Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: lifted functor on Met preserves isometries

Ralph Sarkis (Jan 19 2022 at 17:45):

Let $\mathbf{Met}$ be the category of metric spaces and nonexpansive maps. If $\hat{F}: \mathbf{Met} \to \mathbf{Met}$ is a functor lifting $F:\mathbf{Set} \to \mathbf{Set}$ and $F$ preserves monomorphisms (i.e. injections), does $\hat{F}$ preserve isometries ?

I haven't been able to prove this nor find a counterexample so I am looking for help.

Morgan Rogers (he/him) (Jan 19 2022 at 18:23):

Aren't isometries isomorphisms?

Ralph Sarkis (Jan 19 2022 at 18:24):

I think some authors assume isometries are bijection, for me they are only distance preserving maps (they are automatically injective).

Morgan Rogers (he/him) (Jan 19 2022 at 18:30):

Okay. When you say "lifting $F:\mathbf{Set} \to \mathbf{Set}$", do you mean lifting along the forgetful/underlying set functor?

Ralph Sarkis (Jan 19 2022 at 18:30):

Yes!

Morgan Rogers (he/him) (Jan 19 2022 at 18:36):

Given an isometry $f:X \to Y$ which is not an isomorphism, could we have some silly lifting of the identity functor which is the identity almost everywhere on $\mathbf{Met}$ except that spaces under $Y$ have their distance function multiplied by a factor of $1 - \epsilon$?

Ralph Sarkis (Jan 19 2022 at 18:41):

Sorry, one last precision, I am working with metric spaces with distance bounded by 1, or equivalently extended metric spaces.

In your example, the coprojection $Y \hookrightarrow Y + \mathbf{1}$ (coproducts put any two elements in disjoint parts at distance $1$ or $\infty$ in the extended case) is not nonexpansive after doing the liting.

Morgan Rogers (he/him) (Jan 19 2022 at 18:42):

No no, $Y$ is also reduced in distance (it's under itself with the identity morphism)

Ralph Sarkis (Jan 19 2022 at 18:43):

But $Y+\mathbf{1}$ is not reduced right?

Morgan Rogers (he/him) (Jan 19 2022 at 18:44):

Huh? You've just given a morphism from $Y$ to it, so it would be

Ralph Sarkis (Jan 19 2022 at 18:45):

Oh I didn't understand spaces under $Y$ like that, so it is any space with a morphism from $Y$ to it ?

Morgan Rogers (he/him) (Jan 19 2022 at 18:46):

Yes :D

Morgan Rogers (he/him) (Jan 19 2022 at 18:46):

(I don't know if my idea actually works, but I don't think you've broken it yet)

Ralph Sarkis (Jan 19 2022 at 18:47):

Yeah, but I feel like your example does work :smile:

Ralph Sarkis (Jan 19 2022 at 18:47):

Thanks

Morgan Rogers (he/him) (Jan 19 2022 at 18:47):

(it doesn't work because there are morphisms between every pair of non-empty metric spaces!!)

Morgan Rogers (he/him) (Jan 19 2022 at 18:48):

But hopefully it at least illustrates why isometries could fail to be preserved?