  We introduce the notion of Betti category for graded modules over suitably
graded polynomial rings, and more generally for modules over certain small
categories. Our categorical approach allows us to treat simultaneously many
important cases, such as monomial ideals and toric rings. We prove that in
these cases the Betti category is a finite combinatorial object that completely
determines the structure of the minimal free resolution. For monomial ideals,
the Betti category is the same as the Betti poset that we studied in a previous
article. We describe in detail and with examples how the theory applies to the
toric case, and provide an analog for toric rings of the lcm-lattice for
monomial ideals.
