Myles Tierney

Forcing Topologies and Classifying Topoi

Publisher Summary This chapter discusses forcing topologies and classifying topoi. It presents an introduction to the concept of forcing topology, which is used to define the Zariski topology associated to a commutative ring A in an arbitrary topos E. Discrete fibrations, with a fixed base, are functors satisfying a condition described by certain finite limits. The chapter presents diagrams of the form DT in relation to the models of the theory T. It is important to classify objects for the classification of finite diagrams.

On the Spectrum of a Ringed Topos

Publisher Summary This chapter provides an overview of the spectrum of a ringed topos. One of the original reasons for introducing the axioms of an elementary topos, though by no means the only one, was to do topos theory intrinsically. Before approaching the problem of the spectrum, some basic results of commutative algebra need to be established in an arbitrary ringed topos. A prime of A , as distinct from a prime ideal of A , is defined to be a subobject P → A satisfying those conditions which, in the case E = S, define the complement of a prime ideal. The chapter also discusses the application of the general concept of Zariski topologies in a ringed topos.

Classifying spaces for sheaves of simplicial groupoids

We study the problem of the construction of classifying spaces for sheaves of simplicial groupoids, showing that these exist iff the groupoid is amenable. We show further that any groupoid has an amenable completion, which in the case of the abelian group K(π, n), yields a representation theorem for sheaf cohomology in terms of isomorphism classes of certain principal bundles.

On the theory of path groupoids

Abstract Functorial path groupoids P ( X ) are constructed for each simplicial set X generalizing the loop groups G ( X ) defined by Kan for each reduced simplicial set. Locally transitive simplicial groupoids and adequate refelexive simplicial graphs are introduced in order to prove that the free simplicial groupoid on an adequate graph is a path groupoid. A calculus of extension and restriction for simplicial groupoids and graphs is developed to construct path groupoids from covers of a simplicial set.

Two-descent, two-torsors and local equivalence

Abstract We extend to dimension 2 the well-known equivalence in dimension 1 between objects locally isomorphic to a given one, descent data, Cech one-cocycles and torsors.