Skip to main content Accessibility help
×
Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T01:27:18.012Z Has data issue: false hasContentIssue false

4 - Presheaves: Externally

Published online by Cambridge University Press:  25 April 2019

Denis-Charles Cisinski
Affiliation:
Universität Regensburg, Germany
Get access

Summary

We study the analogues of Grothendieck fibrations with discrete fibres in the setting of ∞-categories, which is one of the ways to describe presheaves with values in the category of ∞-groupoids: these are the right fibrations. We construct Joyal’s contravariant model category structures, whose fibrant objects precisely are these right fibrations. There is also a dual notion of left fibrations, which are the fibrant objects of the covariant model structures. We prove an external version of the Yoneda lemma in this context: we compute the fibres of right fibrations as certain mapping spaces. We introduce final functors as well as final objects, and extend Grothendieck’s theory of smooth functors and of proper functors to ∞-categories. We prove Grothendieck’s base change formulas (which are instances of the Beck-Chevalley property), and explain how to deduce from these (generalisations of) Quillen’s theorem A and theorem B.
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×