Automorphisms of pro-objects

Question

Let $C$ be a small category and let $Pro(C)$ denote its pro-completion. Is it then true that $Aut(X) \cong \varprojlim_{i \in I} Aut(X_i)$ for all objects $X := \{X_i\}_{i \in I} \in Pro(C)$ ? What about when $C$ is a Galois category ?

Answer 1

It is not the case. Take $X=\mathbb{Z} \cup \{\frac{+}{-}\infty\}$ a two point compactification of $\mathbb{Z}$.

You can write it as the limit $\varprojlim \limits_{n \in \mathbb{N}} \{-n ,\cdots ,n\}$, where the transition map from $\{-m, \cdots ,m\} $ to $\{-n,\cdots , n\}$ sends $k$ to $-n$ if $k<-n$, to $n$ if $k>n$ and to itself otherwise.

Take $f \in Aut(X)$ the translation by $1$. Then you cannot decompose it as a limit of automorphisms on $[-n,n]$, since $n-1$ would have to be sent to $n$ and the same goes for $n$, so resulting projection of $f$ would not be injective.

If we however assume that $Aut(X)$ is a profinite group for its open compact topology on $X$, then you can find some family $X_i$ such that $X \cong\varprojlim \limits_{i \in I} X_i$ and such that $Aut(X)\cong \varprojlim \limits_{i \in I} Aut(X_i)$, but it is not true for every choice of finite sets.