(When) Do Essential Subtoposes Form an Internal Locale? · learning: questions

you actually do get an "internal sublocale of levels" and it induces a(n essential) geometric surjection from the toposes of globular sets, semi-simplicial sets, plain semi-cubical sets, and semi-cubical sets with diagonals, respectively. However, in maximal generality it isn't clear to me either that meets distribute over joins, or that you even get a sheaf.

James Deikun (Jan 27 2024 at 18:08):

The standard reference for this is Kelly-Lawvere but their arguments don't seem to internalize easily, nor do they attack the problem of meets distributing over joins.

James Deikun (Jan 28 2024 at 00:36):

One issue making this difficult is that there does not seem to have been any intrinsic characterization discovered of the meets of essential subtoposes; they are known only in the form of the join of all mutual lower bounds of the arguments of the meet. This means that infinite distributivity depends on the "density" of essential subtoposes of

\mathcal{E}

and hence of orthogonal factorization systems in

\mathcal{E}

. (Essential subtoposes correspond to classes of morphisms that are the left class of one factorization system while being the right class of another!)

James Deikun (Jan 28 2024 at 11:50):

It seems that there actually is a "sheaf"

\it{Ess}

of essential subtoposes, at least, although I'm not sure it's small in general. The morphisms

A \to \it{Ess}

are the essential subtoposes of

\mathcal{E}/A

Proof sketch (is a presheaf)

The functorial action of a presheaf on $\mathcal{E}$ is given by a straightforward modification of Elephant A4.1.3. For a morphism $f : A \to B$ and an essential embedding $g : \mathcal{F} \to \mathcal{E}/B$ :

Inverse image on objects: $h : C \to A \mapsto g^*(h) : g^*(hf) \to g^*(f)$ , action on arrows is as $g^*$
Direct image is $h : C \to g^*(f) \mapsto (\eta^{g^*_*}_f)^*(g^*(h)) : \mathrm{dom}(g_*C \times_{g_*(g^*(f))} f) \to B$ , action on arrows uses the same formula
Essentiality is $h : C \to B \mapsto \epsilon^{g_!^*}_f \circ g_!(h)$ , action on arrows is as $g_!$
Importantly, the counit of the inverse-direct adjunction is $h : C \to g^*(f) \mapsto \epsilon^{g^*_*}_C \circ (g^*g_*(h))^*g^*(\eta^{g^*_*}_f)$ . This is an isomorphism because $\epsilon^{g^*_*}$ is a natural isomorphism and so is $g^*\eta^{g^*_*}$ as the inverse of $\epsilon^{g^*_*}g^*$ , meaning we still land in embeddings.

James Deikun (Jan 28 2024 at 11:55):

Proof sketch (gluing in canonical topology)

James Deikun (Jan 28 2024 at 20:15):

Proof sketch (smallness)

A Lawvere-Tierney topology on $\mathcal{E}/A$ is determined by its underlying map $j : \Omega_{/A} \to \Omega_{/A}$ . But $\Omega_{/A}$ is just the second projection of $\Omega \times A$ . But this is a cofree object of the slice category on $\Omega$ so we have $\mathcal{E}/A(\Omega_{/A},\Omega_{/A}) \cong \mathcal{E}(\Omega \times A, \Omega) \cong \mathcal{E}(A, \Omega^\Omega)$ . Thus $\it{Ess}$ is a subsheaf of $\Omega^\Omega$ and is small.

Peter Arndt (Jan 29 2024 at 03:48):

What are those factorization systems where essential subtoposes show up? I think those are new to me.

James Deikun (Jan 29 2024 at 11:24):

I was referring to Theorem 3.2 of Kelly-Lawvere, which characterizes "essential reflective subcategories" in general, hence essential subtoposes in particular, as the full subcategories of objects right orthogonal to classes of morphisms

\mathscr{E}

such that there exist orthogonal factorization systems

(\mathscr{E},\mathscr{M})

and

(\mathscr{N},\mathscr{E})

Jonas Frey (Jan 30 2024 at 03:22):

\Delta\dashv\Gamma\dashv\nabla:S\to E

is an "adjoint cylinder" between finitely complete and cocomplete categories (with

\Delta

and

\nabla

fully faithful), the arrows that are inverted by

\Gamma

are the left class of an OFS whose right class are the arrows whose unit-naturality-square for the

\Gamma\dashv\nabla

adjunction is a pullback. Dually, the

\Gamma

-inverted arrows are the right class of an OFS whose left class are the arrows whose counit-naturality-square for

\Delta\dashv\Gamma

is a pushout.

Jonas Frey (Jan 30 2024 at 03:25):

For example, if

\Gamma

is the forgetful functor from topological spaces to sets, the "middle" arrows are continuous maps which are bijective on points, the right maps are those where the domain topology is the initial topology wrt the codomain, and the left maps are those where the codomain topology is the final topology wrt the domain.

Peter Arndt (Jan 30 2024 at 13:26):

James Deikun (Feb 01 2024 at 05:44):

I claim that for a Lawvere-Tierney topology

j

on a Grothendieck topos, the following are equivalent:

James Deikun (Feb 03 2024 at 22:14):

As every open subtopos has its complementary closed subtopos, so every essential subtopos has a complementary reflective subcategory. Say

f

is the inclusion of an essential subcategory in

\mathcal{E}

. Then

{f^*}^{-1}(1)

is a reflective subcategory of

\mathcal{E}

; the reflector is given by the following pushout:

\begin{CD} f_! f^* A @>{\epsilon}>> A \\ @V{f_! f^* !}VV @VV{\eta}V \\ f_! f^* 1 @>>> f_\& A \end{CD}

where

\epsilon

is the counit of the coreflector

f_! f^*

and

\eta

becomes the unit of the reflector

f_\&

. These objects are "sheaves" in that they are local with respect to subobjects that include all the non-skeletal parts of an object; however the complementary closure operator (union with the skeleton) does not commute with arbitrary pullbacks and so the reflector is not left exact. (Or rather, when it is, it is the reflector for a closed subtopos and

f

is open.)

James Deikun (Feb 03 2024 at 22:19):

I hope that, even though the lattice of arbitrary reflective subcategories of a topos is not itself very well-behaved, the existence of these complements (or maybe "complements") serves to regulate the behaviour of essential subtoposes.

Jonas Frey (Feb 04 2024 at 00:15):

@James Deikun Interesting! To make sure I understand correctly: in the case of the level

n

of simplicial sets, the complementary reflective subcategory would be the simplicial sets with exactly one simplex in all dimensions up to

n

James Deikun (Feb 04 2024 at 01:19):

That's exactly right! The entire thing parallels the discussion of closed subtoposes in A4.5.3 of the Elephant, with appropriate replacements (

- \times U

becomes the coreflector and

\mathcal{E}/U

becomes the coreflective subcategory of skeletal objects).

Jonas Frey (Feb 04 2024 at 01:24):

That's a very nice observation, I'm not aware of that appearing anywhere in the Lawverian "dialectical" interpretation of these things. Menni might now ...

Jonas Frey (Feb 04 2024 at 01:27):

But are you hoping that the lattice of essential sub-toposes might always be distributive? Have you tried finding counterexamples in simple presheaf toposes?

James Deikun (Feb 04 2024 at 01:33):

I kind of am hoping that, but even if it's not true I want to understand the boundaries of when it is true. You make a good point though; after putting this much time in without finding a general principle yet I should probably try giving more time to that kind of "experiment".

Jonas Frey (Feb 04 2024 at 01:41):

Are you aware of the characterization of essential sub-toposes in presheaf toposes in terms of idempotent "ideals" (Kelly-Lawvere, Thm 4.4)? I was thinking about similar things, and my next step would be to try to spell that out for some very simple index categories. I'm not even sure if the levels are always of presheaf type for finite, Cauchy-complete index categories?

James Deikun (Feb 04 2024 at 01:42):

They are for any finite index categories. The simplest example of a non-presheaf essential subtopos of a presheaf category I've come up with is presheaves on the poset of rational numbers, with strict inequality as the idempotent ideal.

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Topic: (When) Do Essential Subtoposes Form an Internal Locale?

James Deikun (Jan 27 2024 at 18:05):