Checkbox array in WordPress widget?

I’m trying to create a widget that enables user to select which categories will be displayed. The following is piece of codes I created but the changes of the checkbox state can’t be saved. There are only two values : the title and list of selected categories.

function form($instance) {
    $instance = (array)$instance;
    if( empty($instance['title']) ) $instance['title'] = 'Category';
    $selected_categories = (array)$instance['category'];
    var_dump($selected_categories);
    ....
    $categories = get_categories( array('exclude'=> 1) );
    foreach($categories as $category) : ?>
    <div>
        <input type="checkbox" name="<?php echo $this->get_field_name('category'); ?>"
            value="<?php echo $category->cat_ID; ?>"
            <?php echo $category->cat_name; ?>
    </div>
    <?php endforeach; ?>
}

function update($new_instance, $old_instance) {
    $instance = $old_instance;
    $instance['title'] = strip_tags($new_instance['title']);
    $instance['category'] = $instance_new['category'];

    return $instance;
}

I observe the changes through var_dump($selected_categories). The value is always array(size=0) ignoring how many checkbox I checked.

I have no idea how to passing array in $instance variable.

Thanks in advance.