Cat back exhaust..
 

What is the difference between a cat back exhaust and a regular exhaust?

A cat back exhaust is one that is so loud that when you start up your car, any nearby cats(and dogs)jump back. Hence we have the name cat back. I think they were originaly invented to scare away cats that curl up and sleep on the hood where it's warm.
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A 'cat back exhaust system' is one which replaces the stock (OEM) exhaust system from the catalytic converter on back. As such, it doesn't modify the original emissions equipment, and is legal to install in all fifty states in the USA, including California, which has the toughest emissions laws in the country.The purpose of most 'cat back' systems is to increase power and/or gas mileage by providing a less restrictive exhaust system; as a result, most of these system are more noisy than their OEM counterparts. The idea is that if the engine can exhale more easily followingcombustion, e.g. with less backpressure, it can provide more power.
If the "regular" exhaust is also installed BEHIND the CAT there isn' any difference

So if the exhaust doesnt specify its Cat back, does that mean it replaces the stock cat converter? For example, the AutoExe Exhaust on the Mazdaparts.com page doesnt say catback, so does that mean it comes with a non-stock cat converter? thanks

The first photo is the Autoexe exhaust; the second is the Greddy “Cat Back”.

Both look the same to me. Why one is $400 more I don’t know. The only item on that page that has anything to do with the catalytic converter is the B&B Race pipe and it specificallysays so in the description. So when shopping for an exhaust system ‘Cat Back’ doesn’t mean much of anything.

Also, the term cat-back indicates you're buying more than just a muffler with exhaust tips that needs to get welded on... it's a complete kit, from the cat back.

A complete race exhaust would also come with a mid-pipe to replace the cat, and would run from the headers back... but that is not street legal, and on a rotary can be painfully loud.