Categorical Approach to Orbit–Stabilizer Theorem

Let $X$$X$ be a $G\text{-}\mathsf{Set}$$G\text{-}\mathsf{Set}$. We could view it as a groupoid by take $\mathrm{Ob}(\hat{X})=X,\mathrm{Mor}(\hat{X})=G$$\mathrm{Ob}(\widehat X)=X,\mathrm{Mor}(\widehat X)=G$. i.e. $x⟼g\cdot x$$x\longmapsto g\cdot x$ as morphism.

Then we have

$$\coprod _{y\in X}\mathrm{Hom}(x,y)=\coprod _{y\in G\cdot x}\mathrm{Hom}(x,y)$$

Now we have

$$\mathrm{\forall }y\in G\cdot x,x\cong y⟹\mathrm{Hom}(x,y)\cong \mathrm{Hom}(x,x)=G_x$$

Thus

$$\coprod _{y\in G\cdot x}\mathrm{Hom}(x,y)\cong \coprod _{y\in G\cdot x}\mathrm{Hom}(x,x)$$

The LHS is $G$$G$ and the RHS is $\begin{array}{c}\coprod _{y\in G\cdot x}{G}_x\end{array}$$\coprod_{y\in G\cdot x}G_x\\$.

Hence we have

$$|G|=|G\cdot x||G_x|$$

 

How to understand Natural Transformation?

We can view homotopy by analogy with natural transformations.

Let $f,g:X\to Y$$f,g:X\to Y$ be continuous maps. A homotopy between them is a continuous map

$$H:X\times [0,1]\to Y$$

with $H(x,0)=f(x)$$H(x,0)=f(x)$ and $H(x,1)=g(x)$$H(x,1)=g(x)$ for all $x\in X$$x\in X$.

Consider the small category $[0\to 1]$$[0\to 1]$ whose objects are two points $0,1$$0,1$ and whose arrows are the two identity arrows and a single arrow $0\to 1$$0\to 1$.
A natural transformation between functors $F,G:\mathcal{C}\to \mathcal{D}$$F,G:\mathcal C\to\mathcal D$ can be described exactly by a functor

$$H:\mathcal{C}\times [0\to 1]⟶\mathcal{D}$$

such that for every object $c\in \mathcal{C}$$c\in\mathcal C$ we have

$$H(c,0)=F(c),H(c,1)=G(c).$$

More precisely, let $i_0,i_1:\mathcal{C}\hookrightarrow \mathcal{C}\times [0\to 1]$$i_0,i_1:\mathcal C\hookrightarrow\mathcal C\times[0\to 1]$ be the two canonical embeddings. Then the condition is

$$H\circ i_0=F,H\circ i_1=G.$$

In the category $\mathcal{C}\times [0\to 1]$$\mathcal C\times[0\to 1]$ we have the obvious commutative square coming from an arrow $f:X\to Y$$f:X\to Y$ in $\mathcal{C}$$\mathcal C$:

$$\begin{array}{ccc}(X,0)& \stackrel{\mathrm{id}\times (0\to 1)}{\to }& (X,1)\\ f\times {\mathrm{id}}_0\downarrow & & \downarrow f\times {\mathrm{id}}_1& \\ (Y,0)& \stackrel{\mathrm{id}\times (0\to 1)}{\to }& (Y,1)\end{array}$$

The functor $H$$H$ preserves that square, and under the identifications $H(X,0)=F(X)$$H(X,0)=F(X)$, $H(X,1)=G(X)$$H(X,1)=G(X)$ the preservation is exactly the usual naturality square:

$$\begin{array}{ccc}F(X)& \stackrel{{\alpha }_X}{\to }& G(X)\\ F(f)\downarrow & & \downarrow G(f)& \\ F(Y)& \stackrel{{\alpha }_Y}{\to }& G(Y)\end{array}$$

If you replace $[0\to 1]$$[0\to1]$ by $[0\leftrightarrow 1]$$[0\leftrightarrow 1]$ (the category in which $0$$0$ and $1$$1$ are isomorphic), then the resulting $H$$H$ gives a natural isomorphism.

Conversely, given a natural transformation $\alpha =({\alpha }_X{)}_{X\in \mathrm{Ob}(\mathcal{C})}$$\alpha=(\alpha_X)_{X\in\mathrm{Ob}(\mathcal C)}$, you can build a functor

$$H:\mathcal{C}\times [0\to 1]⟶\mathcal{D}$$

with $H(c,0)=F(c)$$H(c,0)=F(c)$ and $H(c,1)=G(c)$$H(c,1)=G(c)$, and define $H$$H$ on the canonical arrow ${\mathrm{id}}_X\times (0\to 1)$$\mathrm{id}_X\times(0\to1)$ by $H({\mathrm{id}}_X\times (0\to 1))={\alpha }_X$$H(\mathrm{id}_X\times(0\to1))=\alpha_X$.

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Passing to fundamental groupoids

The fundamental groupoid functor

$${\mathrm{\Pi }}_1:\mathbf{Top}⟶\mathbf{Grpd}$$

turns spaces into groupoids and continuous maps into functors. A homotopy therefore induces a functor at the ${\mathrm{\Pi }}_1$$\Pi_1$ level: a homotopy $H:X\times [0,1]\to Y$$H:X\times[0,1]\to Y$ gives

$$H:{\mathrm{\Pi }}_1(X)\times {\mathrm{\Pi }}_1([0,1])⟶{\mathrm{\Pi }}_1(Y).$$

We can embed the isomorphism-category $[0\leftrightarrow 1]$$[0\leftrightarrow 1]$ into ${\mathrm{\Pi }}_1([0,1])$$\Pi_1([0,1])$ (the two endpoints are connected by a canonical path class). Thus we have an embedding

$$\mathrm{id}\times \iota :{\mathrm{\Pi }}_1(X)\times [0\leftrightarrow 1]\hookrightarrow {\mathrm{\Pi }}_1(X)\times {\mathrm{\Pi }}_1([0,1]).$$

By definition, the composite

$$H\circ (\mathrm{id}\times \iota ):{\mathrm{\Pi }}_1(X)\times [0\leftrightarrow 1]⟶{\mathrm{\Pi }}_1(Y)$$

is exactly a natural isomorphism: at each object $x\in {\mathrm{\Pi }}_1(X)$$x\in\Pi_1(X)$ the component is the path class $[t\mapsto H(x,t)]$$[t\mapsto H(x,t)]$, giving the component of a natural isomorphism between ${\mathrm{\Pi }}_1(f)$$\Pi_1(f)$ and ${\mathrm{\Pi }}_1(g)$$\Pi_1(g)$.