Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homotopy theory of modules over a commutative $S$-algebra: some tools
  and examples   Modern categories of spectra such as that of Elmendorf et al equipped with
strictly symmetric monoidal smash products allows the introduction of symmetric
monoids providing a new way to study highly coherent commutative ring spectra.
These have categories of modules which are generalisations of the classical
categories of spectra that correspond to modules over the sphere spectrum;
passing to their derived or homotopy categories leads to new contexts in which
homotopy theory can be explored. In this paper we describe some of the tools
available for studying these `brave new homotopy theories' and demonstrate them
by considering modules over the $K$-theory spectrum, closely related to
Mahowald's theory of $bo$-resolutions. In a planned sequel we will apply these
techniques to the much less familiar context of modules over the $2$-local
connective spectrum of topological modular forms.

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The categories of this paper are: cs
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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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: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex