Homotopy theoretic models of identity types

Abstract
Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such as Coq and Agda (cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature. This paper provides a precise indication of this connection between homotopy theory and logic; a more detailed discussion of these and further results will be given in [20].
Keywords No keywords specified (fix it)
Categories (categorize this paper)
Options
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
 
Download options
PhilPapers Archive


Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 9,360
External links
  •   Try with proxy.
  •   Try with proxy.
  •   Try with proxy.
  • Through your library Only published papers are available at libraries
    References found in this work BETA

    No references found.

    Citations of this work BETA
    S. Awodey & M. A. Warren (2013). Martin-Löf Complexes. Annals of Pure and Applied Logic 164 (10):928-956.
    Similar books and articles
    Analytics

    Monthly downloads

    Added to index

    2009-04-15

    Total downloads

    27 ( #54,527 of 1,089,155 )

    Recent downloads (6 months)

    1 ( #69,735 of 1,089,155 )

    How can I increase my downloads?

    My notes
    Sign in to use this feature


    Discussion
    Start a new thread
    Order:
    There  are no threads in this forum
    Nothing in this forum yet.