Error reporting is turned on. Getting nothing in the back office when I try to delete the two categories. It's turned on if you want to play with it. I suppose we had better find out what is happening.
Thanks for your question on category deletion. I checked your site and see "Dog Sweaters" and "Dog Leashes". Are these the same categories you mentioned or were you able to get the other two deleted from your store?
Yes I can see them and they are indeed not deleting via normal means. I will need to see what it takes to remove them via the database itself.
I won't mess with it, but rather find out what you may need to delete should you decide to do so. Have you enabled the error reporting to see if it is throwing an error code? If so, I may be able to assist once we know what the error is.
The reason that those items aren't deleting is because there are products associated with the categories you're deleting. When you try to delete from the list at the bottom, you may not see that PrestaShop is asking you a question at the top. Check out the options and either use them to make the deletion, or first find all the products that are in the categories you're trying to delete, and remove them from the categories FIRST.
I hope that makes sense! If you have any further questions or comments, please let us know.