[Vuejs]-Laravel vue Column not found: 1054 Unknown

Its because your categories array is a two dimensional array. Now when your syncing that data your code looks like this.

$product->categories()->sync([0] => ['0' => ''], false);

that is why the error says, Column not found: 1054 Unknown column '0' in 'field list'.

Now, to solve your problem, first your categories value must like this

$categories = [2, 1]; // like this one
$categories = [[2], [1]]; // not this one

You may re structure first your categories array before you sync, by doing this

$categories = [];
foreach ($request->categories$key => $category) {
       $categories [] = $category[0];
}

$product->categories()->sync($categories, false);