A platform for research: civil engineering, architecture and urbanism
A platform for research: civil engineering, architecture and urbanism
The comprehension construction
In this paper we construct an analogue of Lurie's "unstraightening" construction that we refer to as the comprehension construction. Its input is a cocartesian fibration $p \colon E \to B$ 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 $f \colon A \to B$ 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.
The comprehension construction
In this paper we construct an analogue of Lurie's "unstraightening" construction that we refer to as the comprehension construction. Its input is a cocartesian fibration $p \colon E \to B$ 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 $f \colon A \to B$ 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.
The comprehension construction
Riehl, Emily (author) / Verity, Dominic (author)
2018-11-08
Higher Structures; Vol 2, No 1 (2018)
Article (Journal)
Electronic Resource
English
DDC:
690
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