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.
    • CommentAuthorUrs
    • CommentTimeDec 26th 2022

    a stub, for the moment just to have a place for recording Aczel’s original note

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 21st 2023

    wrote an Idea-section (here) with some indications of what’s in the literature, as far as I am aware.

    and then an “Examples”-section (here) with a discussion that I was hoping to find in the literature but didn’t.

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 31st 2023

    What should we say about this topic from the perspective of Higher Observational TT where laws are definitional, e.g., identity of pairs and pairs of identities (slide 3 and 13 of Mike’s slides) and this goes for dependent pair and function too (slide 40)? Might one say definitional univalence (slide 26) leads to structural definitional equality?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 31st 2023

    My understanding is that this is the whole motivation and point of HiObsTT: To make all identification types be definitionally respectful of their type’s structure.

    One wouldn’t be able to tell this from what “Anonymous” has written at higher observational type theory (this entry would deserve a complete overhaul), and it is not immediate to extract this from writings available elsewhere. But that’s my impression.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 31st 2023

    Thanks. It’s a shame we don’t get to hear more of the motivation:

    Homotopy theory is emergent rather than explicit; all rules have a convincing philosophical justification. (slide 8)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 31st 2023

    To decompose this: I suppose that

    1. the whole motivation and point of (non-higher) “observational type theory” is to make equality of types be the structure-preserving equalities,

    2. the “higher” analog is meant to do just the same but with equality replaced by identification types.

    So your main question is whether point (1.) is the case, and if so whether it is admitted anywhere.

    So let’s see:

    In the original article “Towards observational type theory” the motivation for the approach and terminology is:

    two functions are equal, if there are equal pointwise or to put it differently, if all observations about them agree.

    It seems that “observations” is not the best word here, since it is specific to the case of function types.

    The actual rule for equality of (a) dependent function types, (b) dependent pair types and (c) well-founded inductive types are stated in the very bottom left of p. 3.

    After realizing that what looks like norms are meant to be equalities (!, deduced from inside two paragraphs above in that left column) one sees that this bottom left formula says that equality of all such types is to imply component-wise equaities — and the converse is declared in section 2.3.

    In summary, this is to declare that the equalities of compound types are componentwise, hence what the algebraist would (and did) call: homo-morphically, i.e. structure-preservingly.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2023

    added pointer to:

    diff, v9, current

  1. Adding reference

    Anonymouse

    diff, v10, current