How to retrieve categories name or Id for avatar purposes

I have excerpts of my posts instead of complete posts at my homepage.. I want to add a avatar next to them indicating the proper category.

I created a folder with the pictures. The pictures are named as the categories ID.

I added this to my index.php

<img src=”http://site.com/wp-content/avatars/&lt;?php get_cat_name( $cat_id ) ?>.jpg”/>

I tried with “the_category” and get_the_category and nothing seems to work. How can I call the proper category to use it to call my images?