Papers

Quotient toposes of discrete dynamical systems (Joint work with Yuhi Kamio) (Journal of Pure and Applied Algebra) 

• This 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.

Internal Parameterizations of Hyperconnected Quotients (arXiv) 

• In this paper, we have introduced the notion of a local state classifier, which is defined as a colimit of all monomorphisms, (if it exists).

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

• This is a partial solution to the first question of Lawvere's open problems.