Problems

Background: In (TAC), I have defined the notion of a local state classifier Ξ as the colimit of all monomorphisms. Although this is originally motivated by the classification of quotient topoi, I believe this notion has much more potential:

1.1 Relationship with Étendue

• 1.1.1. 🟦 Is it true that a Grothendieck topos E is an étendue if and only if its local state classifier, which is an E-internal semilattice, has the (global) bottom element \bot: 1→Ξ? (cf. A Grothendieck topos is localic if and only if its local state classifiers terminal. ) I have shown that this holds for all presheaf topoi as well.  (This question is also posed by Prof. Matias Menni.) 

• 1.1.2. 🟦 The local state classifier of a Grothendieck topos E is the colimit of all local monomorphisms in the sense of [Kock Moerdijk] (pointed out by Ryo Suzuki). Is it true that a morphism f:A → B is local mono if and only if f commutes with ξ, (ξ_B f = ξ_A) ? (updated: 2025/03/25)

• 1.1.3. If the above proposition is true, we can define a relative local state classifier for any geometric morphism. Does this also classify all (relative) hyperconnected geometric morphisms just like (TAC)?

• 1.1.4.🟦 Related to the above question, does an elementary topos have a local state classifier if it is bounded over a topos with a local state classifier? (cf. Every Grothendieck topos has a local state classifier.)

1.2 The Normalizer Operator in a Topos

• 1.2.1. Typically, the canonical morphism ξ:Ξ→Ξ is not identity. It's more interesting! In the case of the topos of G-actions with a group G, Ξ is the G-set of subgroups of G (equipped with the conjugation action). The map ξ:Ξ→Ξ is the normalizer operator. So let's call ξ:Ξ→Ξ (in a general Grothendieck topos) the Normalizer operator. How does it look like in other topoi?

• 1.2.2. We can also define" the object consisiting of all normal subgroups" in an arbitrary topos, using the normalizer operator. What does it classify? (Note that, in the original case of the group action topos, normal subgroups are in bijective correspondence with essential and connected geometric morphism from G-Set.)

1.3 Local state classifier of slice topos

• 1.3.1. 🟦 Is the local state classifier of the slice topos E/X given by the pullback of the order projection pr₂︎: ≦ → Ξ×Ξ→Ξalong the canonical morphism ξ:X →Ξ? Or more informally, is it "the object of all local states that are more unfolded than X"?

1.4 Local state classifier of a sheaf topos

• 1.4.1. Although (TAC) provides a construction of a local state classifier of every Grothendieck topos, it is not easy to calculate in general (except presheaf topoi). Is there a simple way to describe Ξ when the defining site (C,J) is given by a special form?  For example, is there a nice description of Ξ in the sheaf topos over a regular/coherent/atomic/local/locally connected/extensive site?

1.5 Generalization

• 1.5.1. Is there a common generalization of the classification by LSC and Gabriel's classification theorem of localizing subcategory of module category? (Possibly done by a generalization to the enriched topos theory.)

1.0 Other questions

• 1.0.1 How does the local state classifier of "a topos of spaces" look like? For example, how does the local state classifier of the topos of simplicial set look like? Is there a chatacterization of local (hyperonnected, or cohesive) topoi in terms of a local state classifier? (cf: This problem is partially answered by Menni's paper: Non-singular maps in toposes with a local state classifier.)

• 1.0.2 For some Galois group action (ringed) topos, the correspondence between internal filters of Ξ and hyperquotient (especially its counit at the internal ring) provides a Galois-theoretic correspondence. How can this phenomenon be generalized to a broader context? (I've been working on this.)

• 1.0.3 What does Ξ contravariantly represent? (cf. In every presheaf topos PSh(C), the morphisms y(c) → Ξ are in bijection with quotient objects of y(c). )

• 1.0.4 It is known that the collection of all ∧-semilattice endomorphisms of the subobject classifier Ω form a (non-commutative) semiring R ([Khanjanzadeh, Madanshekaf]). By the theory of Lawvere-Tierney topology ,the idempotents of R are in bijective correspondence with the subtopoi of E.  By the way, as an immediate corollary of (TAC), the ∧-semilattice (thus commutative monoid) of hyperconnected quotients of a topos E (with a local state classifier) form an R-module. How are they, subtopoi and hyperconnected quotients, related?  How is it related to the second problem? (From this point of view, a LT-topology is like a "clopen" in "Spec R", and the module of hyperquotients is a kind of (quasi-coherent) sheaves on it. So we can say something like: the "support" of the hyperconnected quotients is in the "clopen" labeled by a subtopos, ...)

• 1.0.5 Is there a direct connection with copower objects? This question was independently asked by Richard Garner and Peter Johnstone, but I don't know the answer.

• 1.0.6 Is there a direct connection with the isotropy group of a topos?

• 2.0.1. Is there a syntactic way to construct the (Boolean ringed) topos of regular languages from the syntax of regular expressions?

• 2.0.2 🟦  What is the number of quotient topoi of PSh(Σ*)?  Is it small or not?

  • If |Σ|=1, it's small. (j.w.w. Yuhi Kamio)

  • If |Σ| is infinite, it becomes proper-class size. (Another j.w.w. Yuhi Kamio)

  • How about the case where |Σ|=2?

• 2.0.3. 🟦 What is the topos cohomology of the topos of regular languages, or the topos of star-free languages?

• 2.0.4.  🟦 Can we describe all points of the topos of regular languages? (remark: The points of the topos of Σ-sets can be fully described. There is one canonical point. Non canonical points are in bijective correspondence with infinite words up to finite edit distance! ) (Or more importantly, what theory does this topos classify?)

• 2.0.5. What are exponential objects in the topos of regular languages in terms of automata theory? (Possibly related to 2.0.1.)

• 2.0.6. 🟦 The automorphism group of the topos Σ-Set, which we will denote Aut(Σ-Set), is the symmatric group 𝔖(Σ).  What is Aut (Σ-Set_o.f.)? What do the symmetries imply to the structure of regular languages?

•  3.0.1. 🟦 We say that a Grothendieck topos E is totally disconnected, if every connected geometric morphism from E is an equivalence. If E is totally disconnected, then it is a sheaf topos over a totally disconnected locale. (A locale is totally disconnected, if every connected sublocale is a point.) How about the converse? (cf. There is a totally disconnected topos that is neither zero-dimensional nor totally separable. Consider ℚ+ℚ quotiented by ℚ*.)

• 4.0.1. What is the right definition of completely connected geometric morphisms?

• 4.0.2. Is there an elementary definition of "container object" that makes sense in arbitrary elementary topos? (cf. The definition of local topos can be rephrased in an elementary way.)

• 4.0.3. A presheaf topos PSh(C) is completely connected, if and only if PSh(C^op) is local. Is there a broader duality correspondence between completely connected topoi and local topoi? For example, is the category of the Lawvere distributions on a local topos a completely connected topos (just like the presheaf case)? If not, is this described in terms of the monoidal closed structure of presentable categories?

•  5.0.1. 🟦 Classify all the monoidal closed structure on the locally finitely presentable category of combinatorial games, (which wil appear in my ongoing paper.)

• 5.0.2. Is there a double category Games, whose vertical part is our category of games as recursive coalgebras, and whose horizontal part is (in some sense) Joyal's compact closed category of games and strategies?

•  🟦 6.0.1. For a small category C, when is its finite presheaf category fPSh(C) = [C^op, Set] an elementary topos (or a locally cartesian closed category)? If C is a finite category or a group, then fPSh(C) is known to be a topos. More generally, if every slice C/c for every object c\in C is essentially finite, then fPSh(C) is a topos. (I've been thinking about this problem with Jérémie Marquès. My personal motivation of this problem is enumerative combinatorics, especially the theory of "Burnside riegs" in the context of rieg theory, which will be described below.)

• 7.0.1. Let F1 denote the free rieg generated by one element x. Can we interpret elements of F1 as a kind of combinatorial games (just like https://golem.ph.utexas.edu/category/2006/10/classical_vs_quantum_computati_3.html). If so, can we interpret the morphism F1 → N that sends x to 0 in terms of games? 

• 🟦 7.0.2.  When can a function ℕ→ℕ be described by an element of the free algebra F1? 

• 🟦 7.0.3. What's the subrieg of $[0,\infty)$ generated by non-negative rational numbers? Are there any non-trivial equations? Is it isomorphic to the \dq{free rieg} generated by the rig $ℚ_{\geq 0}?$

• 🟦 7.0.4. Higgs Prime conjecture. https://en.wikipedia.org/wiki/Higgs_prime