Incidence hypergraphs: Injectivity, uniformity, and matrix-tree theorems

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