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  1. starting page on equivalence extensionality

    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2023
    • (edited Jan 8th 2023)

    let’s give all HoTT-related entries the context menus for “Type theory” and for “Homotopy theory”:

      +-- {: .rightHandSide}
      +-- {: .toc .clickDown tabindex="0"}
      ### Context
      #### Type theory
      +-- {: .hide}
      [[!include type theory - contents]]
      =--
      #### Homotopy theory
      +--{: .hide}
      [[!include homotopy - contents]]
      =--
      =--
      =--
    

    By the way, I like a minimum of whitespace in type-theoretic formulas to avoid them looking like symbol soup, with standard equivalence- and equality-symbols used in a way that non-insiders cannot parse.

    In the case at hand I have introduced whitespace as follows:

      \mathrm{equivext}(f, g)  
      \;\;\colon\;\; 
      \big(f =_{A \simeq B} g\big) 
      \;\simeq\; 
      \prod_{x \colon A} 
      \;
      f(x) =_{B} g(x)
    

    But if this were my entry, I would go further, since I find that using sub-scripted equality signs for identification types is both conceptually ill-conceived as well as a ill-typeset (at least as soon as the subscript has more than one symbol). What I would type is:

      \mathrm{equivext}(f, g)  
      \;\colon\;
      Equiv
      \bigg(
        Id_{A \simeq B}\big(
          f,\,g
        \big) 
        ,\,
        \prod_{x \colon A} 
        \,
        Id_B\big(
          f(x)
          ,\,
          g(x)
        \big)
      \bigg)
    

    which renders intelligibly as:

    equivext(f,g):Equiv(Id AB(f,g), x:AId B(f(x),g(x))) \mathrm{equivext}(f, g) \;\colon\; Equiv \bigg( Id_{A \simeq B}\big( f,\,g \big) ,\, \prod_{x \colon A} \; Id_B\big( f(x) ,\, g(x) \big) \bigg)

    diff, v3, current