(j.w.w. Yuhi Kamio) Quotient toposes of discrete dynamical systems (Journal of Pure and Applied Algebra)
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 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 paper gives a solution to the first question of Lawvere's open problems!
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.
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 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.
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)!
Topoi of automata II: Hyperconnected geometric morphisms, syntactic monoids, and language classes
Subtopoi of free monoid actions (j.w.w. Morgan Rogers)
Normalization of a subgroup, in a category, and of a word-congruence
Demystifying local state classifiers: local state classifier in a total category with a factorization system
Notes on Rieg theory: semiring with exponentials in logic, profinite arithmetic, enumerative combinatorics, and category theory
Games as recursive coalgebras (or Categorical Origin of Grundy number)
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.
Topoi with enough projectives
Topoi of automata III: Geometry of Σ-sets
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)
An enriched-categorical origin of ε-transition
Category Theoretic Ordinal Invariants
Totally disconnected topoi
A topos-theoretic view on Gabriel's theorem
The lattice of hyperconnected quotients is a module of the semiring of productive weak topologies
On limits in FinSet
Twisted Regular Tetrahedra and Eisenstein Integers
When do finite presheaves form a topos? (j.w.w. Jérémie Marquès)