Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 16th 2018

    A stub so far.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2018
    • (edited Apr 16th 2018)

    Notice that we already have an entry Homotopy Type System, which is the same kind of idea.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 16th 2018

    Ah ok. The second reference’s abstract says

    The idea of two-level type theory is heavily inspired by Voevodsky’s Homotopy Type System (HTS). Two-level type theory can be thought of as a version of HTS without equality reflection. We show that the lack of equality reflection does not hinder the development of the ideas that HTS was designed for.

    Does this merit a distinct entry then?

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeApr 16th 2018

    Sorry, I forgot about that page.

    I would lean towards folding Homotopy Type System into two-level type theory. Unfortunately, “2LTT” seems to be being used both as a general term for type theories with both exact and path equality types, and also a name for the specific ACK system. The general term includes not just ACK and HTS but also the “computational cubical type theory” of Harper-Angiuli et. al. Nearly everything we currently have at Homotopy Type System applies more generally to all two-level type theories, and right now it seems that ACK and CCTT are receiving more active work than HTS. (In particular, HTS has the disadvantage of undecidable type-checking, which ACK restores by leaving out equality reflection in favor of UIP – for CCTT I’m not sure, that may not be a relevant question for the sort of type theory that it is.)

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 17th 2018

    I added a link to Homotopy Type System. I’ll leave it to experts to arrange what they take to be the proper relation between these pages.

    diff, v2, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeApr 17th 2018

    folded HTS into this page

    diff, v3, current

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 17th 2018

    I also added some discussion of the issue of fibrant vs non-fibrant nat.

  1. removed duplicate redirects (fibrant types used to redirect to here and to fibrant type)

    Anonymouse

    diff, v10, current