  This paper is the fourth in a series whose goal is to develop a fundamentally
new way of building theories of physics. The motivation comes from a desire to
address certain deep issues that arise in the quantum theory of gravity. Our
basic contention is that constructing a theory of physics is equivalent to
finding a representation in a topos of a certain formal language that is
attached to the system. Classical physics arises when the topos is the category
of sets. Other types of theory employ a different topos. The previous papers in
this series are concerned with implementing this programme for a single system.
In the present paper, we turn to considering a collection of systems: in
particular, we are interested in the relation between the topos representation
for a composite system, and the representations for its constituents. We also
study this problem for the disjoint sum of two systems. Our approach to these
matters is to construct a category of systems and to find a topos
representation of the entire category.
