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: learning: questions

Topic: Partial fibration

Ralph Sarkis (Apr 20 2023 at 07:24):

I am not familiar with fibrations, so my question might not fit the philosophy of fibered CT.

I believe I have a fibration $p:\mathbf{E} \to \mathbf{B}$, but I am also interested in a category $\mathbf{E}'$ that contains $\mathbf{E}$ and I cannot naturally lift the fibration. So I view $p$ as a kind of partial fibration on $\mathbf{E}'$. Does this thing have a name?

Then, I have another fibration $q: \mathbf{D} \to \mathbf{B}$ and an inclusion $\mathbf{D} \hookrightarrow \mathbf{D}'$ and the end goal will be to have equivalences $\mathbf{E} \simeq \mathbf{D}$ and $\mathbf{E}' \simeq \mathbf{D}'$ that commute with the fibrations and inclusions. Would this be the right notion of equivalence between the thing we just named above?

Sam Speight (Apr 20 2023 at 08:52):

Ralph Sarkis said:

I am not familiar with fibrations, so my question might not fit the philosophy of fibered CT.

I believe I have a fibration $p:\mathbf{E} \to \mathbf{B}$, but I am also interested in a category $\mathbf{E}'$ that contains $\mathbf{E}$ and I cannot naturally lift the fibration. So I view $p$ as a kind of partial fibration on $\mathbf{E}'$. Does this thing have a name?

Is it a latent fibration?

Sam Speight (Apr 20 2023 at 08:55):

Wait sorry, latent fibrations are fibrations between categories with a notion of partial map.

Matteo Capucci (he/him) (Apr 20 2023 at 11:02):

Not a real answer, but these things can be presented as spans whose left leg is a fibration (the other leg is the inclusion of the total category). Me and @David Jaz call these 'fSpans'.

Matteo Capucci (he/him) (Apr 20 2023 at 11:04):

There might be a better answer in terms of cofree fibrations but to tell this story I need to understand better the situation. In which way does $E' \to B$ fail to be a fibration in your case of interest?

Ralph Sarkis (Apr 20 2023 at 12:07):

Spans do work so that is why I felt the term partial fibration was adequate (partial functions are sometimes presented with spans).

Here is my situation (modulo notions of finitariness which are not completely understood yet):

• $\mathbf{B}$ is the category of monads on $\mathbf{Set}$.
• $\mathbf{E}$ is the category of liftings of monads on $\mathbf{Set}$ to $\mathbf{Met}$ the category of metric spaces. In other words, monads on $\mathbf{Met}$ such that applying the forgetful functor yields a monad on $\mathbf{Set}$.
• $\mathbf{E}'$ is the category of all monads on $\mathbf{Met}$.

The fibration $p: \mathbf{E} \to \mathbf{B}$ sends a monad lifting to the underlying monad (I am not 100% sure it is a fibration). To send an arbitrary monad $M$ on $\mathbf{Met}$ to a monad on $\mathbf{Set}$, I can do $U \circ M \circ F$ where $F \dashv U: \mathbf{Met} \to \mathbf{Set}$ is the free-forgetful adjunction. While it is compatible with what I do for $p$, I don't think it will be a fibration because, morally, a $\mathbf{Set}$ monad morphism $T \to UMF$ will only see what happens on discrete metric spaces. Again, my gut feelings might be wrong because I am not comfortable with fibrations.