Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-commutative counting and stability   The second author and Katzarkov introduced categorical invariants based on
counting of full triangulated subcategories in a given triangulated category
$\mathcal T$ and they demonstrated different choices of additional properties
of the subcategories being counted, in particular - an approach to make
non-commutative counting in $\mathcal T$ dependable on a stability condition
$\sigma \in {\rm Stab}(\mathcal T)$. In this paper, we focus on this approach.
After recalling the definitions of a \textit{stable non-commutative curve in
$\mathcal T$} and related notions, we prove a few general facts, and study
fairly completely an example: $\mathcal T = D^b(Q)$, where $Q$ is the acyclic
triangular quiver. In previous papers was shown that there are two
non-commutative curves of non-commutative genus $1$ and infinitely many
non-commutative curves of non-commutative genus $0$ in $D^b(Q)$. Our studies
here imply that for an open and dense subset in ${\rm Stab}(D^b(Q))$ the stable
non-commutative curves in $D^b(Q)$ are finite. Counting such curves as $\sigma$
varies in ${\rm Stab}(D^b(Q))$ and for genus $0$ produces all the even
non-negative integers, $1$, and $+\infty$, whereas for genus $1$ it produces
the integers $0,1,2$. Counting semi-stable derived points (explained in the
paper) produces $\{3,4,5,+\infty\},$ and the subset of ${\rm Stab}(D^b(Q))$
with finitely many such points is open but not dense.

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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
3
The categories of this paper are: physics
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The categories of this paper are: stat
10
The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th