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I have been adding some material to matroid. I haven’t gotten around to defining oriented matroid yet (and of course there’s much besides to add).
Just a boring typesetting issue: I have changed
+-- {: .num_propoposition}
to
+-- {: .num_prop}
Unfortunately the first one does not come out as expected.
On the recent addition to matroid: graph-theorists reading “embedded subgraph” in this context are likely to find this malapropistic to the point of being wrong: “embedded graph” or “embedded graph” are usually reserved to discussions of, well, embedded graphs, i.e., discussions where there is some surface in the background, and some embedding of the abstract graph into the surface. The usual definition of “graphic matroid” does not need this. It would improve the paragraph to change “embedded subgraph” just to “subgraph”.
A more serious issue is defining a graphic matroid on the vertex set: while this can be done (somehow), by far the most usual definition of graphic matroids has the edge set as the ground, see e.g. Oxley, 2nd ed, p. 11.
Moreover, “it turns out” reads as if there is at least something to check, while in fact it is immediate from the usual definitions that dualization is involutive, and involutive with the isomorphism being a strict equality. I have much appreciation for the idea of avoiding equalities whenever possible, and it might be good to do so in this case, too, but just would like to remark that in matroid theory as it exists at the moment, the dual of the dual being the matroid itself is both strict and immediate from the definitions.
Not knowing where you are heading, I will not change it for the time being.
Peter wrote:
It would improve the paragraph to change “embedded subgraph” just to “subgraph”.
Todd, perhaps you meant “induced subgraph”?
I did mean “induced subgraph”, yes.
The word “embedded”, which you rightly point out would convey the wrong idea to graph theorists: I had in mind an analogy with (full) embeddings in category theory. But let’s change that to avoid misunderstanding. But not to just “subgraph” which might mean that not all the edges between vertices in the subgraph are actually in the subgraph.
Please feel free to make appropriate changes to the description of graphic matroid and dual matroid. I was writing in a hurry there; sorry.
By the way, Peter: any expertise you can bring to bear on graph theory matters is very appreciated! Thanks again.
@Todd, I see that last year you indicated that you had possibly done some original research on categories of matroids. Are there any updates on this?
I thought for a while that geometries (in the sense of section 4 of the page matroid) were naturally a reflective subcategory of pregeometries with reflector given by geometrification (delete the closure of $\emptyset$ and quotient by $x \sim y \iff \operatorname{cl}(x) = \operatorname{cl}(y)$). But if you use closure-preserving maps as morphisms, there doesn’t seem to be a natural way to define a counit…
Okay, the mistakes kindly pointed out by Peter have been fixed (I believe).
(J)esse: no, alas, there are no updates on this. My main interest in working on matroid at that time was more in view of model theory than anything else. But with luck I’ll get myself re-interested. :-)
Added reference
Moved the original article
out of the list of “Other references” to before Heunen’s article.
added publication data for:
Added a reference to Bruhn–Diestel–Kriesel–Pendaving–Wollan’s Axioms for infinite matroids. (Working their definition into the main text would require some restructuring; it’s the current Definition 2.1 plus the requirement that every independent set is contained in a maximal independent set.)
Anonymous
Following on from the discussion on another thread, we were wondering about the efficacy of matroids. Just to jot down some relevant things:
There’s the idea that matroids can be understood within tropical geometry, and then a relevant cohomology deployed.
Matthieu Piquerez: Hodge theory for tropical fans
On the introduction to the present symposium, one can read “But the [Heron-Rota-Welsh] conjecture was for an arbitrary matroid, which might not be associated to any type of geometry at all! The proof by Adiprasito–Huh–Katz builds an object from combinatorics, which ought to play the role of the cohomology ring, and proves Poincaré duality, Hard Lefschetz and the Hodge–Riemann bilinear relations for this object directly.”
In this talk, I will show that, on the contrary, these results has a geometric interpretation for any matroid… in the tropical world. Indeed, one can show that the tropical cohomology of the canonical compactification of so-called tropically shellable quasi-projective fans verifies the three above properties. In particular, Bergman fans of matroids belong to those fans, hence we get a generalization of the result of Adiprasito–Huh–Katz. This is a joint work with Omid Amini.
From Piqurez’s thesis
The cohomology of a tropical variety mimics that of a complex variety. A theory of great importance in complex geometry is Hodge theory. It shows numerous interesting properties of cohomologies of complex varieties, and over the decades it has plenty of applications. In this thesis, we achieve to establish a tropical Hodge theory, that is a Hodge theory for tropical varieties.
By the way, Urs, on the other thread what did you mean by
such a basic modal-type-theoretic concept as matroids ?
re #18:
By the definition here, and remembering that “closure operator” is another term for modality, a matroid is a type satisfying a curious condition with respect to a given modality.
It would be good to understand this condition on more general abstract grounds. If we consider pointed types, so that the formation of complements used in the condition is expressible as cofibers, then the condition involves a variant of the modal axiomatization of “determinate negation”. But I haven’t further thought about it yet.
From the paper in #15, the category of free pointed matroids with pointed strong maps is equivalent to the category of finite pointed sets (Prop 5.7) (the brackets round the dot seem to be a typo), so equivalent to $\mathbb{F}_1$ vector spaces.
Then there’s an adjoint quadruple between free pointed matroids and pointed matroids (Thrm 6.9).
If I had leisure to dig into matroids now, I would warm-up with spelling out the basic elementary examples in more detail – the nLab entry leaves enormous room for improvement in this respect.
The first example to spell out would be that of vector spaces. The entry should state and prove (all completely elementary, but still) that the functor taking $k$-vector spaces to their matroids is not full (for example, after fixing any norm $\vert-\vert$ on a finite dimension vector space $V$, the map $v \mapsto \vert v\vert \cdot v$ should be an endomorphism of the matroid of $V$ which is not a linear map) and preferably comment on what this “means”.
(I don’t know what it means, but it seems like a first indication of an important point of the whole subject. Or maybe one of the many extra conditions on matroids that are being considered does make it become full?)
Then to incrementally generalize from this example to that of linear tropical geometries. This to get a foot on the ground of what the subject matter of matroids really is like.
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