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nLab > Latest Changes: line type

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  1. starting page on the homotopical reals in dependent type theory

    Anonymouse

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2023
    • (edited Nov 6th 2023)

    It is dubious that this type deserves to be called anything with “real numbers” in it. It’s the groupoidification of ℤ\mathbb{Z} regarded as a poset, and nobody would call that the anything real numbers.

    Also, the HoTT book which you reference does not call this type such.

    Incidentally, you point to 8.1 in the HoTT book, which is mostly a discussion of related issues – the definition you want to point to is a couple of lines in section 8.1.1 only.

  3. It is dubious that this type deserves to be called anything with “real numbers” in it. It’s the groupoidification of ℤ\mathbb{Z} regarded as a poset, and nobody would call that the anything real numbers.

    This type probably should be called the “continuum line type” or “line continuum type” and denoted with A 1A^1, rather than the “homotopical real numbers type” and denoted with RR.

    Assuming the definition in the HoTT book, the ring structure is inherited from the integers via the first constructor - which means that it is a line in the sense of linear algebra, and the second constructor implies that all points in the type, while not judgmentally equal to each other, are connected to each other by unique paths up to homotopy - and hence that it is contractible and homotopically a continuum.

    While it is true that the real numbers could be used to model the line continuum in homotopy theory, one also sees the use of other objects to model the line continuum, such as the affine line in motivic homotopy theory. The whole point of synthetic homotopy theory is to abstract away the non-homotopical content, and one ends up with an abstract line continuum, rather than anything specific about the real numbers.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2023
    • (edited Nov 6th 2023)

    Thanks, these are good suggestions, I think.

    Another option might be to find a name reflecting the fact that this type is the (-1)-truncation of the type of integers, for a specific but standard construction of (-1)-truncation.

    [edit: typo fixed now, thanks Madeleine]

    Maybe the “bracket type of integers”?

  5. Another option might be to find a name reflecting the fact that this type is the 0-truncation of the type of integers, for a specific but standard construction of 0-truncation.

    I think you meant (-1)-truncation here; the 0-truncation of the integers is still the integers, since the type of integers is a set.

    Maybe the “bracket type of integers”?

    The interval type is the bracket type of the boolean domain, but not many people refer to the interval type as the “bracket type of booleans”.

  6. there is discussion on nForum which says line type is better name than homotopical real numbers type

    Anton Ivanov

    diff, v2, current

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