Closed Set Sheaves and Their Categories

Abstract

With this chapter we examine categorial sheaves over the closed sets of a topological space. Aside from the historical interest that sheaves seem to have been defined over closed sets first before the more usual definition over open sets, there are a number of other reasons for developing the theory of sheaves over closed sets. First of all having a base topology of closed sets gives us a working concept of boundary that does not exist for the open set sheaf notion. One area in which this may work for us is the mathematics of physics. Lawvere in the introduction to Categories in Continuum Physics [21] mentions the speculation that there is a role for a closed set sheaf in thermodynamics as a functor from a category of parts of a body to a category of “abstract thermodynamical state-and-process systems” (p.9). Another reason for closed set sheaves is their effect on categorial logic. A closed set topology ordered by set inclusion is a paraconsistent algebra. Via the sheaves we can introduce this paraconsistency to toposes.