Symmetric Auslander and Bass categories

We define the symmetric Auslander category A^s® to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right tails of totally acyclic complexes of projective modules. The symmetric Auslander category contains A®, the ordinary Auslander category. It is well known that A® is intimately related to Gorenstein projective modules, and our main result is that A^s® is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories: \underline{GMor}® --> A^s® / K^b(Prj R). Here \underline{GMor}® is the stable category of Gorenstein projective objects in the abelian category Mor® of homomorphisms of R-modules, and K^b(Prj R) is the homotopy category of bounded complexes of projective R-modules. This result is set in the wider context of a theory for A^s® and B^s®, the symmetric Bass category which is defined dually.