  An oriented hypergraph is an oriented incidence structure that allows for the
generalization of graph theoretic concepts to integer matrices through its
locally signed graphic substructure. The locally graphic behaviors are
formalized in the subobject classifier of incidence hypergraphs. Moreover, the
injective envelope is calculated and shown to contain the class of uniform
hypergraphs -- providing a combinatorial framework for the entries of incidence
matrices. A multivariable all-minors characteristic polynomial is obtained for
both the determinant and permanent of the oriented hypergraphic Laplacian and
adjacency matrices arising from any integer incidence matrix. The coefficients
of each polynomial are shown to be submonic maps from the same family into the
injective envelope limited by the subobject classifier. These results provide a
unifying theorem for oriented hypergraphic matrix-tree-type and
Sachs-coefficient-type theorems. Finally, by specializing to bidirected graphs,
the trivial subclasses for the degree-$k$ monomials of the Laplacian are shown
to be in one-to-one correspondence with $k$-arborescences.
