How to obtain a nested HTML list from object’s array recordset?

First of all map the objects onto a new hash (array) in which the index is the id:

// map the array onto hash
$hash = array();
foreach($array as $object)
{
    $hash[$object->id] = array('object' => $object);
}

Then transpose this flat hash into a tree-like structure, see this answer for another code example, it’s merely the same here:

// build tree from hash
$tree = array();
foreach($hash as $id => &$node)
{
    if ($parent = $node['object']->top_id)
        $hash[$parent]['children'][] =& $node;
    else
        $tree[] =& $node;
}
unset($node, $hash);

Finally you can output this tree-like structure as HTML. This can be done with either a stack or recursive. This is one variant with recursion:

// render tree
function render_tree($tree)
{
    echo '<ul>', "\n";
    foreach($tree as $node)
    {
        render_node($node);
    }
    echo '</ul>';
}

// render tree node
function render_node($node, $level = 0)
{
    $inset = str_repeat('    ', $level) . '  ';
    echo $inset, '<li>', $node['object']->name;
    if (isset($node['children']))
    {
        echo "\n", $inset, '  <ul>', "\n";
        foreach($node['children'] as $node)
        {
            render_node($node, $level+1);
        }
        echo $inset, '  </ul>', "\n", $inset;
    }
    echo '</li>', "\n";
}

// output
render_tree($tree);

Output:

<ul>
  <li>Cat 1</li>
  <li>Cat 2
    <ul>
      <li>Subcat 1</li>
      <li>Subcat 2
        <ul>
          <li>Subcat 4</li>
        </ul>
      </li>
    </ul>
  </li>
  <li>Cat 3
    <ul>
      <li>Subcat 3</li>
    </ul>
  </li>
</ul>

Full code Example + HTML Demo.