| Title:
|
The comprehension construction (English) |
| Author:
|
Riehl, Emily |
| Author:
|
Verity, Dominic |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
2 |
| Issue:
|
1 |
| Year:
|
2018 |
| Pages:
|
116-190 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
$\infty$-categories |
| Keyword:
|
straightening |
| Keyword:
|
unstraightening |
| Keyword:
|
comprehension |
| Keyword:
|
Yoneda embedding |
| MSC:
|
18G55 |
| MSC:
|
55U35 |
| idZBL:
|
Zbl 1409.18021 |
| idMR:
|
MR3917428 |
| DOI:
|
10.21136/HS.2018.05 |
| . |
| Date available:
|
2026-03-10T16:51:31Z |
| Last updated:
|
2026-03-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153404 |
| . |
| Reference:
|
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| . |
