\documentclass{article}
\usepackage{setspace}
\def\C{\mathcal{C}}
\def\I{\mathcal{I}}
\def\J{\mathcal{J}}
\def\K{\mathcal{K}}
\def\N{\mathcal{N}}
\def\F{\I\stackrel{F}{\rightarrow}\C}
\def\G{\J\stackrel{G}{\rightarrow}\C}
\def\H{\K\stackrel{H}{\rightarrow}\C}
\def\Z{\N\stackrel{Z}{\rightarrow}\C}
\def\Hom{\mathrm{Hom}}
\def\f{\mathbf{f}}
\def\g{\mathbf{g}}
\def\h{\mathbf{h}}
\def\zero{\mathbf{i}^0}
\def\one{\mathbf{i}^1}
\def\Fzig{\I\times\N\stackrel{F\times Z}{\rightarrow}\C}

\begin{document}
\onehalfspace
\paragraph{Pro-objects in $\C$\\}
Let $\I$ be a cofiltered category - it must satisfy \\ (i) $\forall i, j \ \exists k, \ k\stackrel{\alpha}{\rightarrow}i, k\stackrel{\beta}{\rightarrow}j$ \\ and \\ (ii) $\forall i\stackrel{\alpha,\beta}{\rightarrow}j \exists k\stackrel{\gamma}{\rightarrow}i, \alpha\circ\gamma=\beta\circ\gamma$.\\\\
A pro-object in $\C$ is a functor $\F$.\\
These pro-objects form the objects of category $pro-\C$.\\\\
Morphisms in $pro-\C$, from objects $\F$ to $\G$, are of the following form:\\
The set $\Hom\left(F\left(i\right),G\left(j\right)\right)$ is contravariantly indexed in $i$ and covariantly indexed in $j$. Take FIRST the colimit in $i$ and SECOND the limit in $j$ to get the set $\Hom\left(\F,\G\right)$.\\\\
So, this is non-empty iff $\forall j \ \exists i, \ \exists F\left(i\right)\stackrel{f}{\rightarrow}G\left(j\right)$.\\\\
And an element $\f$ of $\Hom\left(\F,\G\right)$ (a `pro-morphism') is a set of equivalence classes $\{F\left(i\right)\stackrel{\f_j}{\rightarrow} G\left(j\right)\}$ indexed over all $j$, where $F\left(i\right)\stackrel{F\left(\alpha\right)}{\rightarrow}F\left(i'\right)\stackrel{\f_j}{\rightarrow}G\left(j\right)$ is equivalent to $F\left(i\right)\stackrel{\f_j\circ F\left(\alpha\right)}{\longrightarrow}G\left(j\right)$, such that for $j\stackrel{\beta}{\rightarrow}j'$, assuming $f_j$ and $f_{j'}$ have the same domain choice $i$ (which can do b/c $\I$ is cofiltered), $G\left(\beta\right)\circ f_j=f_{j'}$.\\\\
Composition of pro-morphisms: given pro-objects $\F$, $\G$, $\H$, and pro-morphisms $\F\stackrel{\f}{\Rightarrow}\H$ and $\G\stackrel{\g}{\Rightarrow}\H$, define $\h:=\g\circ\f$ as follows. For each $k\in \K$ have some $G\left(j\right)\stackrel{\g_k}{\rightarrow}H\left(k\right)$ and some $F\left(i\right)\stackrel{\f_j}{\rightarrow}G\left(j\right)$; define $\h_k:=\g_k\circ\f_j$. By the equivalence relation and by definition of lim and colim, this is independent of the choice of $j$ and of $i$, and also satisfies the necessary composition rules. \\

Identity pro-morphism:
Given pro-object $\F$, its identity map is the pro-morphism $\f$ consisting of identity maps $F\left(i\right)\stackrel{\f_i:=id_{F\left(i\right)}}{\longrightarrow}F\left(i\right)$ for each $i\in\I$.\\


\pagebreak
\paragraph{Pro-homotopy\\}
Let $\C$ be the category of finite topological spaces, with morphisms being continuous maps. We want to define what it means for two pro-morphisms $\F\stackrel{\f,\g}{\Longrightarrow}\G$ to be pro-homotopic.\\

Let $\N$ be the indexing category of the nonnegative integers, with a unique morphism $n\rightarrow m$ iff $n\geq m$. Define $\Z$ as sending $n$ to the finite space $\{a_0,b_1,a_1,b_2,\dots b_n,a_n\}$ with relations $a_i>b_{i-1},b_i$ for all $i$ (a contractible `zigzag') and for $n\geq m$, $Z\left(n\rightarrow m\right)$ collapses $\{b_i,a_i \ | \ i>m\}$ to the point $a_m$.\\

This gives a pro-object $\Fzig$. Define two pro-morphisms $\F\Rightarrow\Fzig$, $\zero$ and $\one$. (Note that as $\N$ is cofiltered, so is $\I\times\N$). \\

For $i\times n\in \I\times\N$ choose $\zero_{i\times n}$ to be inclusion of $F\left(i\right)$ to $F\left(i\right)\times a_0 \subset F\left(i\right)\times Z\left(n\right)$. Choose $\one_{i\times n}$ to be inclusion of $F\left(i\right)$ to $F\left(i\right)\times a_n \subset F\left(i\right)\times Z\left(n\right)$. (Note that all composition rules hold).\\

Now define a pro-homotopy between $\f$ and $\g$ to be a pro-morphism $\Fzig\stackrel{\h}{\Rightarrow}\G$ such that $\h\circ\zero=\f$ and $\h\circ\one=\g$.\\

\paragraph{Connection to fundamental group of finite spaces\\}
For a finite topological space $X$, let $\J=\{\ast\}$ and let $\G$ map the point to $X$. This gives a pro-object representing $X$.
Let $\I=\N$ as above and let $\F$ send $n$ to $Y^{\left(n\right)}$ and send morphisms to $\zeta$ as in my notes on fundamental groups of A-spaces.\\

Now consider pro-homotopy classes of the pro-morphisms $\F\Rightarrow\G$. The resulting set is isomorphic to $\pi_1\left(X\right)$ (by the work in those notes, etc ... need to fill in the details here).



\end{document}
