Lawvere's open problem on quotient toposes has been solved for boolean Grothendieck toposes but not for non-boolean toposes. As a simple and non-trivial example of a non-boolean topos, this paper ...provides a complete classification of the quotient toposes of the topos of discrete dynamical systems, which, in this context, are sets equipped with an endofunction. This paper also offers an order-theoretic framework to address the open problem, particularly useful for locally connected toposes.
Our result is deeply related to monoid epimorphisms. At the end of this paper, utilizing the theory of lax epimorphisms in the 2-category Cat, we explain how (non-surjective) monoid epimorphisms from N correspond to (non-periodic) behaviors in discrete dynamical systems.
One of the most fundamental facts in topos theory is the internal parameterization of subtoposes: the bijective correspondence between subtoposes and Lawvere-Tierney topologies. In this paper, we ...introduce a new but elementary concept, "a local state classifier," and give an analogous internal parameterization of hyperconnected quotients (i.e., hyperconnected geometric morphisms from a topos). As a corollary, we obtain a solution to the Boolean case of the first problem of Lawvere's open problems.
This paper solves the first problem of the open problems in topos theory posted by William Lawvere, which asks the existence of a Grothendieck topos that has a proper class many quotient topoi. This ...paper concretely constructs such Grothendieck topoi, including the presheaf topos of the free monoid generated by countably infinite elements \(\mathbf{PSh}(M_\omega)\). Utilizing the combinatorics of the classifying topos of the theory of inhabited objects and considering pairing functions, the problem is reduced to making rigid relational structures. This is accomplished by using Kunen's theorem on elementary embeddings in set theory.
This paper gives a classification of classes of discrete dynamical systems (a set equipped with an endofunction) closed under finite limits and small colimits. The conclusion is simple: they ...bijectively correspond to the ideals of the product poset \(\mathbb{N} \times \mathbb{N}\), where the first \(\mathbb{N}\) is ordered by the usual order and the second is by the divisibility. Our method is based on a detailed analysis of the behaviors of states, especially non-periodic behaviors, in discrete dynamical systems. Specifically, extending the fundamental quantity, time until entering a loop, to even those states that do not enter a loop plays a crucial role. Our classification is closely related to epimorphisms from \(\mathbb{N}\) in the category of monoids. There are countably many injective epimorphisms from \(\mathbb{N}\), including \(\mathbb{N} \to \mathbb{Z}\). Those injective epimorphisms correspond to the non-periodic behaviors of states of discrete dynamical systems. We discuss this point at the end of the paper. This fun puzzle is motivated by an open problem in topos theory. Lawvere left open problems in topos theory on his webpage, and the first problem is called quotient toposes. The main theorem of this paper provides a non-trivial example of this problem, which is not implied by any known results. This paper also provides a theoretical framework to address the open problem. We define a preorder among the objects of a Grothendieck topos (which we have named generative order), which enables us to reduce calculations of quotient toposes to calculations of objects. Our method is its application to the topos of discrete dynamical systems.
