Re: [HoTT] "Identifications" ?

[Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>]

On 04/05/2020, 12:00, "stre...@mathematik.tu-darmstadt.de" <stre...@mathematik.tu-darmstadt.de> wrote:

    > I am just reading a paper which uses the word “Identification” instead
    > of equality. I think this has been proposed by Bob Harper. Can anybody
    > enlighten me what is the difference between identifications and equality?
    > Maybe there is an identification between them but they are not equal? Are
    > the real numbers 0.999… identified or are they equal?

    You may identify things which are not equal.

Hence my question wether 0.999... and 1 are equal or identified. 

My understanding is that 0.999... and 1 as real numbers are equal. That is what we mean by equality in Mathematics. The difference which is introduced here is intensional and doesn't make any sense. We may even ask whether 3+1 and 4 are identified or equal?

   So ''identify'' seems to mean
    consider as equal.
    In a sense this is very descriptive for what is going on in HoTT where one
    never performs quotients but rather widens equivalence relations.

I don't understand this. Surely we quotient in HoTT even though the notion of a HIT or even a QIT (i.e. a set level HIT) are more general than ordinary set-quotients.

    This is really different in the respective topos models as eg simplicial
    sets or already the topos of reflexive graphs as exemplified by N. Kraus's
    puzzling counterexample.

What puzzling counterexample is this?

Thorsten

On 04/05/2020, 12:00, "stre...@mathematik.tu-darmstadt.de" <stre...@mathematik.tu-darmstadt.de> wrote:

    > I am just reading a paper which uses the word “Identification” instead
    > of equality. I think this has been proposed by Bob Harper. Can anybody
    > enlighten me what is the difference between identifications and equality?
    > Maybe there is an identification between them but they are not equal? Are
    > the real numbers 0.999… identified or are they equal?

    You may identify things which are not equal. So ''identify'' seems to mean
    consider as equal.
    In a sense this is very descriptive for what is going on in HoTT where one
    never performs quotients but rather widens equivalence relations.
    This is really different in the respective topos models as eg simplicial
    sets or already the topos of reflexive graphs as exemplified by N. Kraus's
    puzzling counterexample.

    Thomas





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