Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Grothendieck ring of pretriangulated categories   We consider the abelian group $PT$ generated by quasi-equivalence classes of
pretriangulated DG categories with relations coming from semi-orthogonal
decompositions of corresponding triangulated categories. We introduce an
operation of "multiplication" $\bullet$ on the collection of DG categories
which makes this abelian group into a commutative ring. A few applications are
considered: representability of "standard" functors between derived categories
of coherent sheaves on smooth projective varieties and a construction of an
interesting motivic measure.

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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess