This is my version:

path_no_dup () 
{ 
    local IFS=: p=();

    while read -r; do
        p+=("$REPLY");
    done < <(sort -u <(read -ra arr <<< "$1" && printf '%s\n' "${arr[@]}"));

    # Do whatever you like with "${p[*]}"
    echo "${p[*]}"
}

Or, a version which uses no external programs:

path_no_dup () 
{ 
    local o=() r= 
    declare -A s=()
    IFS=: read -a e <<< "$1"
    for i in "${e[@]}"; do
        [[ ! ${s[$i]} ]] && { s[$i]=1;o+=("$i"); }
    done

    printf -v r '%s:' "${o[@]}"

    # Do whatever you like with "${r%:}"
    echo "${r%:}"
}

Usage: path_no_dup "$PATH"

Sample output:

rany$ v='a:a:a:b:b:b:c:c:c:a:a:a:b:c:a'; path_no_dup "$v"
a:b:c
rany$

This is my version:

path_no_dup () 
{ 
    local IFS=: p=();

    while read -r; do
        p+=("$REPLY");
    done < <(sort -u <(read -ra arr <<< "$1" && printf '%s\n' "${arr[@]}"));

    # Do whatever you like with "${p[*]}"
    echo "${p[*]}"
}

Usage: path_no_dup "$PATH"

Sample output:

rany$ v='a:a:a:b:b:b:c:c:c:a:a:a:b:c:a'; path_no_dup "$v"
a:b:c
rany$

This is my version:

path_no_dup () 
{ 
    local IFS=: p=();

    while read -r; do
        p+=("$REPLY");
    done < <(sort -u <(read -ra arr <<< "$1" && printf '%s\n' "${arr[@]}"));

    # Do whatever you like with "${p[*]}"
    echo "${p[*]}"
}

Usage: path_no_dup "$PATH"

Sample output:

rany$ v='a:a:a:b:b:b:c:c:c:a:a:a:b:c:a'; path_no_dup "$v"
a:b:c
rany$

This is my version:

path_no_dup () 
{ 
    local IFS=: p=();

    while read -r; do
        p+=("$REPLY");
    done < <(sort -u <(read -ra arr <<< "$1" && printf '%s\n' "${arr[@]}"));

    # Do whatever you like with "${p[*]}"
    echo "${p[*]}"
}

Or, a version which uses no external programs:

path_no_dup () 
{ 
    local o=() r= 
    declare -A s=()
    IFS=: read -a e <<< "$1"
    for i in "${e[@]}"; do
        [[ ! ${s[$i]} ]] && { s[$i]=1;o+=("$i"); }
    done

    printf -v r '%s:' "${o[@]}"

    # Do whatever you like with "${r%:}"
    echo "${r%:}"
}

Usage: path_no_dup "$PATH"

Sample output:

rany$ v='a:a:a:b:b:b:c:c:c:a:a:a:b:c:a'; path_no_dup "$v"
a:b:c
rany$