...I want a camera that I can ALWAYS carry around in my purse and one can't really do that with a 60D. Also, I will be buying the Fuji X-ex1 for $800 with a lense from a family member. So does this change anyones thoughts on Fuji X-EX1 vs Canon SL-1?
I borrowed a NEX-6 and used it on vacation. The EVF is very good, video is decent, low light capability is good, panorama and built-in HDR good, an the 16-50mm kit lens is extremely compact. It is one of the few APS-C sensor cameras with a zoom lens that can stored in a purse or large pocket.
That said I didn't like it. The UI is extremely menu oriented, with the Setup menu having so many pages the scroll bar is just a tiny sliver. While I got pretty good results, it felt like a gadget not a camera. A number of NEX-6's have a "camera error" flaw which locks up the camera if the electronic front curtain shutter is used. It can only be fixed by returning it for service.
I haven't used a X-E1 but have used a friend's X20, which has a roughly similar control layout. The Fuji felt like a camera and was pleasant to use.
The SL1 is very small for a DSLR and is a good alternative to enthusiast mirrorless cameras. It is much less expensive than the NEX-6 but will definitely not fit in a pocket or purse with the kit lens.
You will probably be very happy with the X-E1. Another pocketable alternative is the Sony RX100 II. Despite any possible UI issues, it is very small, has a 1" sensor, and I've seen many great shots from them. However it does not have a built-in EVF or OVF.