The basics of creating an internal wordpress ajax request.

wordpress-ajax.php

<?php
// Inside functions.php
function get_category_posts(){

    $nonce = $_POST['nonce'];

    // check to see if the submitted nonce matches with the
    // generated nonce we created earlier
    if ( ! wp_verify_nonce( $nonce, 'my-nonce' ) )
        die ( 'Requested resource is not accessible.' );

    $category_id = $_REQUEST['categoryId'];

    // args
    $args = array(
        'posts_per_page' => -1,
        'post_type' => 'hozer',
        'post_status' => 'publish',
        'cat' => $category_id
    );

    $return_pages = array();

    $the_query = new WP_Query( $args );
    $posts = $the_query->posts;

    if ( $posts ) {

        foreach ( $posts as $page ) {

        $id = $page->ID;

          $return_pages[] = array(
              'title' => get_the_title( $id )
          );

        }

        // Reset wordpress loop
        wp_reset_query();

    }

    header( "Content-Type: application/json" );
    echo json_encode( $return_pages );

    exit();

}

add_action("wp_ajax_get-category-posts", "get_category_posts");
add_action("wp_ajax_nopriv_get-category-posts", "get_category_posts");

// When adding a script via wordpress add the nonce as well
// See https://gist.github.com/hitautodestruct/8912648#file-add-scripts-php-L31
// For example of how to add nonce and js scripts

?>

<script>
var sendParams = {
  action: 'get-category-posts',
  nonce: mainScriptParams.Nonce
};

var getPosters = function( options ) {

html = [];

$.post( '/wp-admin/admin-ajax.php', params )
  .done(function( data ){

      data.forEach(function(item) {

          html.push( '<li>'+ item.title +'</li>' );

      });

      $('#resultWrapper').html( html.join('') );

      showLoading(false);

  })
  .fail(function( err ){ showLoading(false); });
</script>