1 9 D ec 1 99 6 GRAPH - BASED LOGIC AND SKETCHES II : FINITE - PRODUCT CATEGORIES AND EQUATIONAL LOGIC ( PRELIMINARY REPORT )

In [Wells, 1990] the second author introduced the notion of form, a graph-based method of specification of mathematical structures that generalizes Ehresmann’s sketches. In [Bagchi and Wells, 1996], the authors produced a structure for forms which provides a uniform proof theory based on finite-limit constructions for many types of forms, including all types of sketches and also forms that can specify higherorder structures in cartesian closed categories and toposes (among many others). The parameter in the proof theory that determines the types of constructions that can be made is the constructor space. For example, the constructor space for cartesian closed categories (with specified structure) is the finite-limit theory CCC for cartesian closed categories. In particular for the concerns of the present paper, the constructor space for structures that can be specified by finite products is a finitelimit theory FinProd for categories with specified finite products. This theory is described explicitly in [Bagchi and Wells, 1996]. Each finite product form F is given by a syntactic category denoted by SynCat[FinProd, F ]. The logical structure in [Bagchi and Wells, 1996] identifies a statement as a potential factorization in SynCat[FinProd, F ], which is a diagram of the form