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: practice: terminology & notation

Topic: simple order theory concept

Nathaniel Virgo (Aug 07 2024 at 02:16):

Suppose I have a poset $P$ and an order-preserving map $F:P\to P$. Is there a name for an element $p\in P$ such that $F(p)\le p$, and similarly is there a name for an element $p$ such that $p\le F(p)$?

If we were talking about categories this would be an algebra and a coalgebra of the functor $F$, but it seems silly to use those terms for posets and preorders, where it's really just a property of $p$. Are there standard terms for them in the context of partial orders?

Ralph Sarkis (Aug 07 2024 at 05:45):

Pre-fixed point/pre-fixpoint and post-fixed point/post-fixpoint, but I get confused all the time about which is which.

Mike Shulman (Aug 07 2024 at 23:46):

I don't think "algebra" and "coalgebra" are silly. They certainly have the advantage that it's easier to remember which is which (for a category theorist).

Kevin Carlson (Aug 08 2024 at 00:33):

The lax limit and oplax limit of the functor give the collections of all such points (though you have to figure out which is which again) so lax and oplax fixpoints would also work pretty well for a category-theoretic audience.

Matteo Capucci (he/him) (Aug 09 2024 at 15:50):

Kevin Carlson said:

The lax limit and oplax limit of the functor give the collections of all such points (though you have to figure out which is which again) so lax and oplax fixpoints would also work pretty well for a category-theoretic audience.

I like this! Fits in the pattern of 'laxifying' notions defined for sets: a lax fixpoint is like a fixpoint except equality is replaced by a morphism. As per the direction... my brain tells me $p \leq Fp$ ought to be a lax fixpoint, but if lax fixpoints are points in the lax limit of $F:P \to P$, then actually such a limit comes with a 2-cell $Fp \leq p$, so it's the other way around than I expected. I'm fine with that!

Mike Shulman (Aug 09 2024 at 17:12):

Of course, the category of algebras for a monad is also a lax limit.