Results 1 to 5 of 5
Like Tree1Thanks
  • 1 Post By Deveno

Math Help - why is power set functor contravariant?

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    17

    why is power set functor contravariant?

    I have read that the powerset constuction when viewed as an endofunctor is contravariant but I cant find any explanation of why it is contravariant and not covariant. Any clues?

    ie: if f is an arrow A->B why is the arrow P(f) an arrow PB->PA

    I did wonder if the author was saying if you view it as contravariant as opposed to covariant then various other things follow. So is it necessarily contravariant?

    Thanks,
    Chad.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,513
    Thanks
    769

    Re: why is power set functor contravariant?

    If P(f)(A) is defined as f[A], then P is covariant. If P(f)(A) is defined as f^{-1}[A], then it is contravariant.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2010
    Posts
    17

    Re: why is power set functor contravariant?

    So it can be either covariant or contravariant depending on which way you want to define it...?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,513
    Thanks
    769

    Re: why is power set functor contravariant?

    Yes.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,314
    Thanks
    695

    Re: why is power set functor contravariant?

    the covariant power-set functor arrow P(f) is not a lattice isomorphism (considering meet as intersection, and join as union) from P(A) to P(B), whereas the contravariant P(f) is from P(B) to P(A). naively, you can think of this as: "functions may lose information" (they may identify distinct elements), whereas "pre-images preserve information" (more precisely, they respect the partition of A induced by f).

    this is why it is more natural to consider pre-images of topologies, than images of topologies (topologies are lattices, defined by bases...if we wish say that continuous functions are "nearness-morphisms" we have to account for the fact that two points "not near" in A may indeed be "near" in the image set f(A), for example: constant functions).

    monomorphisms (or as functors: faithful functors) are "nicer" than regular morphisms. often these have special names in any given category (injection, embedding, sub-thingy, full rank) to indicate their distinguished status.

    the contravariant power-set functor is an example of a Hom-functor: since we can regard P(A) as 2A = Hom(A,2) (where "2" is any two-element set, typically
    { {},{{}} }, or its more "usual" name {0,1}) and Hom is contravariant in the first argument.

    some people use opposite (or dual) categories to make everything "covariant". for example, when creating a partial order, we get to choose what "up" ("bigger", "join") means, usually we do so to aid our intuition, or following convention (in a complemented lattice, you can see how arbitrary this choice actually is: for example, "smaller open set" means "bigger closed complement", and vice-versa).
    Thanks from emakarov
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Contravariant Vector (newbie in trouble)
    Posted in the Higher Math Forum
    Replies: 11
    Last Post: August 8th 2012, 07:20 AM
  2. Forgetful Functor
    Posted in the Advanced Math Topics Forum
    Replies: 5
    Last Post: September 8th 2011, 04:23 PM
  3. Covariant & Contravariant differentiation
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: May 14th 2011, 11:25 AM
  4. Pure contravariant tensor
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 16th 2009, 09:03 AM
  5. Homotopy Functor
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 28th 2009, 01:33 AM

Search Tags


/mathhelpforum @mathhelpforum