233 research outputs found
On the topology of a boolean representable simplicial complex
It is proved that the fundamental groups of boolean representable simplicial complexes (BRSC) are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension >= 2. In the case of dimension 2, it is shown that BRSC have the homotopy type of a wedge of spheres of dimensions 1 and 2. Also, in the case of dimension 2, necessary and sufficient conditions for shellability and being sequentially Cohen-Macaulay are determined. Complexity bounds are provided for all the algorithms involved
Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity
This paper is motivated by questions such as P vs. NP and other questions in
Boolean complexity theory. We describe an approach to attacking such questions
with cohomology, and we show that using Grothendieck topologies and other ideas
from the Grothendieck school gives new hope for such an attack.
We focus on circuit depth complexity, and consider only finite topological
spaces or Grothendieck topologies based on finite categories; as such, we do
not use algebraic geometry or manifolds.
Given two sheaves on a Grothendieck topology, their "cohomological
complexity" is the sum of the dimensions of their Ext groups. We seek to model
the depth complexity of Boolean functions by the cohomological complexity of
sheaves on a Grothendieck topology. We propose that the logical AND of two
Boolean functions will have its corresponding cohomological complexity bounded
in terms of those of the two functions using ``virtual zero extensions.'' We
propose that the logical negation of a function will have its corresponding
cohomological complexity equal to that of the original function using duality
theory. We explain these approaches and show that they are stable under
pullbacks and base change. It is the subject of ongoing work to achieve AND and
negation bounds simultaneously in a way that yields an interesting depth lower
bound.Comment: 70 pages, abstract corrected and modifie
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, MatÃas. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentin
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