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Keywords:
$\infty$-categories; straightening; unstraightening; comprehension; Yoneda embedding
$\infty$-categories; straightening; unstraightening; comprehension; Yoneda embedding
Summary:
In this paper we construct an analogue of Lurie’s “unstraightening” construction that we refer to as the {\it comprehension construction}. Its input is a cocartesian fibration $p : E \twoheadrightarrow B$ between $\infty$-categories together with a third $\infty$-category $A$. The comprehension construction then defines a map from the quasi-category of functors from $A$ to $B$ to the large quasi-category of cocartesian fibrations over $A$ that acts on $f : A \rightarrow B$ by forming the pullback of $p$ along $f$. To illustrate the versatility of this construction, we define the covariant and contravariant Yoneda embeddings as special cases of the comprehension functor. We then prove that the hom-wise action of the comprehension functor coincides with an “external action” of the hom-spaces of $B$ on the fibres of $p$ and use this to prove that the Yoneda embedding is fully faithful, providing an explicit equivalence between a quasi-category and the homotopy coherent nerve of a Kan-complex enriched category.
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