Results 1 to 3 of 3

Math Help - Category Theory

  1. #1
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721

    Category Theory

    What are \mathcal{AB}^{op} and \mathcal{GRP}^{op} naturally equivalent to?

    Okay I'm stuck in this one, I know what a natural equivalence is, but my problem is I don't really understand what (.)^{op} does, I know it reverses all arrows in the Category but I don't get what changes by doing that.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by Jose27 View Post
    What are \mathcal{AB}^{op} and \mathcal{GRP}^{op} naturally equivalent to?

    Okay I'm stuck in this one, I know what a natural equivalence is, but my problem is I don't really understand what (.)^{op} does, I know it reverses all arrows in the Category but I don't get what changes by doing that.
    In wiki (link), an opposite category or dual category of C^{op} of a given category C is formed by reversing the morphisms. It means the objects of C^{op} are just the objects of C and the arrows of C^{op} are arrows f^{op} in a one to one correspondence manner (if an arrow of C is f:x \rightarrow y, then f^{op} in C^{op} is f^{op}:y \rightarrow x. I assume you already know this one.

    Now consider some examples. Let S be a functor from the category of C^{op} to category of B such that S:C^{op} \rightarrow B. It assigns to each object c \in C^{op} an object Sc of B and to each arrow f^{op}:b \rightarrow a of C^{op} an arrow Sf^{op}:Sb \rightarrow Sa of B.
    Here is another example. If T:C \rightarrow B is a functor from the category of C to the category of B, then we can define a functor from T^{op}:C^{op} \rightarrow B^{op}. In this case, c \mapsto Tc can be applied to both, where c is an object of both C and C^{op} (Remember that objects remain same in the dual category). However, f \mapsto Tf should be changed to f^{op} \mapsto (Tf)^{op} for the latter one.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Jose27 View Post
    What are \mathcal{AB}^{op} and \mathcal{GRP}^{op} naturally equivalent to?
    this question is not easy to answer! right now i have no idea what category \mathcal{GRP}^{op} is equivalent to but it is known that \mathcal{AB}^{op} is equivalent to the category of compact abelian groups (continuous

    homomorphisms are the morphisms of this category). the idea is to associate to any abelian group G the abelian group of its characters, i.e. \hat{G}=\text{Hom}_{\mathbb{Z}}(G, \mathbb{T}), where \mathbb{T}=\mathbb{R}/\mathbb{Z} is the circle group.

    we also associate to any homomorphism f: G_1 \longrightarrow G_2 the morphism \hat{f}: \hat{G_2} \longrightarrow \hat{G_1} defined by \hat{f}(g)=g \circ f. the topology on \hat{G} is the pointwise convergence topology on \mathbb{T}^G, where \mathbb{T}^G is the set

    of all functions G \longrightarrow \mathbb{T}. there are so much details here to be proved and you probably can find find them in some textbooks.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Groups in Category Theory
    Posted in the Advanced Algebra Forum
    Replies: 12
    Last Post: November 28th 2011, 07:37 AM
  2. Question on Category Theory
    Posted in the Advanced Math Topics Forum
    Replies: 6
    Last Post: June 20th 2010, 02:10 PM
  3. Help on Category Theory
    Posted in the Advanced Algebra Forum
    Replies: 16
    Last Post: November 18th 2009, 09:24 PM
  4. Help w/ Proof in Category Theory
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 6th 2009, 05:53 PM

Search Tags


/mathhelpforum @mathhelpforum