system
Given the following post, respond as if you're commenting on Reddit:
human
Is there an industrial or structural reason that any recyclable plastic bottle most always has a non-recyclable plastic cap?

I have looked pretty far and wide for an answer. The only reason I could think of would be that bottle makers want a harder plastic cap, either for the threading or to prevent it breaking, but if high density polyethylene is strong enough for the container, why isn't it strong enough for the cap?

Can you not dye HDPE?

Is polypropaline just more durable?
gpt
Both polyethylene and polypropylene are thermoplastic polymers. That is, they can be heated up and reformed, thus 100% recyclable. This is in comparison to thermosetting polymers like Bakelite, which cannot be heated up and reformed, as they will break down when heated.

The main reasons that you are told not to recycle the polypropylene cap:

Polyethylene and polypropylene have different melting temperatures. They need to be processed separately. 

Also, the bottles are crushed by the recycling facility before shipment (if they're sent out to melt at another facility) to save space. Leaving the cap on would cause problems with injuries and the crushing process. Polypropylene is harder, and would likely shatter instead of compressing nicely like the bottles.

Finally, the human element is the last line of defense against making sure extraneous materials aren't traveling with the polyethylene bottles down the line towards the melt. Workers would be a lot more likely to miss the small caps, which would then remain solid as the bottles are melted. 

Edit: Forgot to answer why the cap is polypropylene instead of polyethylene. One, polypropylene is cheaper. Two, polypropylene is one of the next best things to Teflon for "slipperyness," so the cap comes off a lot easier than it would compared to PE on PE.