🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

The definition already gives us the mathematical objects of a category. We only need to show composition satisfies associativity and identity law.

We first show associativity. Let $f^{\mathrm{o}\mathrm{p}}:A\to B,g^{\mathrm{o}\mathrm{p}}:B\to C,h^{\mathrm{o}\mathrm{p}}:C\to D$ be morphisms in ${𝒞}^{\mathrm{o}\mathrm{p}}$. Correspondingly, we have morphisms $f:B\to A,g:C\to B,h:D\to C$ in $𝒞$. We then have the following reasoning (composition subscript omitted): $${\displaystyle \begin{array}{cc}\hfill & (h^{\mathrm{o}\mathrm{p}}\circ g^{\mathrm{o}\mathrm{p}})\circ f^{\mathrm{o}\mathrm{p}}\hfill \\ \hfill =& (g\circ h{)}^{\mathrm{o}\mathrm{p}}\circ f^{\mathrm{o}\mathrm{p}}\hfill \\ \hfill =& (f\circ (g\circ h){)}^{\mathrm{o}\mathrm{p}}\hfill \\ \hfill =& ((f\circ g)\circ h{)}^{\mathrm{o}\mathrm{p}}\hfill \\ \hfill =& h^{\mathrm{o}\mathrm{p}}\circ (f\circ g{)}^{\mathrm{o}\mathrm{p}}\hfill \\ \hfill =& h^{\mathrm{o}\mathrm{p}}\circ (g^{\mathrm{o}\mathrm{p}}\circ f^{\mathrm{o}\mathrm{p}})\hfill \end{array}}$$

Lastly we show the existence of identity morphism along with their desired property. Let $A,B\in {𝒞}^{\mathrm{o}\mathrm{p}}$, we choose the identity morphism of this object to be $1_A^{\mathrm{o}\mathrm{p}},1_B^{\mathrm{o}\mathrm{p}}$ respectively.

Let $f^{\mathrm{o}\mathrm{p}}:A\to B$ be a morphism in $𝒞$. We can show the following: $${\displaystyle \begin{array}{cc}\hfill f^{\mathrm{o}\mathrm{p}}\circ 1_A^{\mathrm{o}\mathrm{p}}=& (1_A\circ f{)}^{\mathrm{o}\mathrm{p}}\hfill \\ \hfill =& (f{)}^{\mathrm{o}\mathrm{p}}\hfill \\ \hfill 1_B^{\mathrm{o}\mathrm{p}}\circ f^{\mathrm{o}\mathrm{p}}=& (f\circ 1_B{)}^{\mathrm{o}\mathrm{p}}\hfill \\ \hfill =& (f{)}^{\mathrm{o}\mathrm{p}}\hfill \end{array}}$$

Thus all desired properties are shown, we conclude ${𝒞}^{\mathrm{o}\mathrm{p}}$ is a category.

For each $(X,x_0)\in ob\left({Top}_0\right)$ we map it to ${\pi }_1(X,x_0)$ which is the object functor. We call this function ${\pi }_1^{ob}:ob\left({Top}_0\right)\to ob(Grp)$

Now we will construct our morphism functor, that is given $f\in hom((X,x_0),(Y,y_0))$ we map it to a function , defined by $\left[\gamma \right]\mapsto [f\circ \gamma ]$ we call this mapping $h$, that is: $F_{(X,x_0),(Y,y_0)}$ where $F_{(X,x_0),(Y,y_0)}\left(f\right)=h$

We now show that $h\in hom({\pi }_1(X,x_0),{\pi }_1(Y,y_0))$, to do this, we must prove that it is a group homomorphism, and also that it is well defined. For the well definedness we note that if $\left[\alpha \right]=\left[\beta \right]$ then we have that $[f\circ \alpha ]=[f\circ \beta ]$. For ease of notation recall that $f_{*}\left(\left[\gamma \right]\right):=[f\circ \gamma ]$, and we prove that our function $h$ is a group homomorphism, that is we show that $${\displaystyle f_{*}\left(\left[\alpha \right]\left[\beta \right]\right)=f_{*}\left(\left[\alpha \right]\right)f_{*}\left(\left[\beta \right]\right)}$$ but on the left we have $${\displaystyle \begin{array}{cc}\hfill f_{*}\left(\left[\alpha \right]·\left[\beta \right]\right)& =f_{*}\left(\left[\alpha ·\beta \right]\right)\hfill \\ \hfill & =[f\circ \left(\alpha ·\beta \right)]\hfill \\ \hfill & =\left[(f\circ \alpha )·(f\circ \beta )\right]\hfill \\ \hfill & =[f\circ \alpha ]·[f\circ \beta ]\hfill \\ \hfill & =f_{*}\left(\alpha \right)·f_{*}\left(\beta \right)\hfill \end{array}}$$ which is the right, so we are done.

We now verify the functor properties: $${\displaystyle F_{(X,x_0),(X,x_0)}\left({id}_{(X,x_0)}\right)={id}_{{\pi }_1^{ob}(X,x_0)}={id}_{{\pi }_1}(X,x_0)\}}$$ We have that $F_{(X,x_0),(X,x_0)}\left({id}_{(X,x_0)}\right)=h$ where $h\left(\left[\gamma \right]\right)=[{id}_{(X,x_0)}\circ \gamma ]=\left[\gamma \right]$, in other words $h$ takes an element from ${\pi }_1(X,x_0)$ and returns that same element, so that $h={id}_{{\pi }_1(X,x_0)}$ as needed.

We now verify that $${\displaystyle F_{(X,x_0),(Y,y_0)}(g\circ f)=F_{(Y,y_0),(Z,z_0)}\left(g\right)\circ F_{(X,x_0),(Y,y_0)}\left(f\right)}$$ If we focus on the right hand side and give the two functions simpler names ; $h_g:{\pi }_1(Y,y_0)\to {\pi }_1(Z,z_0)$ and $h_f:{\pi }_1(X,x_0)\to {\pi }_1(Y,y_0)$ respectively, and thus on the right hand side we have: $${\displaystyle \begin{array}{cc}\hfill (h_g\circ h_f)\left(\left[\gamma \right]\right)& =h_g\left(h_f\left(\left[\gamma \right]\right)\right)\hfill \\ \hfill & =h_g\left([f\circ \gamma ]\right)\hfill \\ \hfill & =[g\circ (f\circ \gamma )]\hfill \\ \hfill & =[(g\circ f)\circ \gamma ]\hfill \\ \hfill & =F_{(X,x_0),(Y,y_0)}(g\circ f)\hfill \end{array}}$$ so our original equation holds true.

Thus since we've verified all the conditions, we conclude that ${\pi }_1$ is a functor.