Why would you want users to search for a category and then let them search for a specific item within this category?
If I search for "TV" I want to browse the TVs, compare them, so I can make an informed decision. If I want a 40inch LED TV I would change my search query accordingly or use filters to select 40inch and LED.
So I don't really see the use for a global search function and a local search function.
The Amazon search bar with a select box for category is useful if you want to search for (for example) the Percy Jackson The Lightning Thief book. Entering 'Percy Jackson and the lightning thief' will result in DVDs and books. Selecting 'books' as a category will refine the scope for the search.
If you have a long list of categories, too long to show in a select box, I can perhaps see the use of a second autocomplete input field that allows user to type in 'bo' so they can easily select the category 'books'.
This question reminds me of a daring concept I once suggested to my boss when I worked at an ecommerce company. I suggested to remove all navigation and just show people a search bar. Every search query they made would remain inside the search bar so people could easily refine on their search query or remove the previous search query.
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Perhaps it can be of inspiration to you.