Proudfoot, NicholasMiyata, Dane2024-01-092024-01-092024-01-09https://hdl.handle.net/1794/29172Graphs and matroids are two of the most important objects in combinatorics.We study invariants of graphs and matroids that behave well with respect to certain morphisms by realizing these invariants as functors from a category of graphs (resp. matroids). For graphs, we study invariants that respect deletions and contractions ofedges. For an integer $g > 0$, we define a category of $\mathcal{G}^{op}_g$ of graphs of genus at most g where morphisms correspond to deletions and contractions. We prove that this category is locally Noetherian and show that many graph invariants form finitely generated modules over the category $\mathcal{G}^{op}_g$. This fact allows us to exihibit many stabilization properties of these invariants. In particular we show that the torsion that can occur in the homologies of the unordered configuration space of n points in a graph and the matching complex of a graph are uniform over the entire family of graphs with genus $g$. For matroids, we study valuative invariants of matroids. Given a matroid,one can define a corresponding polytope called the base polytope. Often, the base polytope of a matroid can be decomposed into a cell complex made up of base polytopes of other matroids. A valuative invariant of matroids is an invariant that respects these polytope decompositions. We define a category $\mathcal{M}^{\wedge}_{id}$ of matroids whose morphisms correspond to containment of base polytopes. We then define the notion of a categorical matroid invariant which categorifies the notion of a valuative invariant. Finally, we prove that the functor sending a matroid to its Orlik-Solomon algebra is a categorical valuative invariant. This allows us to derive relations among the Orlik-Solomon algebras of a matroid and matroids that decompose its base polytope viewed as representations of any group $\Gamma$ whose action is compatible with the polytope decomposition. This dissertation includes previously unpublished co-authored material.en-USAll Rights Reserved.Electronic Thesis or Dissertation