Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Monads and theories   Given a locally presentable enriched category $\mathcal{E}$ together with a
small dense full subcategory $\mathcal A$ of arities, we study the relationship
between monads on $\mathcal E$ and identity-on-objects functors out of
$\mathcal A$, which we call $\mathcal A$-pretheories. We show that the natural
constructions relating these two kinds of structure form an adjoint pair. The
fixpoints of the adjunction are characterised as the $\mathcal A$-nervous
monads---those for which the conclusions of Weber's nerve theorem hold---and
the $\mathcal A$-theories, which we introduce here.
  The resulting equivalence between $\mathcal A$-nervous monads and $\mathcal
A$-theories is best possible in a precise sense, and extends almost all
previously known monad--theory correspondences. It also establishes some
completely new correspondences, including one which captures the globular
theories defining Grothendieck weak $\omega$-groupoids.
  Besides establishing our general correspondence and illustrating its reach,
we study good properties of $\mathcal A$-nervous monads and $\mathcal
A$-theories that allow us to recognise and construct them with ease. We also
compare them with the monads with arities and theories with arities introduced
and studied by Berger, Melli\`es and Weber.

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The categories of this paper are: eess
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th