What hose is this supposed to connect to? Emissions?

Got a new battery and it started great then died. Started it up again and very rough idle. I turned it off, plugged the hose (blue tape) with a plastic wine cork and it ran better and stayed running but idle is still bad. Moved it up and down the driveway but it kept wanting to stall... Ideas?

Previous owner stated he thought the TPS was bad.. cat was cut off...or stolen... before I got it

 

I believe that hose with the blue thing on it is the split air hose to the cat from the acv. A hard line which has that rubber hose is bolted onto the catalytic converter and gets air from the air control valve.

 

I believe this hose will get connected to the third cat..(CA car) . . . There will be a check valve in-between that only allows flow to go back to the engine.. . .. This is an emission related item I believe. ..

 

So, safe to leave plugged? (Obviously not with a wine cork). It's exempt from emissions and I wasn't planning on putting the cat back on anyways.