presentable category is a generalisation of that of a locally presentable category: In a class-locally presentable category one may need a proper class of objects (instead of just a small set) to build all the others by suitable colimits.

Definition

Let $\lambda$ be a regular cardinal, and let the category $C$ be complete and cocomplete. Then $C$ is called class $\lambda$-presentable if there is a class $A \subset C$ of $\lambda$-presentable objects such that every object of $C$ is a $\lambda$-filtered colimit of objects in $A$. A category is class-locally presentable if it is class $\lambda$-presentable for some $\lambda$.

If $C$ merely has $\lambda$-filtered colimits, rather than all colimits (and limits), then it is called class $\lambda$-accessible, and class-accessible if it is class $\lambda$-accessible for some $\lambda$.

Related concepts

• macrotopos?

References

• Boris Chorny, Jiří Rosický, Class-locally presentable and class-accessible categories, Journal of Pure and Applied Algebra

  216, Issue 10 (2012) pp 2113–2125 (doi:10.1016/j.jpaa.2012.01.015)