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.
Fawzi Hreiki (Oct 16 2020 at 12:58):One of the first things we prove about toposes is that the subobject classifier is an internal Heyting algebra object. Dually, there is the notion of a co-Heyting algebra (https://ncatlab.org/nlab/show/co-Heyting+algebra) which is a (bounded) lattice equipped with a subtraction operator which is (element-wise) left adjoint to taking sums (joins). The prototypical example of such a thing is the poset of closed subsets in any topological space.
A bi-Heyting algebra is then defined as a Heyting algebra which is also a co-Heyting algebra. Any Boolean algebra is a bi-Heyting algebra - although in an uninteresting way. Are there any examples of 'naturally occurring' non-Boolean toposes whose subobject classifiers are bi-Heyting algebras?
Fawzi Hreiki (Oct 16 2020 at 13:03):The nLab page for bi-Heyting toposes (https://ncatlab.org/nlab/show/bi-Heyting+topos) mentions that any presheaf topos and any essential subtopos of a presheaf topos is bi-Heyting. Has anyone studied the co-Heyting structure in any concrete examples (e.g. simplicial sets, monoid actions, etc...)?
Morgan Rogers (he/him) (Oct 16 2020 at 13:55):When you say
Has anyone studied the co-Heyting structure in any concrete examples (e.g. simplicial sets, monoid actions, etc...)?
what kind of studying are you seeking? Certainly people have studied the subobject classifiers of these toposes, but I haven't personally seen a focus on the co-Heyting structure. A more experienced logician might be able to tell you whether anyone has leveraged the co-Heyting structure for logical purposes.
Matteo Capucci (he/him) (Oct 16 2020 at 17:46):I reckon people discussing it here some time ago
Matteo Capucci (he/him) (Oct 16 2020 at 17:46):And a paper
Matteo Capucci (he/him) (Oct 16 2020 at 17:47):But really no details
Matteo Capucci (he/him) (Oct 16 2020 at 17:47):Did google failed to spit out anything on that?
Matteo Capucci (he/him) (Oct 16 2020 at 17:48):There it goes https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category.20theory/topic/Bi-Heyting.20algebras
Matteo Capucci (he/him) (Oct 16 2020 at 17:49): Fawzi Hreiki (Oct 16 2020 at 18:06):Ah thank you. I’ll check those out
Eduardo Ochs (Oct 17 2020 at 15:38):Fawzi Hreiki said:
One of the first things we prove about toposes is that the subobject classifier is an internal Heyting algebra object. Dually, there is the notion of a co-Heyting algebra (https://ncatlab.org/nlab/show/co-Heyting+algebra) which is a (bounded) lattice equipped with a subtraction operator which is (element-wise) left adjoint to taking sums (joins). The prototypical example of such a thing is the poset of closed subsets in any topological space.
A bi-Heyting algebra is then defined as a Heyting algebra which is also a co-Heyting algebra. Any Boolean algebra is a bi-Heyting algebra - although in an uninteresting way. Are there any examples of 'naturally occurring' non-Boolean toposes whose subobject classifiers are bi-Heyting algebras?
The planar Heyting algebras of http://angg.twu.net/math-b.html#zhas-for-children-2 are bi-Heyting algebras, and they look natural to me - possibly because I learned to like finite things that are easy to visualize...
Fawzi Hreiki (Oct 17 2020 at 19:30):Funnily enough, I actually just came by this paper on the arXiv and downloaded it to read it later. I'll check it out.