The identity type weak factorisation system

We show that the classifying category C(T) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids.

[1]  N. Strickland,et al.  MODEL CATEGORIES (Mathematical Surveys and Monographs 63) , 2000 .

[2]  Maria Emilia Maietti,et al.  Modular correspondence between dependent type theories and categories including pretopoi and topoi , 2005, Mathematical Structures in Computer Science.

[3]  Roy Dyckhoff.  PRACTICAL FOUNDATIONS OF MATHEMATICS (Cambridge Studies in Advanced Mathematics 59) , 2000 .

[4]  Edwin D. Mares,et al.  On S , 1994, Stud Logica.

[5]  Peter Dybjer,et al.  Internal Type Theory , 1995, TYPES.

[6]  May,et al.  A Concise Course in Algebraic Topology , 1999 .

[7]  André Joyal,et al.  Strong stacks and classifying spaces , 1991 .

[8]  Martin Hofmann,et al.  Extensional Constructs in Intensional Type Theory , 1997, CPHC/BCS Distinguished Dissertations.

[9]  Bart Jacobs,et al.  Categorical Logic and Type Theory , 2001, Studies in logic and the foundations of mathematics.

[10]  M. Hofmann.  Extensional concepts in intensional type theory , 1995 .

[11]  Per Martin-Löf,et al.  Intuitionistic type theory , 1984, Studies in proof theory.

[12]  Stephen Lack,et al.  Homotopy-theoretic aspects of 2-monads , 2006 .

[13]  G. M. Kelly,et al.  Two-dimensional monad theory , 1989 .

[14]  Richard Garner,et al.  Two-dimensional models of type theory , 2008, Mathematical Structures in Computer Science.

[15]  Martin Hofmann,et al.  On the Interpretation of Type Theory in Locally Cartesian Closed Categories , 1994, CSL.

[16]  A. K. Bousfield,et al.  Constructions of factorization systems in categories , 1977 .

[17]  William A. Howard,et al.  The formulae-as-types notion of construction , 1969 .

[18]  Giovanni Sambin,et al.  Twenty-five years of constructive type theory. , 1998 .

[19]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[20]  D. W. Anderson,et al.  Fibrations and geometric realizations , 1978 .

[21]  Bengt Nordström,et al.  Programming in Martin-Lo¨f's type theory: an introduction , 1990 .

[22]  Samson Abramsky,et al.  Handbook of logic in computer science. , 1992 .

[23]  Paul Taylor,et al.  Practical Foundations of Mathematics , 1999, Cambridge studies in advanced mathematics.

[24]  Ross Street,et al.  Fibrations and Yoneda's lemma in a 2-category , 1974 .

[25]  A. Pitts.  INTRODUCTION TO HIGHER ORDER CATEGORICAL LOGIC (Cambridge Studies in Advanced Mathematics 7) , 1987 .

[26]  R. Seely,et al.  Locally cartesian closed categories and type theory , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[27]  F. Ramsey.  The foundations of mathematics , 1932 .

[28]  John Cartmell,et al.  Generalised algebraic theories and contextual categories , 1986, Ann. Pure Appl. Log..

[29]  Andrew M. Pitts,et al.  Categorical logic , 2001, LICS 2001.

[30]  F. W. Lawvere,et al.  FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[31]  de Ng Dick Bruijn,et al.  Telescopic Mappings in Typed Lambda Calculus , 1991, Inf. Comput..

[32]  S. Awodey,et al.  Homotopy theoretic models of identity types , 2007, Mathematical Proceedings of the Cambridge Philosophical Society.