Another question about Kan

Another question about Kan

[Michael Barr]

Is the following known?

An equational category has the property that every simplicial object is
Kan iff it is a Mal'cev category.  This means that there is a ternary
operation I call <-,-,-> such that <x,y,y> = x and <x,x,y> = y.  In a
sense this is not surprising.  The Kan condition makes homotopy an
equivalence relation.  The degeneracies make homotopy reflexive and
Mal'cev categories are characterized by the fact that every reflexive
binary relation is an equivalence relation.

Michael


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Re: Another question about Kan

[Aurelio Carboni]

Yes, it is in my paper with Kelly and Pedicchio
'Some Remarks on Maltsev and Goursat Categories',
(Applied Mathematical Structures 1, 385-421, 1993)
where it is proved more generally for regular categories
(Theorem 4.2, p. 404).
                             Aurelio Carboni

At 16.28 13/09/2011, you wrote:
>Is the following known?
>
>An equational category has the property that every simplicial object is
>Kan iff it is a Mal'cev category.  This means that there is a ternary
>operation I call <-,-,-> such that <x,y,y> = x and <x,x,y> = y.  In a
>sense this is not surprising.  The Kan condition makes homotopy an
>equivalence relation.  The degeneracies make homotopy reflexive and
>Mal'cev categories are characterized by the fact that every reflexive
>binary relation is an equivalence relation.
>
>Michael
>


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Re: Another question about Kan

[Michael Barr]

Since it is a Springer journal, there is no way I can see the article.

I would point out that every paper of mine since 1986 along with many
earlier ones are available on my ftp site:
ftp.math.mcgill.ca/pub/barr/pdffiles.  Perhaps I will go ahead and try to
publish my proof anyway, since it will be accessible.

Michael

On Wed, 14 Sep 2011, Aurelio Carboni wrote:

> I am sorry, the name of the journal is wrong: it is
> 'Applied Categorical Structures' and not 'Applied
> Mathematical Structures'. Everything else is correct.
>         Aurelio.
>
>
> At 15.33 14/09/2011, you wrote:
>> Google can find no journal called Applied Mathematical Structures.  Can
>> you send me a copy of the paper, or a link.
>>
>> Michael
>


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Re: Another question about Kan

[Aurelio Carboni]

I am sorry, the name of the journal is wrong: it is
'Applied Categorical Structures' and not 'Applied
Mathematical Structures'. Everything else is correct.
          Aurelio.


At 15.33 14/09/2011, you wrote:
>Google can find no journal called Applied Mathematical Structures.  Can
>you send me a copy of the paper, or a link.
>
>Michael



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