Papers

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.

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.

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.