Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $C(\mathcal{X^{*}})$-Cover and $C(\mathcal{X^{*}})$-Envelope   Let $R$ be any associative ring with unity and $\mathcal{X}$ be a class of
$R$-modules of closed under direct sum (and summands) and with extension
closed. We prove that every complex has an $C(\mathcal{X^{*}})$-cover
($C(\mathcal{X^{*}})$-envelope) if every module has an $\mathcal{X}$-cover
($\mathcal{X}$-envelope) where $C(\mathcal{X^{*}})$ is the class of complexes
of modules in $\mathcal{X}$ such that it is closed under direct and inverse
limit.

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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: physics
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: cs
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The categories of this paper are: nucl-ex