I'm having trouble understanding how the following are exactly related to one another:
$\bullet$ The Temperley-Lieb Category TL
$\bullet$ The idempotent completion (Karoubi Envelope) of TL, Kar(TL)
$\bullet$ The TL$_{n}$ algebra.
I'm attempting to use the Schur-Weyl duality to show that the functor $F:Kar(TL)\to Rep \mathfrak{sl}_{2}\mathbb{C}$ is full. Although, I think I can see how we could possibly use the TL$_{n}$ algebra to do this, I can't see how to do this for Kar(TL). In fact, I don't think I understand Kar(TL) at all.
If anyone could provide helpful links, suggestions or answers, anything would be appreciated, Thanks!