Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Brown representability does not come for free   We exhibit a triangulated category T having both products and coproducts, and
a triangulated subcategory S of T which is both localizing and colocalizing,
for which neither a Bousfield localization nor a colocalization exists. It
follows that neither the category S nor its dual satisfy Brown
representability. Our example involves an abelian category whose derived
category does not have small Hom-sets.

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The categories of this paper are: hep-lat
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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: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: cs
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The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex