Homotopy Theoretic Models of Identity Types

Quillen [16] 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 [14] or the work of Joyal [10, 11] and Lurie [12] on quasicategories seems 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 [13] due to Hofmann and Streicher [8]. 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. [2] and [4]), this promise of applications is of a practical, as well as theoretical, nature. The present 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 [19].