Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The moduli space of matroids   In the first part of the paper, we clarify the connections between several
algebraic objects appearing in matroid theory: both partial fields and
hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are
compatible with the respective matroid theories. Moreover, fuzzy rings are
ordered blueprints and lie in the intersection of tracts with ordered
blueprints; we call the objects of this intersection pastures.
  In the second part, we construct moduli spaces for matroids over pastures. We
show that, for any non-empty finite set $E$, the functor taking a pasture $F$
to the set of isomorphism classes of rank-$r$ $F$-matroids on $E$ is
representable by an ordered blue scheme $Mat(r,E)$, the moduli space of
rank-$r$ matroids on $E$.
  In the third part, we draw conclusions on matroid theory. A classical
rank-$r$ matroid $M$ on $E$ corresponds to a $\mathbb{K}$-valued point of
$Mat(r,E)$ where $\mathbb{K}$ is the Krasner hyperfield. Such a point defines a
residue pasture $k_M$, which we call the universal pasture of $M$. We show that
for every pasture $F$, morphisms $k_M\to F$ are canonically in bijection with
$F$-matroid structures on $M$.
  An analogous weak universal pasture $k_M^w$ classifies weak $F$-matroid
structures on $M$. The unit group of $k_M^w$ can be canonically identified with
the Tutte group of $M$. We call the sub-pasture $k_M^f$ of $k_M^w$ generated by
``cross-ratios' the foundation of $M$,. It parametrizes rescaling classes of
weak $F$-matroid structures on $M$, and its unit group is coincides with the
inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if
its foundation is the regular partial field, and a non-regular matroid $M$ is
binary if and only if its foundation is the field with two elements. This
yields a new proof of the fact that a matroid is regular if and only if it is
both binary and orientable.

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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th