PHP: database output some things twice logic

Here's where I think you're going wrong...

It looks like you have a database structure looking kind of like this:

objects
-------
object_id, name

categories
----------
category_id, name

materials
---------
material_id, name

category_objects
----------------
category_id, object_id

object_materials
----------------
material_id, object_id

In order to load all of the data you want to display, you're probably performing a single query like this:

$results = query(
    'SELECT * FROM objects o
    INNER JOIN object_materials om ON om.object_id = o.object_id
    INNER JOIN materials m ON om.material_id = m.material_id

    INNER JOIN category_objects co ON co.object_id = o.object_id
    INNER JOIN categories c ON c.category_id = co.category_id'
);

Not only will this give you duplicate results for materials, but it will also give you duplicate results when you start putting objects in multiple categories. Try breaking this up into multiple queries:

$objects = query('SELECT * FROM objects');

// Iterate over each object and load the category/materials for 
// each object individually
    $categories = query(
        'SELECT * FROM categories c
            INNER JOIN category_objects co ON co.category_id = c.category_id
        WHERE co.object_id = ?',
        $object['id']
    );

    $materials = query(
        'SELECT * FROM materials m
            INNER JOIN object_materials om ON om.material_id = m.material_id
        WHERE mo.object_id = ?',
        $object['id']
    );

Then you can individually iterate over each of those result sets and display them however you like. You may be able to get away with using a single query to do this, but it won't be at all clear to the guy coming in behind you later, or even to you 6 months from now when/if a bug is discovered.

Update

It's not exactly clear from your question how you're querying or what markup you're displaying this in, but here's some pseudo-code that should work for you if you want to keep going the way you are. I'm using a table element to display the results here, but you can modify it to suit your needs.

echo '<table>';
$currentObjectId = null;
$currentCategoryId = null;
$currentMaterialId = null;

// Iterate over your result set
// Each object here is a tuple representing joined entities from the 
// objects, categories and materials tables
foreach($objects as $object) { 
    echo '<tr>';

    if(!$currentObjectId || $currentObjectId != $object['id']) {
        echo '<td>' . $object['name'] . '</td>';
    } else {
        echo '<td></td>';
    }

    if(!$currentCategoryId || $currentCategoryId != $object['category_id']) {
        echo '<td>' . $object['category_name'] . '</td>';
    } else {
        echo '<td></td>';
    }

    if(!$currentMaterialId || $currentMaterialId != $object['material_id']) {
        echo '<td>' . $object['material_name'] . '</td>';
    } else {
        echo '<td></td>';
    }

    echo '</tr>';
}

echo '</table>';