Equivalence of categories
May 30, 2007  Posted by John Armstrong  Category theory
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Minor typo in the paragraph that begins “The upshot of all this …” You’re missing a backslash on a mathcal. Great stuff, keep it coming!
Comment by Cotton Seed  May 31, 2007 
[…] on a category constitute a full subcategory of all contravariant valued functors on , which is equivalent to the category […]
Pingback by What does Yoneda’s Lemma mean? « The Unapologetic Mathematician  June 7, 2007 
[…] The functor that we described from to is an equivalence. […]
Pingback by The Category of Matrices IV « The Unapologetic Mathematician  June 24, 2008 
Can you show that there is an equivalence of categories between PreSet( the category of preordered sets) and PoSet (the category of partialy ordered sets) ? :)
Comment by Lucia  February 1, 2009 
Yes, I can. I even mentioned the important bit once before.
Comment by John Armstrong  February 1, 2009 
I read the post about Divisibility, but it still isn’t very clear to me why every preordered set is equivalent to a partially ordered set…when you have time, please detail, it’s important for me to understand.
Comment by lucia  February 1, 2009 
Every partial order is also a preorder, so to go from partial orders to preorders you just forget that it’s a partial order.
In that post I tell how to go from a preorder to a partial order.
All that’s left is to show that these constructions are functors, and check whether they furnish an equivalence of categories.
Comment by John Armstrong  February 1, 2009 
Thanks :)
Comment by Lucia  February 1, 2009 
The equivalence of PreSet and PoSet that you have indicated may be in error. If F is the forgetful functor from PoSet to PreSet, and G is the functor that you indicated, then G is not faithful. Moreover, while GF = Id, there cannot be any isomorphism between a finite proper PreSet X and FG(X), since these sets have different cardinalities.
Is there some other equivalence that I am missing? It seems to me that these categories are not equivalent.
Comment by Chris  December 5, 2010 