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Videos: The videos of lectures 2-4 are [[http://www.mpi-sws.org/seminars/higher_dimensional_type_theory/|available online]]. Altogether they're around 850mb. Use [[http://www.videolan.org/vlc/|VLC]] to play them if you're having trouble. They're password protected; email Scott for the credentials.

Higher-Dimensional Type Theory Seminar

WHAT? This lecture series conducted by Bob Harper will cover various topics in higher-dimensional type theory, including higher category theory and homotopy theory. It is not for the faint of heart.

WHO? This is some hardcore PL stuff. You should know dependent type theory and at least some category theory.

WHEN? See schedule below.

WHERE? MPI-SWS Kaiserslautern/Wartburg

HOW? Mailing list for this series (and other type theory discussion): type-theory@mpi-sws.org

Organizers: Bob Harper and Scott Kilpatrick

Videos: The videos of lectures 2-4 are available online. Altogether they're around 850mb. Use VLC to play them if you're having trouble. They're password protected; email Scott for the credentials.



2011-08-03, 2PM


Intro to HDTT and homotopy.


2011-08-05, 1PM


Intro to dependent type theory and equality types.


2011-08-10, 1PM


More on equality in dependent type theory.


2011-08-19, 1PM


Univalence, 2DTT, inductive definitions, and concluding remarks.


Blogs and Websites



Meeting #1

  • Intro to higher-dimensional type theory.
  • Intro to homotopy theory.
  • Intro to (higher) category theory.
  • Basic interpretation of type theory in category theory.
  • Different notions of equivalence (definitional and propositional).

  • What's going on in the field right now.

Meeting #2

  • Basics of dependent type theory.
  • Type-indexed families of types.
  • Natural numbers type and recursion.
  • Identity types as "least reflexive relation."
  • Elimination of identity types -- the J operator.
  • Substitution operator (think Coq's rewrite tactic), and how to encode it with J.
  • Discreteness axioms for adding extensionality; cutting off discreteness, e.g., for uniqueness of equality proofs of other equality types.

Meeting #3

  • Groupoid structure of identity types.
  • Encoding of transitivity and symmetry terms of identity type.
  • More on discreteness conditions. All types discrete in NuPRL; only identity types in 2-D type theories.
  • The Set universe. Individual sets are discrete, but Set is indiscrete.

  • More structure in the type A means fewer identities in Id_A(-,-).

  • Identify sets up to isomorphism. Introduction of iso term of type Id_Set(-,-) but no computational interpretation!

Meeting #4

  • Alternative notion of Id: Paulin's "one-sided elimination."

  • 2-dimensional type theory (see Bob and Dan's paper above) as an extension to NuPRL-like semantics.
  • Internalize identity as judgment Γ ⊢ α : M ≃ N : A and recover an Id type.

  • Now we can define subst on isos gotten from univalence.

  • Brief look at inductive definitions in 2DTT.

hdtt (last edited 2011-08-24 15:58:47 by skilpat)