There's an operation in matroid theory which is called "relaxation".
To keep things simple, let's consider a matroid $M$ with set of bases $\mathcal{B}$. If $M$ has a subset $H$ of $M$ that is both a circuit and a hyperplane, then one may construct a matroid $\widetilde{M}$ whose set of bases is $\widetilde{\mathcal{B}} = \mathcal{B} \cup \{H\}$.
Notice that both $M$ and $\widetilde{M}$ are matroids on the same ground set and of the same rank. This says that the identity map $i:\widetilde{M} \to M$ is indeed a weak map (every independent set in $M$ is independent in $\widetilde{M}$).
I want to know the following:
Do connected matroids with no relaxations have a name? Have they been studied? Do they have any nice properties?
Of course, many matroids are of that type: for instance uniform matroids are connected and do not admit any relaxation (they are just as big as possible: they don't have any circuit+hyperplane).
