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Stream: learning: questions

Topic: When are subobjects nice?

Jade Master (Sep 15 2021 at 01:38):

For a category C and for each object X of C, we can form the poset Sub(X) of subobjects of X as in nlab

Jade Master (Sep 15 2021 at 01:38):

I'm wondering what conditions on C make Sub(X) a boolean algebra?

Jade Master (Sep 15 2021 at 01:41):

The top element should be 1_X and if C has an initial object 0, then the map 0 -> X becomes the bottom element.

Jade Master (Sep 15 2021 at 01:42):

If C has pullbacks and pushouts then you get unions and intersections. The one I'm struggling with is negation.

Jade Master (Sep 15 2021 at 01:46):

A negation of a subobject $A$ is a subobject $\neg A$ with $A \cap \neg A = \bot$ and $A \cup \neg A = \top$

Mike Shulman (Sep 15 2021 at 02:35):

Have you read about Heyting categories and Boolean categories?

Steve Awodey (Sep 15 2021 at 03:12):

Jade Master said:

I'm wondering what conditions on C make Sub(X) a boolean algebra?

In an elementary topos, one condition that implies boolean is well-pointed: for f, g : A --> B, if fa = ga for all a : 1 --> A, then f = g.

Zhen Lin Low (Sep 15 2021 at 03:29):

Jade Master said:

If C has pullbacks and pushouts then you get unions and intersections. The one I'm struggling with is negation.

No, this is more subtle than you are making it out to be. The unique morphism from the initial object may fail to be a monomorphism. Pushouts of monos may not be mono. etc.

Jade Master (Sep 15 2021 at 04:18):

@Mike Shulman Oh cool. Im looking for boolean categories I think then.

Jade Master (Sep 15 2021 at 04:24):

@Steve Awodey interesting...Thanks!

Jade Master (Sep 15 2021 at 04:31):

@Zhen Lin Low oh yeah I see, you need some compatibility between limits and colimits as well?

Zhen Lin Low (Sep 15 2021 at 04:48):

You could say that. Or you could just directly specify hypotheses on the subobject posets.

Jade Master (Sep 15 2021 at 15:47):

Right but I'm looking for conditions on C

Mike Shulman (Sep 15 2021 at 17:19):

A condition on the subobject posets of C is a condition on C...

Mike Shulman (Sep 15 2021 at 17:21):

But I know what you mean. For unions (including the empty union, i.e. the bottom element) what you need are coproducts and images. You probably want them to be pullback-stable, in which case you get a coherent category.

Mike Shulman (Sep 15 2021 at 17:21):

To make it a Heyting category, it suffices to be locally cartesian closed.

Ben MacAdam (Sep 15 2021 at 17:39):

Hey, I think something like this would be addressed in Boolean and Classical Restriction Categories by Cockett and Manes. I haven't checked the paper in ages, but there should be a characterization of when a class of monics $\mathcal{M}$ in a category $\mathbb{C}$ gives rise to a boolean/classical restriction categories (which, if I remember correctly, correspond to the sorts of lattices you're talking about).

Edit: I don't have access to a university library account at the moment, so I can only work off of a pretty vague memory and the abstract of the paper.