foreach my $category (sort { ($list->{$_}->{cat_name} cmp $list->{$_}->{cat_name}) } keys %$list) {
  say $list->{$category}{cat_name};
}

##</code><code>##

Use of uninitialized value $_ in hash element at ./facebook.pl line 34.
Use of uninitialized value $_ in hash element at ./facebook.pl line 34.
Use of uninitialized value in string comparison (cmp) at ./facebook.pl line 34.
Use of uninitialized value in string comparison (cmp) at ./facebook.pl line 34.

##</code><code>##

$list = 
{
          '174972792548177' => {
                               'places' => [
                                             {
                                               '264651060353078' => {
                                                                    'name' => 'WHIP City Towing',
                                                                    'link' => 'https://www.facebook.com/WHIP-City-Towing-264651060353078/'
                                                                  }
                                             },
                                             {
                                               '177166582312074' => {
                                                                    'link' => 'https://www.facebook.com/Center-City-Service-177166582312074/',
                                                                    'name' => 'Center City Service'
                                                                  }
                                             },
                                             {
                                               '965119126910980' => {
                                                                    'link' => 'https://www.facebook.com/Michaels-Towing-965119126910980/',
                                                                    'name' => 'Michael\'s Towing'
                                                                  }
                                             }
                                           ],
                               'cat_name' => 'Towing Service'
                             },
          '199850580046875' => {
                               'cat_name' => 'Garage Door Services',
                               'places' => [
                                             {
                                               '233152426881482' => {
                                                                    'link' => 'https://www.facebook.com/413doors/',
                                                                    'name' => '413 Doors & Home Services'
                                                                  }
                                             },
                                             {
                                               '432129326916445' => {
                                                                    'link' => 'https://www.facebook.com/Jfcsbose/',
                                                                    'name' => 'Jfcsbose'
                                                                  }
                                             },
                                             {
                                               '471289059572856' => {
                                                                    'link' => 'https://www.facebook.com/Conte-Door-Service-Inc-471289059572856/',
                                                                    'name' => 'Conte Door Service, Inc.'
                                                                  }
                                             }
                                           ]
                             },
          '127806747288007' => {
                               'places' => [
                                             {
                                               '152911074751051' => {
                                                                    'link' => 'https://www.facebook.com/pages/Pioneer-Valley-Rider-Training/152911074751051',
                                                                    'name' => 'Pioneer Valley Rider Training'
                                                                  }
                                             }
                                           ],
                               'cat_name' => 'Traffic School'
                             },

};