Please is it possible to make web site like this one with k2 catalog.
www.specialized.net/Specialized/

Main problem is to connect categories between each other. For example main categories will be like on this page in top menu. But when you chose one category from main menu catalog will list all items from this big main category but on the left side (column or div) of web page must be listed connected categories (like brands of products). When you click on one brand the catalog must filer only items from this brand.

You need to enable the catalog on the parent category and enable the subcategories blocks to show. Keep in mind that for this layout you need to assign items only to the subcategories and have only one level of subcategories.

I am not sure that you understand me. I need to assign one product to more categories. So I can combine them.
In top menu will be the main categories. For example drill machines, and when visitor click on drill machines k2 catalog will list all drill machines in database. But in left column then must be another filter with manufacturers of drill machines. And then if visitor want can click on one manufacturer on the left side and k2 must filter only drill machines of that manufacturer.
You can see on web page which I found as example.
www.specialized.net/Specialized/fiber-optic-cable-slitter-fiber-optic-cable-sheath-cutter-6753.aspx

I would recommend using Additional Categories for K2. It's a free extension (I personally made a donation to show support)
extensions.joomla.org/profile/extension/extension-specific/k2-extensions/additional-categories-for-k2

With the extension, you can assign each item to multiple categories.
So in your case you can have a main menu and then secondary menu in which an item can be assigned to both.

Main Menu:
- Home
- Drill Machines
- About
- Contact

Manufacturers (Sidebar Menu):
- Duradrill
- Drillmaster
- Toughneck
- Odyssey Press

Note: the extension only allows you to select additional categories that the item will be listed in, when you click on the item from any category, it links to the item (which resides in the main category, and the main category only - this is ideal for not creating duplicate content)

So for this example the item (Heavy Duty Drillmaster 3000) lives in the main category "Drill Machines"
domain.com/drill-machines/heavy-duty-drillmaster-3000

But with the use of the extension, it can be displayed in main category, as well as:
- products
- drillmaster
- etc.

I hope this helps.

I have now installed this plugin and it works fine. However, I use a module to list all items in a certain category, and this module does not looks in additional categories.
So this plugin makes another table to store the additional categories and does not use the original table from K2.
And that's not good :-(

I think you can use extra fields to assign brands (and other searchable fields) to your products.

Then you can use JA K2 filter available from Joomlart

I think you need to pay for it, but its a very good/powerful extension in my opinion!

There is an example on the Joomlart site but I cannot add the link due to spam protection!

So within a category you can filter the results with this.

If you need to show all products (from multiple categories) in a brand, then perhaps using the tagging system could work (menu item to show specific tag).

If you want to add e-commerce capabilities to your site, why not try K2Store?

I resolve your problem by modifying the following modules file / mod_k2_content / helper.php
in line 96
comment $query = "SELECT i.*,";
add $query = "SELECT DISTINCT i.*,";
in line 112
add $query .=" LEFT JOIN #__k2_additional_categories AS ca ON ca.`itemID` = i.`id` ";
in line 150
add $query .= " OR ca.catid IN ({$sql})";
in line 150:
add $query .= " OR ca.catid IN(".implode(',', $cid).")";
in line 172
add $query .= " OR ca.catid IN ({$sql})";

hello dear friends
I have used your solution but on our site, some limitation does not work. For example in mod_k2_content i have set show only feature item in a specific category, but the module shows all items in the selected category. what I have to do?
please help me to solve the problem

Hello,

Was it working prior to these edits?

Yes it was work correctly

Can you share the entire edited file?
You can use Github's gists.

Thank you, yes I can.
here you are
gist.github.com/alimirarab/26aefb548ef9c942dde000ecdf8708f5