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Stream: deprecated: topos theory

Topic: pointed subobjects

Naso (Mar 20 2022 at 05:51):

if $1 \to X$ is a pointed object in a topos, does its set of pointed subobjects form a heyting algebra?

Zhen Lin Low (Mar 20 2022 at 06:32):

Well, any morphism with domain $1$ is monic, so it is a subobject. If you have a distributive lattice and take the set of elements above a certain element, you get another distributive lattice which should be considered the lattice of subobjects of the complement of the chosen element. (Whether it is genuinely complementary or not is a separate question.) I think in a topos these will even be Heyting algebras – certainly this is so for Grothendieck toposes.

Naso (Apr 16 2022 at 12:23):

A follow up questions:

In Maclane-Moerdijk III.9 Prop. 2, it says in a Groth. topos, for a morphism $f : E \to F$ the pullback functor $f^* : \mathsf{Sub} F \to \mathsf{Sub} E$ has left and right adjoints $\exists_f \dashv f^* \dashv \forall_f$.

If $f$ is a pointed map between pointed objects, does it still give rise to this adjoint triple of maps between Heyting algebras of pointed subobjects? I am mainly interested in the case of pointed simplicial sets and if it is true there.

And either way, do the maps $\exists_f$, $f^*$, $\forall_f$ generally preserve the Heyting structure? In MM they only seemed to mention that $f^*$ is order-preserving.

Naso (Apr 27 2022 at 03:43):

Ok, I have tried to work this one out and my conclusion is that in a topos, while $f^*$ is always a Heyting algebra morphism, the maps $\exists_f$ and $\forall_f$ only preserve resp. joins and meets, as is forced by the adjointness properties.

For pointed objects, the right adjoint $\forall_f$ does not exist because the preimage $f^*$ does not generally preserve the empty join (which is the basepoint).

If someone can confirm this I would be grateful!

Mike Shulman (Apr 27 2022 at 04:35):

Yes, that sounds right.

Naso (Apr 27 2022 at 06:34):

thank you, Mike :pray: