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Stream: deprecated: topos theory

Topic: intersections of subobjects

Naso (Nov 08 2021 at 15:38):

I'm trying to understand this question and its reply by Derek Elikins about intersections of subobjects:

https://math.stackexchange.com/a/2327966

The reply says:

*"...In a topos, or any category with a subobject classifier, we can internalize [taking arbitrary intersecitons of subobjects] into an operation $p=\bigwedge_{i\in I} p_i : A \to \Omega$ for an arbitrary family $\{p_i : A \to \Omega \mid i \in I\}$ such that $\land \circ \langle p,p_i \rangle = p$ for each $i \in I$ and, for all $q : A \to \Omega$, if $\forall i \in I. \land \circ \langle q,p_i \rangle = q$ then $\land \circ \langle q,p \rangle = q$.

I don't think the extra structure of a topos provides any short-cut to having arbitrary intersections. When we talk about "arbitrary" pullbacks/products/limits, we mean set-indexed versions, and the structure of an (elementary) topos gives no tools for talking about this external notion of set or set-indexed families."

Question:
I thought that a lattice of subobjects in a topos is always complete as a set-theoretic lattice, so don't we still 'have' arbitrary set-indexed intersections?

In that case, I think $I$ above is an object of the topos rather than a set---so is Derek saying we can't internalise arbitrary set-indexed intersections, but only intesections indexed by objects of the topos?

More vaguely, what is the benefit of 'internalising'? I think it has something to do with the internal language, but I'm not sure what the practical implications of this are..

Reid Barton (Nov 08 2021 at 15:42):

To clarify, by "topos" do you mean elementary topos or Grothendieck topos?

Naso (Nov 08 2021 at 15:45):

Reid Barton said:

To clarify, by "topos" do you mean elementary topos or Grothendieck topos?

the SE question is about elementary toposes I think, but I'm interested in both cases

Fawzi Hreiki (Nov 08 2021 at 15:47):

In Grothendieck toposes, the answer is yes. In elementary toposes, the answer is no, but you do have internally indexed completeness.

Fawzi Hreiki (Nov 08 2021 at 15:49):

This all generalises to toposes bounded over a base: if $E$ is a topos bounded over a base topos $S$ then $E$ is complete and cocomplete with respect to $S$ (where you need to phrase things in terms of fibred/indexed categories)

Morgan Rogers (he/him) (Nov 08 2021 at 15:50):

Nasos Evangelou-Oost said:

More vaguely, what is the benefit of 'internalising'? I think it has something to do with the internal language, but I'm not sure what the practical implications of this are..

One (emergent) motivation for topos theory is that any elementary topos can provide a base over which to do mathematics. The objects of the base topos replace sets, and it's reasonable to want to perform this replacement consistently throughout the theory, which amounts to internalization of concepts into the base topos.

Naso (Nov 08 2021 at 15:59):

I see, that explains it... I had seen the result about completeness of $\mathsf{Sub} E$ in Maclane-Moerdijk (p146) but did not remember it was talking about Grothendieck toposes only.

Naso (Nov 08 2021 at 16:00):

Is the internal language still applicable to grothendieck toposes? EDIT: Given that grothendieck toposes are elementary toposes, maybe this question is not meaningful.

Fawzi Hreiki (Nov 08 2021 at 16:14):

Yes it's definitely applicable: eg how the existential quantifier becomes existence on an open cover in the sheaf topos of a topological space

Morgan Rogers (he/him) (Nov 09 2021 at 11:01):

Nasos Evangelou-Oost said:

Is the internal language still applicable to grothendieck toposes? EDIT: Given that grothendieck toposes are elementary toposes, maybe this question is not meaningful.

In the Grothendieck topos setting, internalization typically produces concepts which are strictly stronger than their external set-theoretic counterparts. The reason is that there is a copy of Set living faithfully (and fully in the connected case) in any non-degenerate Grothendieck topos via the "constant sheaf" functor which is adjoint to the global sections functor, and external set-indexed constructions end up (closely related to) internal constructions indexed by these constant sheaves, while there are non-constant objects indexing constructions which aren't necessarily expressible in the same framework externally.