So, I've actually wanted to do this for some time now and this was a great occasion...

Grab the new router.php file I've uploaded here: jmp.sh/vG1WXii

Replace the current one and then do the following:
- Make sure there is at least 1 menu item pointing to all K2 categories. This could be your "products" menu or even the root menu item (home) of your site. Doing so will avoid duplicate URLs in Joomla as Joomla will know to "map" to K2 the new URL structure.
- Edit K2's Settings and change to the SEO tab, then scroll down to "Custom SEF URLs".
- In there, make sure that:
a) "Enable custom SEF settings for K2 URLs" is set to "Yes"
b) "Prefix for category URLs" is set to the string "empty" (this is important so we don't have to change the XML files of the component to unset the default values there for empty fields)
c) "Prefix for item URLs" is set to "Use the item's category"
d) "Use the item ID in the URL" is set to "No"
e) "Use the category ID in the URL" is set to "No"

Your settings there should look like this screenshot overall: jmp.sh/RoCBjqK

Clean your caches and check your site.

URLs for both categories and items will now be in the format:

I've now switched the demo site to the new router.

Here's an example URL on 3-level nested categories assigned to the "home" menu item: demo.getk2.org/magazine/design/architecture/morbi-adipiscing-sapien-non-tellus

If tests prove the new router to work as expected, I will adapt the default "Custom SEF" settings so you can get that layout just by enabling the top option and without having to configure the rest of the options (cause, frankly, few people would want more control over that).

Thanks for the feedback.

1. I can only assume that you have a menu item with the alias "prodotti" pointing to a K2 (parent) category with the alias "prodotti" as well. Is that correct?

2. I think I know why this occurs. I'll update the code and get back to you.

I'm also planning on adding some sort of caching to the router for increased performance.

Or you can edit the menu item and point it to link *directly* to the children of the K2 category "Prodotti" (and not to the parent category). Try that too and let me know.

What basically happens is that your menu structure for the Category "Prodotti" starts AFTER the menu item which also happens to be named "Prodotti". But, yes, with some tweaking you can get nice URLs for sure. Unfortunately, the requirement for at least one K2 menu item pointing to at least all categories in K2 is required to get these URLs, otherwise Joomla won't know where to route requests.

I'll keep you posted on the hyphen issue.

Fixed the hyphen issue. Get the updated file from: jmp.sh/pCV9gLc

And let me know if you spot anything else.