Making a category a tab navigation bar

Hi everyone. I need help on something.
I want to make a navigation bar(new page tab) in the header, that coincide to a certain category. Is it posible?
And how can I make posts in a new page as if I make posts for the homepage. Let me put it this way: I want a new page (tab) that acts (has the same atributes) exactly like the homepage.

Many thanks