use 5.014;
use DateTime qw();

my %day2t = (
    Friday    => 'T',
    Saturday  => 'T-1',
    Sunday    => 'T-2',
    Monday    => 'T-3',
    Tuesday   => 'T-4',
    Wednesday => 'T-5',
    Thursday  => 'T-6',
);

{
    my $today = DateTime->now->truncate(to => 'day'); # 2013-11-25T00:00:00
    $today->day_name;   # Monday
    $day2t{$today->day_name};   # T-3
    my $into_the_past = ($day2t{$today->day_name} =~ s/T-?//r) || 0; # 3
    $today->clone->subtract(days => $into_the_past)->strftime('%F'); # 2013-11-22
}

{
    my $beginning_of_november = DateTime->new(year => 2013, month => 11, day => 1); # 2013-11-01T00:00:00
    $beginning_of_november->day_name; # Friday
    $day2t{$beginning_of_november->day_name};   # T
    my $into_the_past = ($day2t{$beginning_of_november->day_name} =~ s/T-?//r) || 0; # 0
    $beginning_of_november->clone->subtract(days => $into_the_past)->strftime('%F'); # 2013-11-01
}