Composer and Cake lib folder

I have a standard Cake project setup on Github. The project root for has the app, lib folders in addition to all the other files like .gitignore. I would like to remove the lib folder from my repo and control which Cake version I am using with a dependency manager. Is this possible with composer?

For example, I could just change the composer json file from Cake 2.4 to 2.6, and the repo will not have to track the lib folder. Then when I want to put the project online, I can just have composer download the lib folder.

Is this possible? What would that composer file look like? I tried adding "cakephp/cakephp": "~2.6" to the composer file but it gives and error saying the package could not be found.

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