The Category Quad

Abstract

This chapter studies the objects introduced in Chapter 1 from the point of view of category theory. One of the main reasons for doing this is so that we can introduce the notion of a matrix ring over a form ring (see §1.10), which will be used to define the hyperbolic co-unitary group (see (5.2.5)) and in the proofs of the main theorems in Chapter 5. Another application will be the definition of the Witt group of representations of a form ring (§4.6). This will be used to define the universal Clifford-Weil group associated with a finite form ring (see §5.4).