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Stream: learning: questions

Topic: Continuous functor in enriched category

Peva Blanchard (Mar 18 2024 at 21:35):

I am parsing the definition of a continuous functor in the context of enriched categories.

In the case of $V = Set$, we have actually two notions of continuity:

• the original definition of continuous functor: preserving all limits, with the trivial weighting $W : (d :D) \mapsto \{\star\}$.
• $Set$-continuous functor: preserving all limits with respect to any weighting $W : D \rightarrow Set$

Of course, the second condition implies the first. But are they equivalent?

Kevin Carlson (aka Arlin) (Mar 18 2024 at 21:45):

Yes. Weighted limits over Set can be reduced to conical limits, which is why they're rarely explicitly discussed before enriched category theory. Kelly's book on enriched category theory will discuss this.

Peva Blanchard (Mar 18 2024 at 21:47):

Great, thank you.