The Unapologetic Mathematician

Mathematics for the interested outsider

Equivalence of categories

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May 30, 2007 - Posted by | Category theory


  1. 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 | Reply

  2. […] 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 | Reply

  3. […] The functor that we described from to is an equivalence. […]

    Pingback by The Category of Matrices IV « The Unapologetic Mathematician | June 24, 2008 | Reply

  4. 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 | Reply

  5. Yes, I can. I even mentioned the important bit once before.

    Comment by John Armstrong | February 1, 2009 | Reply

  6. 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 | Reply

  7. 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 | Reply

  8. Thanks :)

    Comment by Lucia | February 1, 2009 | Reply

  9. 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 | Reply

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