Let be a ring. Previously we characterized the finitely presented projective (right) -modules as the tiny objects in : the objects such that

preserves colimits. We also highlighted the key role that these modules play in Morita theory.

If is a commutative ring, then has a natural symmetric monoidal structure which allows us to describe another finiteness condition called dualizability. Unlike tininess, dualizability makes no reference to colimits; instead, it is a purely equational definition involving the monoidal structure. The dualizable modules are again the finitely presented projective -modules.

Dualizability implies that we can treat finitely presented projective -modules like finite-dimensional vector spaces in many ways: for example, dualizability allows us to define the trace of an endomorphism. Moreover, since dualizability is defined using only a monoidal structure, it makes sense in very general settings, and we’ll look at some more exotic examples of dualizable objects as well.

Duals are also a special case of a 2-categorical notion of adjunction which, in the 2-category of categories, functors, and natural transformations, reproduces the usual notion of adjunction. In a suitable 2-category it will also reproduce another characterization of finitely presented projective modules, this time over noncommutative rings.

This post should, but will not, include diagrams, so pretend that I’ve inserted some string diagrams or globular diagrams where appropriate.