* Another question about Kan
@ 2011-09-13 14:28 Michael Barr
2011-09-14 7:55 ` Aurelio Carboni
[not found] ` <4DE63475093E3CCF@smtp205.alice.it>
0 siblings, 2 replies; 4+ messages in thread
From: Michael Barr @ 2011-09-13 14:28 UTC (permalink / raw)
To: Categories list
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 4+ messages in thread
* Re: Another question about Kan
2011-09-13 14:28 Another question about Kan Michael Barr
@ 2011-09-14 7:55 ` Aurelio Carboni
[not found] ` <4DE63475093E3CCF@smtp205.alice.it>
1 sibling, 0 replies; 4+ messages in thread
From: Aurelio Carboni @ 2011-09-14 7:55 UTC (permalink / raw)
To: Michael Barr; +Cc: categories
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
>
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
^ permalink raw reply [flat|nested] 4+ messages in thread
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2011-09-13 14:28 Another question about Kan Michael Barr
2011-09-14 7:55 ` Aurelio Carboni
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2011-09-14 15:36 ` Michael Barr
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2011-09-14 14:45 ` Aurelio Carboni
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