If we consider metric spaces to be categories enriched over $\mathbb R_{\geq 0}$, the object corresponding to presheaves should be lipschitz-continuous functions $\operatorname{Lip^ 1}(M, \mathbb R_{\geq 0})$. Now there should be an obvious metric on this set; making the Yoneda map $$x\mapsto \operatorname d(-,x)$$ an isometric embedding. What is this metric?
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
2
|
|
||
|
|
|
3
|
It is the usual sup metric. See section 2 of Lawvere's original article. |
||
|
|
