Category theory intersects with philosophy at various levels. But perhaps nowhere is the direct utility of category theory for philosophy more evident than in the area of logic. This chapter examines two important constructions in category theory with particular relevance for logic: adjoint functors (adjunctions) and topoi. These are mathematically precise and quite technical notions, and their generality is astonishing. More so than in previous chapters it will be necessary at points for the sake of an introductory presentation to omit certain formal details that while essential for higher-level mathematical work would only obscure the matter at a first or even fifth encounter. The mathematical and philosophical ramifications of these two concepts are immense, and it will be impossible to survey them all here. The reader is encouraged to pursue these concepts further, especially by considering them in light of recent philosophical proposals such as those of Williamson, Zalamea and Badiou that encourage a strong linkage between formal methods (either logical or mathematical) and metaphysics. Such proposals lend themselves readily in a categorical framework to formulation in terms of diagrammatic immanence.

Logic studies extremely general relations, relations so general that they are meant to structure necessarily anything that may be coherently thought. When logic and metaphysics are brought into alignment, which is quite natural, the general relation between sameness and difference becomes especially salient. Typically, logicians and metaphysicians privilege unity and sameness over multiplicity and difference for reasons that are perhaps endemic to representational models of thought, but it is also possible to work philosophically from the standpoint of the priority or privilege of difference and multiplicity. We saw in the previous chapter how Deleuze shares some of the core philosophical commitments of Spinoza and Peirce – in particular their respective commitments to different types of philosophical immanence – and yet develops on this basis a philosophy of creative difference that makes divergence primary with respect to identity and treats subjectivity as the purely differential spark of communication across incompossible worlds. Most importantly, for Deleuze this differential communication structures philosophy itself ‘in an essential relationship with the No that concerns it’ (like each of the disciplines of art and science, too, in their own distinctive ways).