ct.category theory - Category Theoretic Interpretation of Matroids?

I have come to the conclusion that a lot of mathematical structure on sets (e.g. constructs) can be defined through (combinations of) relations

$(1) quad S_X^Fsubset F(X)times X^{I}$

for some underlying set X, some functor $F$: Rel $rightarrow$ Rel and some set $I$.

Examples:

• Magmas (monoids, groups,...) are defined through functions $Ssubset X^2times X$.


• Graphs are defined through relations $Ssubset Xtimes X$.


• Metric spaces could be defined through relations $Ssubset (mathbb{R}times X)times X$, where $((r,x),x')in SLeftrightarrow d(x,x')=r$. Or better: through $d(x,x')le r$, since the category Met have retractions as morphisms.


• Topological spaces could be defined by $Ssubset 2^Xtimes X$, where $(M,x)in SLeftrightarrow xin Mintau$ or by the closure $xin overline M$.


• Uniform spaces could be defined through relations $Ssubset 2^{Xtimes X}times X^2$, 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 $2^Xoverset{2^f}longrightarrow 2^Y$, where
$(M,M')in 2^fLeftrightarrow M=f^{-1}(M')$ and
$S_X^F$ is defined as $(M,x)in S_X^{F}Leftrightarrow xin overline{M}$.

Then due to $(2)$:

$M=f^{-1}(M')Rightarrow (xin overline{M}Rightarrow f(x)in overline{M,'})$, so $xin overline{f^{-1}(M,')}Rightarrow f(x)in overline{M,'}$. (Continuity).

In case of matroids $(X,mathcal I)$ I can see two possibilities that fits into my scheme:

1. $(A,x)in SLeftrightarrow xin Ainmathcal I$, that gives a condition for morphisms $f^{-1}(A')inmathcal IRightarrow A'inmathcal I'$;


2. $(A,x)in SLeftrightarrow xin cl(A)$, that gives the condition $r(f^{-1}(A'))=r(f^{-1}(A')cup{x})Rightarrow r(A')=r(A'cup{f(x)})$, where $cl(A)={xin X|r(A)=r(Acup{x})}$ 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 $f:Xrightarrow X'$, where $(X,mathcal I)$ and
$(X',mathcal I')$ are matroids, is a morphism if it holds for any set
$A'subseteq X'$ that $f^{-1}(A')inmathcal I Rightarrow A'inmathcal I'$.

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.

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