The problems of representation and reference under conditions of radical immanence are particularly striking, as both of these appear to require at least a minimal transcendence in order to function at all. The category-theoretical framework continued in this chapter addresses these abstract yet still strictly philosophical problems from a purely formal standpoint in which the ‘interiority’ and ‘exteriority’ of categorical determinations serve as a basis for reconceiving the internal and external relations of entities understood as diagrammatically structured. The key insight is a continuation and extension of that begun in Chapter 2, namely that within the framework of category theory categories themselves may appear as objects with welldefined systems of relations to one another via functors. Such systems of objects and relations may themselves constitute categories in their own right. This formal ‘immanence’ – the collapse of meta-systems and meta-relations into the same type of diagrammatically tractable systems and relations – may itself be lifted to the level of functors themselves: whereas functors in category theory are essentially maps between categories, natural transformations are structure-preserving maps between functors (maps between maps of systems of maps). Thus the relation between functors and natural transformations is a useful general model of the distinction between systems of relations and systems of meta-relations that yet treats these in a structurally identical manner. Examples in this chapter are developed in terms of demonstrating the ways in which the diagrammatic and structural methods of category theory thus model immanence, building up to a formal model of diagrammatic signification based in presheaves, a type of functor, and the functor categories – presheaf categories – that are constructed from them. This model will cast the Peircean characterisation of diagrammatic semiotics from the previous chapter in a categorical context.

Consider an abstract ‘slice’ of the cocktail party encountered in Chapter 2, selecting from among its innumerable constituent things, properties and relations two couples in the midst of two conversations. With dots representing people and arrows standing for the relation ‘is speaking to’, we have a diagram like that on the left in Figure 4.1.