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: Fibrations of models

Patrick Nicodemus (Oct 02 2023 at 21:54):

I have a number of observations which seem true but i haven't had time to write down a proof, they could be false. References appreciated for related work.

Let $p:\mathbb{E} \to \mathbb{B}$ be a Grothendieck fibration. It's well known that if the fibers of $p$ have limits of shape $J$ and these are preserved by reindexing functors, $p$ lifts limits of shape $J$.

I claim that a corollary is the following.
Let $T$ be a limit sketch, so it's a category $C$ together with a family of cones $(J_i, F_i:J_i\to C, a_i, \lambda_i :a_i\to F_i)$. I claim that if the fibers of $p$ have all $J_i$ shaped limits and reindexing functors preserve them, then there is a fibration $Mod(T, \mathbb{E}) \to Mod(T,\mathbb{B})$.

Patrick Nicodemus (Oct 02 2023 at 21:56):

moreover something similar is true in the 2-categorical setting, using the notion of 2-fibration.

Patrick Nicodemus (Oct 02 2023 at 22:03):

I need the 2-fibration version for my purposes so if there's a nice formal argument for this i would like to see it.

Kevin Arlin (Oct 03 2023 at 01:46):

What is your notion of 2-limit sketch? I didn't think there was an entirely standard story there.

Patrick Nicodemus (Oct 03 2023 at 12:15):

Kevin Arlin said:

What is your notion of 2-limit sketch? I didn't think there was an entirely standard story there.

Oh yeah I'm sorry. I would say everything is strict? so it's a 2 category endowed with a set of 2cones and functors have to send the 2 cones to strict 2 limits

Kevin Arlin (Oct 03 2023 at 17:25):

OK, but you still have free parameters in describing what your morphisms are. Bourke has a bit about the topic in Accessible Aspects... and that's the only reference I'm aware of. As far as I know there's really very little work that would help with this; it's 2-monads that are the vastly more developed story in 2-universal algebra.

Kevin Arlin (Oct 03 2023 at 17:26):

But this thing is much harder to state for monads, because of the typical thing that a monad doesn't come with a notion of its models in other categories!