nLab conical limit

Idea

A conical limit is an ordinary limit as opposed to a more general weighted limit. (However, note that they still satisfy an enriched universal property, rather than being limits in the underlying category.)

When the base of enrichment is $Set$, every weighted limit can be expressed as a conical limit. However, it is not true that completeness under a class of weights can always be expressed as completeness under a class of diagrams. For instance, every power $A \times A$ is a product, but the class of categories admitting powers cannot be expressed as the class of categories admitting $D$-indexed limits for some class of categories $D$.

Properties

• An enriched category admits all weighted limits if and only if it admits conical limits and powers.

References

• Michael Albert, and Max Kelly. The closure of a class of colimits, Journal of Pure and Applied Algebra 51.1-2 (1988): 1-17. (doi)