Topos Theory and Mereology

A common idea is that logical truth values are fixed independently of the objects to which logic is applied. Topos theory reveals a different possibility. In a topos, the available truth values and logical operations depend upon the structure of its objects’ subobjects (generalized parts). In fact, the familiar first-order semantics of sets appears as one instance of this more general phenomenon: it is a logic arising from a particular algebra of parts (namely, the algebra of subsets). This paper provides a detailed exposition of this idea for philosophers without prior training in category theory. Using sets and directed multigraphs as running examples, it explains how a topos’s internal logic—including its truth values—depends upon the structure of its subobjects.