As the size of a web page is very important for the loading speed (specially on mobile), there are already lightweight alternatives to these full-featured frameworks (Zepto is the most popular one). Minified.js is a very strong alternative to these frameworks and it comes with an impressive size, less than 4kb minified.
It covers the most frequently used features: DOM manipulation, animation, events, cookies and HTTP requests. The reason of this small size is not due to an extraordinary feature or codebase but mostly because Minified.js doesn’t work that hard to hide browser differences like other frameworks do (which can be acceptable for many projects).
A very good documentation exists and Minified.js support IE6 too.