Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the unicity of formal category theories   We prove an equivalence between cocomplete Yoneda structures and certain
proarrow equipments on a 2-category $\mathcal K$. In order to do this, we
recognize the presheaf construction of a cocomplete Yoneda structure as a
relative, lax idempotent monad sending each admissible 1-cell $f :A \to B$ to
an adjunction $\boldsymbol{P}_!f\dashv\boldsymbol{P}^*f$. Each cocomplete
Yoneda structure on $\mathcal K$ arises in this way from a relative lax
idempotent monad "with enough adjoint 1-cells", whose domain generates the
ideal of admissibles, and the Kleisli category of such a monad equips its
domain with proarrows. We call these structures "yosegi". Quite often, the
presheaf construction associated to a yosegi generates an ambidextrous Yoneda
structure; in such a setting there exists a fully formal version of Isbell
duality.

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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ