1. ## 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.

2. Originally Posted by Jose27
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.

3. Originally Posted by Jose27
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.