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We introduce the notion of Betti category for graded modules over suitablygraded polynomial rings, and more generally for modules over certain smallcategories. Our categorical approach allows us to treat simultaneously manyimportant cases, such as monomial ideals and toric rings. We prove that inthese cases the Betti category is a finite combinatorial object that completelydetermines 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 previousarticle. We describe in detail and with examples how the theory applies to thetoric case, and provide an analog for toric rings of the lcm-lattice formonomial ideals.

Betti categories of graded modules and applications to monomial ideals and toric rings