Mathematical Sciences Colloquium

Graphs, point arrangements and sets of vectors can all be described in terms of their closed sets. In each case, the collection of closed sets satisfies certain properties. We define a matroid to be an object whose closed sets satisfy these properties. The matroids in which we are interested are those arising from finite graphs with a single loop. These matroids are called loopless pointed matroids. We consider these objects and special maps, called strong maps, between them. This collection of objects and maps is called the category of loopless pointed matroids and strong maps.

In our research we find a set of axioms satisfied by the category of loopless pointed matroids and strong maps. Our goal is to show any other category satisfying these axioms is equivalent to the category of loopless pointed matroids and strong maps. We use as a model for our research the work done by D. I. Schlomiuk in 1971. Schlomiuk studied the category of topological spaces and continuous mappings and found twelve axioms satisfied by this category. In addition, she proved that any category satisfying these twelve axioms is equivalent to the category of topological spaces and continuous mappings. We begin our research by examining these twelve axioms to determine which hold in our category. Moreover, we describe many matroid notions purely in term of strong maps.

This talk will provide an overview of our research. No prior knowledge of matroid or category theory is assumed!

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