Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Six model structures for DG-modules over DGAs: Model category theory in
  homological action   In Part 1, we describe six projective-type model structures on the category
of differential graded modules over a differential graded algebra A over a
commutative ring R. When R is a field, the six collapse to three and are
well-known, at least to folklore, but in the general case the new relative and
mixed model structures offer interesting alternatives to the model structures
in common use. The construction of some of these model structures requires two
new variants of the small object argument, an enriched and an algebraic one,
and we describe these more generally. In Part 2, we present a variety of
theoretical and calculational cofibrant approximations in these model
categories. The classical bar construction gives cofibrant approximations in
the relative model structure, but generally not in the usual one. In the usual
model structure, there are two quite different ways to lift cofibrant
approximations from the level of homology modules over homology algebras, where
they are classical projective resolutions, to the level of DG-modules over
DG-algebras. The new theory makes model theoretic sense of earlier explicit
calculations based on one of these constructions. A novel phenomenon we
encounter is isomorphic cofibrant approximations with different combinatorial
structure such that things proven in one avatar are not readily proven in the
other.

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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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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: gr-qc
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The categories of this paper are: atom-ph
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The categories of this paper are: adap-org