First time poster, long time lurker here. I have a really basic question that has been bugging me for sometime. Specifically, I'm not exactly sure what the 'correct' category theoretic definition of a matroid should be. The only definition I know involves heavy use of set-theory, and is kind of clumsy:


Given a set $E$, a *matroid* $\mathcal{I} \subseteq 2^E$ is a non-empty collection of subsets which satisfy the following axioms:


1. (Heredity) If $X \in \mathcal{I}$ and $X' \subset X$, then $X' \in \mathcal{I}$.
2. (Exchange) If $X, Y \in \mathcal{I}$ and $|X| > |Y|$, then there exists some $b \in X \backslash Y$ such that $Y \cup \{ b \} \in \mathcal{I}$.


Given that both categories and matroids were introduced around the same time and both were studied by MacLane, it stands to reason that someone ought to have thought about this before. Also it is obvious from the Heredity axiom that each matroid is a category, since the containment relation is reflexive and transitive. The second property is a bit more difficult to model, as I am not sure how to get rid of the ugly element / cardinality operators.


In the optimal solution, it would be nice to get rid of the set $E$ entirely, and instead view the specific interpretation of the abstract matroid as a functor from $\mathcal{I} \to 2^E$, the power-set lattice. This would also suggest a functorial interpretation of the graph theoretic and linear algebra applications of matroids. I strongly suspect that someone has already done this, but am having great difficulty locating any references. (Of course I may also be totally wrong headed here too...)



Category theoretic interpretation of matroids?
If I understand your question correctly, I believe that the problem is still open. That is, if we let $\mathcal{M}$ be the category of (simple) matroids, where the morphism are given by strong maps, then it is still open how to describe $\mathcal{M}$ by a nice set of axioms. However, partial progress has been made in this [paper](http://www.emis.de/journals/HOA/IJMMS/Volume26_12/770.pdf).


