Fri Oct 9, 2009
60 Evans Hall, 4:10–6 PM
Nate Ackerman (University of Pennsylvania)
Trees, Sheaves, and Definition by Recursion
We will show there is a topological space for which presheaves are the same thing as trees. We will further show that there is a sheaf on this topological space which has an important relationship with Baire space. We will then use these connections to show how a definition by transfinite recursion can be thought of as an operation on sheaves, and how the well-definedness of such a definition can be thought of as a property of the sheaf we are working on. This will then allow us to define a second-order tree as a sheaf on a tree and to expand our notion of definition by transfinite recursion to all well-founded second-order trees (even those which are ill-founded as normal trees). We will then mention how these techniques can be used to prove a variant of the Suslin-Kleene Separation theorem.