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.
I have come to the conclusion that a lot of mathematical structure on sets (e.g. constructs) can be defined through (combinations of) relations
$ (1) \quad S_X^F\subset F(X)\times X^{I} $
for some underlying set X, some functor $F$: Rel $\rightarrow$ Rel and some set $I$.
Examples:
- Magmas (monoids, groups,...) are defined through functions $S\subset X^2\times X$.
- Graphs are defined through relations $S\subset X\times X$.
- Metric spaces could be defined through relations $S\subset (\mathbb{R}\times X)\times X$, where $((r,x),x')\in S\Leftrightarrow d(x,x')=r$. Or better: through $d(x,x')\le r$, since the category Met have retractions as morphisms.
- Topological spaces could be defined by $S\subset 2^X\times X$, where $(M,x)\in S\Leftrightarrow x\in M\in\tau$ or by the closure $x\in \overline M$.
- Uniform spaces could be defined through relations $S\subset 2^{X\times X}\times X^2$, where $(U,(x,y))\in S \Leftrightarrow (x,y)$ is U-close. (Wikipedia)
(See Can any construct be characterized as a relation?)
This works for any construct I know and there even seems to be a general rule to generate the morphisms between the constructs, showed by the (in general not commuting, if the relations not are functions) diagram of sets and relations: $\require{AMScd}$ \begin{CD} F(X) @>F(f)>> F(Y)\\ @V S_X^F V V(2) @VV S_Y^F V\\ X^{I} @>>f^{I}> Y^{I} \end{CD} $(2)\quad (\phi_X,\phi_Y)\in F(f)\Rightarrow [(\phi_X,(x_i)_I)\in S_X^F \Rightarrow (\phi_Y,((f(x_i))_I)\in S_Y^F]$.
Example: If $I=1$, $F$ is the (contravariant) functor defined as $2^X\overset{2^f}\longrightarrow 2^Y$, where $ (M,M')\in 2^f\Leftrightarrow M=f^{-1}(M')$ and $S_X^F$ is defined as $(M,x)\in S_X^{F}\Leftrightarrow x\in \overline{M}$.
Then due to $(2)$:
$M=f^{-1}(M')\Rightarrow (x\in \overline{M}\Rightarrow f(x)\in \overline{M\,'})$, so $x\in \overline{f^{-1}(M\,')}\Rightarrow f(x)\in \overline{M\,'}$. (Continuity).
In case of matroids $(X,\mathcal I)$ I can see two possibilities that fits into my scheme:
- $(A,x)\in S\Leftrightarrow x\in A\in\mathcal I$, that gives a condition for morphisms $f^{-1}(A')\in\mathcal I\Rightarrow A'\in\mathcal I'$;
- $(A,x)\in S\Leftrightarrow x\in cl(A)$, that gives the condition $r(f^{-1}(A'))=r(f^{-1}(A')\cup\{x\})\Rightarrow r(A')=r(A'\cup\{f(x)\})$, where $cl(A)=\{x\in X|r(A)=r(A\cup\{x\})\}$ and $r$ is the rank function.
It seems to me as the former definition of a morphism is more natural, given the scheme, since the exchange axiom doesn't have to affect the form of the morphism more than associativity affect the form of the group homomorphism. So my primary candidate is:
A function $f:X\rightarrow X'$, where $(X,\mathcal I)$ and $(X',\mathcal I')$ are matroids, is a morphism if it holds for any set $A'\subseteq X'$ that $f^{-1}(A')\in\mathcal I \Rightarrow A'\in\mathcal I'$.
I don't claim that this is the answer and I can't evaluate the result because of lack of experience of matroids, but this is what I got from the empirical scheme.
The following related article was recently published:
Heunen, C. & Patta, V. Appl Categor Struct (2018) 26: 205. https://doi.org/10.1007/s10485-017-9490-2
In section 9, the authors give a categorical characterization of matroids based on the "greedy algorithm characterization"; very roughly speaking, the optimality of the greedy algorithm for all possible weight functions is encoded as the property that the limits of certain diagrams are all the same. The paper also begins with a nice discussion of the various properties of the category of matroids and strong maps mentioned in Tony Huynh's answer.