How to Call Category Collection on Home Page in Magento 2

I think it should help you http://www.magentocommerce.com/wiki/groups/248/display_products_on_home_page

So, result added string should be something like that

{{block type="catalog/product_new" category_id="ID OF SHOWING CATEGORY" name="home.catalog.product.new" template="catalog/product/new.phtml"}}

You can use the following code at the top :

$category = Mage::getSingleton('catalog/layer')->getCurrentCategory();
if(!$category->getChildren()){
 //$parentcat_id = $category->getParentCategory()->getId();
 //$category = Mage::getSingleton('catalog/category')->load($parentcat_id);
 $category = $category->getParentCategory();
}
$categories = $category->getCollection()
        ->addAttributeToSelect(array('name', 'thumbnail'))
        ->addAttributeToFilter('is_active', 1)
        ->addIdFilter($category->getChildren());