Papers

• This paper introduces my favorite notion, a local state classifier, which is defined as a colimit of all monomorphisms, (if it exists).

• Utilizing this simple tool, I establish an internal parameterization of hyperconnected quotients, which is a bijective correspondence between hyperconnected geometric morphisms and internal semilattice homomorphisms.

• This result is meant to be an analogy of the theory of Lawvere-Tierney topology and to be a partial solution to the first question of Lawvere's open problems.

• This very elementary and combinatorial paper gives a classification of classes of discrete dynamical systems (a set equipped with an endofunction) closed under finite limits and small colimits.

• This puzzle is motivated by the first question of Lawvere's open problems and gives a non-trivial example of the open problem.

• What makes this paper interesting is the relationship with other mathematical concepts, including monoid epimorphisms, lax epimorphismsm, LSC (the previous paper), and periodic behavior of states of discrete dynamical systems.

• This paper studies  topoi that admit a left adjoint to a left adjoint to a left adjoint to the global sections functor, which we call completely connected topoi.

• We give a site characterisation of such class of topoi, and provides many examples.

• What's interesting about this? There are several duality between completely connected topoi and local topoi.

• This paper gives a solution to the first question of Lawvere's open problems!

• This paper solves one of the seven open problems posted by William Lawvere.

• This result may be of interest also from the view point of Joyal's combinatorial species. Symmetric simplicial set is a kind of a species for which we can pullback all structures along all functions. The ``Aufhebung relation" studied in this paper is a study of the interplay of two dual notion of complexities of species.

• This paper introduces a topos-theoretic point of view on formal language theory.

• There will be a follow-up paper, topoi of automata II, which deals with hyperconnected quotients and language classes.

• In 1901, Bouton proved that a winning strategy of the game of Nim is given by the bitwise XOR, called the nim-sum. But, why does such a weird binary operation work? Led by this question, this paper introduces a categorical reinterpretation of combinatorial games and the nim-sum. The main categorical gadget used here is recursive coalgebras, which allow us to redefine games as ``graphs on which we can conduct recursive calculation'' in a concise and precise way.

• For game-theorists, we provide a systematic framework to decompose an impartial game into simpler games and synthesize the quantities on them, which generalizes the nim-sum rule for the Conway addition. To read the first half of this paper, the categorical preliminaries are limited to the definitions of categories and functors.

• For category theorists, this paper offers a nicely behaved category of games, which is a locally finitely presentable symmetric monoidal closed category comonadic over admitting a subobject classifier!

• As this paper has several ways to be developed, we list seven open questions in the final section. 

• This paper provides a new categorical definition of a normalization operator motivated by topos theory and its applications to algebraic language theory.

• We first define a normalization operator Ξ→Ξ in any category that admits a colimit of all monomorphisms Ξ, which we call a local state classifier. In the category of group actions for a group G, this operator coincides with the usual normalization operator, which takes a subgroup H⊂G and returns its normalizer subgroup Nor_G(H)⊂G.

• Using this generalized normalization operator, we prove a topos-theoretic proposition that provides an explicit description of a local state classifier of a hyperconnected quotient of a given topos. 

• We also briefly explain how these results serve as preparation for a topos-theoretic study of regular languages, congruences of words, and syntactic monoids.

I am currently writing the following papers as well. If you're interested in some of them, please let me know (horaryuya38@gmail.com). I am willing to send my draft (after checking with my coauthors)!

1. Topoi of automata II: Hyperconnected geometric morphisms, syntactic monoids, and language classes

2. Subtopoi of free monoid actions (j.w.w. Morgan Rogers)

3. Demystifying local state classifiers: local state classifier in a total category with a factorization system

4. Notes on Rieg theory: semiring with exponentials in logic, profinite arithmetic, enumerative combinatorics, and category theory

5. Differential calculus of impartial combinatorial games (j.w.w. Ryo Suzuki)

+ some ongoing joint works that I am not supposed to mention here.

I am also writing the following (very immature) drafts, though I’m not sure yet whether I will try to publish them. Again, I am very happy to share those ideas.

1. Topoi with enough projectives

2. Topoi of automata III: Geometry of Σ-sets

3. Dynamical systems on pretopological spaces (This is partially available as What is the geometry behind Conway's game of life?  a first step with a relative topos)

4. An enriched-categorical origin of ε-transition

5. Category Theoretic Ordinal Invariants

6. Totally disconnected topoi

7. A topos-theoretic view on Gabriel's theorem

8. The lattice of hyperconnected quotients is a module of the semiring of productive weak topologies

9. On limits in FinSet

10. Twisted Regular Tetrahedra and Eisenstein Integers

11. When do finite presheaves form a topos? (j.w.w. Jérémie Marquès)