Unique (multiple columns) and null in one column

This works as defined by the SQL standard. NULL means unknown. If you have two records of par_cat = NULL and name = 'X', then the two NULLs are not regarded to hold the same value. Thus they don't violate the unique key constraint. (Well, one could argue that the NULLs still might mean the same value, but applying this rule would make working with unique indexes and nullable fields almost impossible, for NULL could as well mean 1, 2 or whatever other value. So they did well to define it such as they did in my opinion.)

As MySQL does not support functional indexes where you could have an index on ISNULL(par_cat,-1), name, your only option is to make par_cat a NOT NULL column with 0 or -1 or whatever for "no parent", if you want your constraints to work.

I see that this was asked in 2014. However it is often requested from MySQL: https://bugs.mysql.com/bug.php?id=8173 and https://bugs.mysql.com/bug.php?id=17825 for example. People can click on affects me to try and get attention from MySQL.

Since MySQL 5.7 we can now use the following workaround:

ALTER TABLE categories 
ADD generated_par_cat INT UNSIGNED AS (ifNull(par_cat, 0)) NOT NULL,
ADD UNIQUE INDEX categories_name_generated_par_cat (name, generated_par_cat), 
ADD UNIQUE INDEX categories_url_generated_par_cat (url, generated_par_cat); 

The generated_par_cat is a virtual generated column, so it has no storage space. When a user inserts (or updates) then the unique indexes cause the value of generated_par_cat to be generated on the fly which is a very quick operation.

Just in case you come from Laravel...

This is Laravel's Migration version for Virtual Column to workaround the UNIQUE issue when one of the columns is NULL in value

$table->integer('generated_par_cat')->virtualAs('ifNull(par_cat, 0)');

$table->unique(['name', 'generated_par_cat'], 'name_par_cat_unique');