Translate direct SQL query into collection model usage

Question

I've got this query:

SELECT * FROM 'catalog_category_product' WHERE 'product_id' IN ('.$IDs.')

where $IDs I get the following way: $IDs = implode(',', $products); and $products is an array of product ids.

Now I want to do the same thing, but through the use of a model collection. I've got so far with these examples:

$collection = Mage::getModel('catalog/product')->getCollection();
        $collection->addAttributeToSelect('product_id');
        $collection->addAttributeToSelect('category_id');
        $collection->addAttributeToSelect('position');
        $collection->addFieldToFilter(array('attribute'=>'product_id','in'=>array($products)));

and

$collection = Mage::getModel('catalog/product')->getCollection();
        $collection->addAttributeToSelect('product_id');
        $collection->addAttributeToSelect('category_id');
        $collection->addAttributeToSelect('position');
        $collection->addFieldToFilter(array(
                array('attribute'=>'product_id','in'=>array($products)),
        ));

But I get exception on the last function call, in both cases.

P.S. I need to acquire these 3 attributes at once (as the SQL query would do), but not iterating for each value in $products because I am dealing with thousands of products.