Group actions as colimit of torsors? I'm learning group theory and I know a little bit about category theory(Mac Lane ch1-3,but have not appreciated what a colimit is).  
I know a group action can be viewed as a functor from a group as a one-element category to $Set$. And I also know the statement of so-called co-Yoneda theorem,which roughly says any Set-valued functor is a colimit of representable functors. Furthermore,if I understand it correctly,the only representable functors in this setting are torsors(roughly,group action acting on this group viewed as a set,see torsor).  
I want to learn that,what's the meaning of co-Yoneda theorem in this example of group action?(roughly,any group action can be viewed as a colimit of "torsors". here "torsors" should be realized as whatever it should be in the co-Yoneda theorem.) Or,I should say I have no idea how to transform it into the group theory language.  
 A: A group can be viewed as a one-object category where all arrows are isomorphisms. (Few categorists would say "one-element category".)  Concretely, given a group $G$, we can make a category $\mathcal{G}$ with one object, call it $*$ though what it is is completely irrelevant, and $\text{Hom}(*,*) = G$.  $id$ corresponds to the unit of $G$ and $\circ$ corresponds to the multiplication of $G$.  A functor $F : \mathcal{G} \to \mathbf{Set}$ gives a group action. The one-object gets mapped to a set $X=F(*)$ and each arrow $g$ in $\mathcal{G}$ (i.e. each element of the group $G$) gets mapped to a function $F(g) : X \to X$.  The functor laws $F(id) = id$ and $F(f \circ g) = Ff \circ  Fg$, say exactly that $F$ is a group homomorphism from $G$ to the automorphism group $X \to X$, i.e. $F$ is a group action.
Generally, a representable functor, is a functor $H : \mathcal{C} \to \mathbf{Set}$ for some category $\mathcal{C}$ that is naturally isomorphic to a $\text{Hom}$ functor, i.e. $H \cong \text{Hom}(Z, -)$ where the object $Z$ is said to represent the functor $H$.  In our case, there's only one object so up to isomorphism the only representable functor is $\text{Hom}(*,-)$.  This functor corresponds to $G$ acting on itself.  Not looking at things only up to isomorphism, we have that a natural transformation $\tau : H \to \text{Hom}(*,-)$ is a natural transformation with only one component since $\mathcal{G}$ has only one object and which concretely looks like $g\cdot\tau(x) = \tau(H(g)(x))$ where $g\cdot g'$ is the multiplication of the group, i.e. composition in $\mathcal{G}$.  For it to be a natural isomorphism it would have an inverse which satisfied $\tau^{-1}(g\cdot g') = H(g)(\tau^{-1}(g'))$.  Combining these we can define the action of $H$ on arrows via $H(g)(x) = \tau^{-1}(g\cdot\tau(x))$.
The co-Yoneda lemma is the following statement $$KX \cong \int^{C:\mathcal{C}} \text{Hom}(C^-, X)\times KC^+ \cong \text{Hom}(-,X)\star K \cong \text{Lan}_{Id}K(X)$$ natural in $X$ where $\int^C$ denotes a coend, $\star$ denotes the weighted colimit, and $\text{Lan}_J D$ is the left Kan extension of $D$ along $J$.  Incidentally, the regular Yoneda lemma is $$KX \cong \int_{C:\mathcal{C}} \mathbf{Set}(\text{Hom}(C^-,X), KC^+) \cong \{\text{Hom}(-,X),K\} \cong \text{Ran}_{Id}K(X)$$ where $\int_C$ denotes the end, the brackets denote the weighted limit, and $\text{Ran}_J D$ is the right Kan extension of $D$ along $J$.  The superscript $+$ and $-$ are my own variation of notation to keep track of whether the bound variable $C$ is in a covariant or contravariant position.  You are probably not familiar with any of these concepts as they are not "beginning of the book" material.  Surprisingly and unfortunately, (co)ends and weighted (co)limits are often nowhere in the book.  This is surprising as these notions are critical for enriched category theory (suggesting they are more natural than (conical) (co)limits), and they dramatically simplify calculations.  Conical (co)limits can be expressed as (co)ends over functors that only use the bound variable covariantly.  With my notation, it looks like $\int^C DC^+ \cong \text{Colim}(D)$ (one of the benefits of tracking variance).
In $\mathbf{Set}$, it is possible to express weighted colimits as (conical) colimits indexed by the category of elements of the weighting functor.  I'm not going to go into detail about the category of elements, since I just want to have a concrete expression of the particular colimit in $\mathbf{Set}$, so I will just say that we get the following rewriting of a coend: $$\int^{C:\mathcal{C}} WC^-\times HC^+ \cong \int^{E:\text{el}W} H(\pi(E^+)) \cong \text{Colim}(H \circ \pi)$$ where $\pi : \text{el}W \to \mathcal{C}$ is the codomain projection of the category of elements.  When both $H$ and $W$ are $\mathbf{Set}$-valued, we can swap them about (which results in indexing over the opposite category) and so choosing $W = F$ and $H = \text{Hom}(-,X)$ we get $$FY \cong \int^{E:(\text{el}F)^{op}} \text{Hom}(\pi^{op}(E^+),Y) \cong \int^{(A,B):(\text{el}F)^{op}} \text{Hom}(A^+,Y) \cong \text{Colim}(\text{Hom}(-,Y) \circ \pi^{op})$$ which is to say, $F$ is a colimit of representables.  In our case, $\text{el}F$ has an object for each element of $X = F(*)$, hence "category of elements", and an arrow from $x\in X$ to $y\in X$ exactly when $y = F(g)(x)$ for some $g\in G$. 
In general, a colimit in $\mathbf{Set}$ of a functor $D : \mathcal{C} \to \mathbf{Set}$, is made by first forming a coproduct (i.e. disjoint union) of $D$ over all the objects $C$ of $\mathcal{C}$, giving $\coprod_{C\in \text{Ob}(\mathcal{C})}DC$ whose elements look like pairs of an object $C$ of $\mathcal{C}$ and an element of $DC$.  From there we quotient it by the equivalence relation which for $(C, x)$ and $(C', y)$ in $\coprod_{C\in\text{Ob}(\mathcal{C})} DC$, we have 
$$(C, x) \sim (C', y) \iff (\exists f:C \to C'. D(f)(x) = y)\lor(\exists g:C' \to C. D(g)(y) = x)$$
so $\text{Colim}(D) \cong (\coprod_{C\in\text{Ob}(\mathcal{C})} DC)/\!\sim$.
In our case, it works out to be $$FY \cong \left(\coprod_{x \in F(*)}\text{Hom}(*, Y)\right)/\!\sim\ \cong (F(*)\times\text{Hom}(*,Y))/\!\sim\ = (X\times\text{Hom}(*,Y))/\!\sim$$ which, at the only object of $\mathcal{G}$ will be $$X = F(*) \cong (X\times \text{Hom}(*,*))/\!\sim\ = (X\times G)/\!\sim$$ where 
$$\begin{align}(x,g_1) \sim (y,g_2) & \iff \exists g \in G.x = F(g)(y) \land g_1\cdot g = g_2 \\& \iff x = F(g_1^{-1}\cdot g_2)(y) \\& \iff F(g_1)(x) = F(g_2)(y)\end{align}$$
There's a touch of subtlety here.  The $\exists g\in G.x = F(g)(y)$ comes from the $\exists f: C \to C'$ part of the general definition, where here an arrow of $\text{el}F$ from $x \to y$ was defined to be a $g$ such that $F(g)(x) = y$.  But wait! That's still not the same thing.  What's happening is that we indexed over the opposite category of the category of elements, so an arrow $x \to y$ in $(\text{el}F)^{op}$ is an arrow $y \to x$ in $\text{el}F$ which is a $g$ such that $F(g)(y) = x$.  The hardest part of categorical calculations is making sure you keep your variances straight, and this probably took me the most time of all of this.
The action of this functor on arrows is $$((X\times\text{Hom}(*,g))/\sim)([(x,g')]) = [(x, g\cdot g')] = [(F(g')(x), g)]$$
To verify, we have $\tau : F \to (X\times\text{Hom}(*,-))/\!\sim$ via $\tau(x) = [(x,1)]$ and $\tau^{-1}([(x,g)]) = F(g)(x)$.  We need to show that these are group action homomorphisms and that they are inverses.  For the former for $\tau$ we need to show that $[(F(g)(x),1)] = [(x,g)]$ which is true because $F(1)(F(g)(x)) = F(g)(x)$.  This is also what's needed to show $\tau^{-1}$ is well-defined (i.e. it doesn't depend on the representative chosen) and to show that $\tau$ and $\tau^{-1}$ are inverses, so we are done.
