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: deprecated: topos theory

Topic: Flat vs Projective Modules in Category of sup-lattices

Ming Ng (Nov 25 2020 at 15:22):

I was revisiting Joyal-Tierney's "An Extension of Galois Theory of Groth.", and I came across Proposition 1 of Section II.4, which states that: Let $A$ be a commutative monoid in the category of sup-lattices. Then, an $A$-module $M\in Mod(A)$ is flat iff $M$ is a projective module.

This result seemed strange to me, because it is well-known in classic algebra that while projective modules are flat, flat modules need not be projective - a standard example would be how $\mathbb{Q}$ is a flat $\mathbb{Z}$-module but not projective. We also know that $\mathbb{Z}$ and $\mathbb{Q}$ can be presented as locales/frames.

I am aware that Joyal-Tierney were porting a lot of the standard ideas about tensor products and modules etc. from the category of abelian groups, and presumably working with $\mathbb{Z}$ and $\mathbb{Q}$ in the category of abelian groups is somewhat different from working with $P(\mathbb{Z})$ and $P(\mathbb{Q})$ in the category of sup-lattices. But I don't have a clear conceptual understanding as to why working in the category of sup-lattices forces all flat modules to be projective. The proof given by Joyal-Tierney is somewhat formal and concise, and I feel like I'm taking their proof on faith rather having a meaningful understanding of it. Does anyone have any useful hints/ideas on what I'm missing?

sarahzrf (Nov 25 2020 at 15:34):

i don't know enough algebra to be comfortable trying to actually answer this, but one thing i'd note is that one of the primary differences between suplattices and more traditional algebraic structures is idempotence

sarahzrf (Nov 25 2020 at 15:35):

so while you wait for somebody qualified to answer, that could be something worth poking at?

Morgan Rogers (he/him) (Nov 25 2020 at 15:39):

Please could you give a precise statement of the result, since that isn't an open access document? :sweat_smile: (e.g. what are the constraints on $A$?)

Ming Ng (Nov 25 2020 at 15:41):

[Mod] Morgan Rogers said:

Please could you give a precise statement of the result, since that isn't an open access document? :sweat_smile: (e.g. what are the constraints on $A$?)

edited!

Ming Ng (Nov 25 2020 at 15:42):

sarahzrf said:

i don't know enough algebra to be comfortable trying to actually answer this, but one thing i'd note is that one of the primary differences between suplattices and more traditional algebraic structures is idempotence

Yes, idempotence is the obvious thing to think about, isn't it? I'll have a little poke at it, as you suggested, and see what comes of it. :)

Jens Hemelaer (Nov 26 2020 at 11:19):

I think an important ingredient is that the category of sup-lattices is equivalent to it's dual category, and that flatness and projectivity are dual concepts: flatness means that tensoring preserves monomorphisms (in their definition), and projectivity means that Hom's preserve epimorphisms.

Below the reason why the category of sup-lattices is self-dual:

A sup-lattice is a poset that has arbitrary suprema... but then it also has arbitrary infima, so it is a complete lattice.
So the category of sup-lattices has

• as objects the complete lattices;
• as morphisms the supremum-preserving functions.

In particular, the morphisms are not necessarily lattice morphisms, they do not necessarily preserve finite infima.

From a categorical point of view, the supremum-preserving functions are precisely the functors $P \to Q$ that preserve colimits.
By some adjoint functor theorem, they admit a right adjoint, $Q \to P$ which preserves limits.
After taking opposites, this gives a colimit-preserving functor $Q^\mathrm{op} \to P^\mathrm{op}$.

Ming Ng (Nov 29 2020 at 23:49):

Jens Hemelaer said:

I think an important ingredient is that the category of sup-lattices is equivalent to it's dual category, and that flatness and projectivity are dual concepts: flatness means that tensoring preserves monomorphisms (in their definition), and projectivity means that Hom's preserve epimorphisms.

Below the reason why the category of sup-lattices is self-dual:

A sup-lattice is a poset that has arbitrary suprema... but then it also has arbitrary infima, so it is a complete lattice.
So the category of sup-lattices has

• as objects the complete lattices;
• as morphisms the supremum-preserving functions.

In particular, the morphisms are not necessarily lattice morphisms, they do not necessarily preserve finite infima.

From a categorical point of view, the supremum-preserving functions are precisely the functors $P \to Q$ that preserve colimits.
By some adjoint functor theorem, they admit a right adjoint, $Q \to P$ which preserves limits.
After taking opposites, this gives a colimit-preserving functor $Q^\mathrm{op} \to P^\mathrm{op}$.

Oops forgot to reply to this - thanks! The observation that the result is due to the category of sup-lattices being self-dual looks about right to me, and was indeed what I had overlooked. :)