Continuous cohesion over sets

A pre-cohesive geometric morphism p : E → S satisfies Continuity if the canonical p!(Xp ∗S) → (p!X) S is an iso for every X in E and S in S. We show that if S = Set and E is a presheaf topos then, p satisfies Continuity if and only if it is a quality type. Our proof of this characterization rests on a related result showing that Continuity and Sufficient Cohesion are incompatible for presheaf toposes. This incompatibility raises the question whether Continuity and Sufficient Cohesion are ever compatible for Grothendieck toposes. We show that the answer is positive by building some examples.Fil: Menni, Matías. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentin

Combinatorial functional and differential equations applied to differential posets

We give combinatorial proofs of the primary results developed by Stanley for deriving enumerative properties of differential posets. In order to do this we extend the theory of combinatorial differential equations developed by Leroux and Viennot.Facultad de Ciencias ExactasLaboratorio de Investigación y Formación en Informática Avanzad

Finite Presheaf categories as a nice setting for doing generic programming

The purpose of this paper is to describe how some theorems about constructions in categories can be seen as a way of doing generic programming. No prior knowledge of category theory is required to understand the paper. We explore the class of nite presheaf categories. Each of these categories can be seen as a type or universe of structures parameterized by a diagram (actually a nite category) C. Examples of these categories are: graphs, labeled graphs, nite automata and evolutive sets. Limits and colimits are very general ways of combining objects in categories in such a way that a new object is built and satis es a certain universal property. When con- centrating on nite presheaf categories and interpreting them as types or structures, limits and colimits can be interpreted as very general operations on types. Theorems on the construction of limits and colimits in arbitrary categories will provide a generic implementation of these operations. Also, nite presheaf categories are toposes. Because of this, each of these categories has an internal logic. We are going to show that some theorems about the truth of sentences of this logic can be interpreted as a way an implementing a generic theorem prover. The paper discusses non trivial theorems and de nitions from category and topos theory but the emphasis is put on their computational content and in what way they provide rich and abstract data structures and algorithms.Eje: Workshop sobre Aspectos Teoricos de la Inteligencia ArtificialRed de Universidades con Carreras en Informática (RedUNCI

A representation theorem for integral rigs and its applications to residuated lattices

We prove that every integral rig in Sets is (functorially) the rig of global sections of a sheaf of really local integral rigs. We also show that this representation result may be lifted to residuated integral rigs and then restricted to varieties of these. In particular, as a corollary, we obtain a representation theorem for pre-linear residuated join-semilattices in terms of totally ordered fibers. The restriction of this result to the level of MV-algebras coincides with the Dubuc-Poveda representation theorem.Comment: Manuscript submitted for publicatio

Cocomplete toposes whose exact completions are toposes

Let ε be a cocomplete topos. We show that if the exact completion of ε is a topos then every indecomposable object in ε is an atom. As a corollary we characterize the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere-Schanuel characterization of Boolean presheaf toposes and Hofstra's characterization of the locally connected Grothendieck toposes whose exact completion is a Grothendieck topos. We also show that for any topological space X, the exact completion of Sh (X) is a topos if and only if X is discrete. The corollary in this case characterizes the Grothendieck toposes with enough points whose exact completions are toposes.Laboratorio de Investigación y Formación en Informática Avanzad

Monic skeleta, Boundaries, Aufhebung, and the meaning of 'one-dimensionality'

Let E be a topos. If l is a level of E with monic skeleta then it makes sense to consider the objects in E that have l-skeletal boundaries. In particular, if p : E o S is a pre-cohesive geometric morphism then its centre (that may be called level 0) has monic skeleta. Let level 1 be the Aufhebung of level 0. We show that if level 1 has monic skeleta then the quotients of 0-separated objects with 0-skeletal boundaries are 1-skeletal. We also prove that in several examples (such as the classifier of non-trivial Boolean algebras, simplicial sets and the classifier of strictly bipointed objects) every 1-skeletal object is of that form.Fil: Menni, Matías. Universidad Nacional de La Plata. Facultad de Informática. Laboratorio de Investigación y Formación en Informática Avanzada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentin

Continuous cohesion over sets

A pre-cohesive geometric morphism p : E → S satisfies Continuity if the canonical p!(Xp ∗S) → (p!X) S is an iso for every X in E and S in S. We show that if S = Set and E is a presheaf topos then, p satisfies Continuity if and only if it is a quality type. Our proof of this characterization rests on a related result showing that Continuity and Sufficient Cohesion are incompatible for presheaf toposes. This incompatibility raises the question whether Continuity and Sufficient Cohesion are ever compatible for Grothendieck toposes. We show that the answer is positive by building some examples.Facultad de Ciencias Exacta

Continuous cohesion over sets

A pre-cohesive geometric morphism p : E → S satisfies Continuity if the canonical p!(Xp ∗S) → (p!X) S is an iso for every X in E and S in S. We show that if S = Set and E is a presheaf topos then, p satisfies Continuity if and only if it is a quality type. Our proof of this characterization rests on a related result showing that Continuity and Sufficient Cohesion are incompatible for presheaf toposes. This incompatibility raises the question whether Continuity and Sufficient Cohesion are ever compatible for Grothendieck toposes. We show that the answer is positive by building some examples.Facultad de Ciencias Exacta