  This is the first article in an upcoming series of papers. They have arisen
through an attempt to answer open questions of Deligne proposed in "Le
determinant de la cohomologie", Contemp. Mathematics 67 (1987). It amounts to
functorial and metrized versions of the Grothendieck-Riemann-Roch theorem, as
well as a Lefschetz-Riemann-Roch formula in the sense of Thomason, cf.
"Lefschetz-Riemann-Roch theorem and coherent trace formula" Ann. Sci. Ec. Norm.
Sup. 18 (1985), Theorem 3.5. In this article we treat various preliminary
results on virtual categories which are the categories where the
Deligne-Riemann-Roch theorem is originally formulated. Finally we compare our
constructions to constructions of Franke on Chern intersection functors and
Chow categories.
