Temperley-Lieb Category 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!
 A: I hope this answer will help, even if right now has passed one year from your question.
The Temperley-Lieb category $TL$ can be viewed as a monoidal category whose objects are natural numbers ($n,m,\ldots$), and arrows certain diagrams as you can see here: https://arxiv.org/pdf/1502.06845.pdf. They satisfy some diagrammatic relations. Composition (resp. monoidal) structure is given by the isotopy class of vertical (resp. horizontal) concatenation of diagrams.
$TL_n$ is just the algebra of endomorphisms of $n$, i.e., $\operatorname{End}_{TL}(n)$. A diagrammatic representation of $TL_n$ contains diagrams only with the same number ($n$) of boundary vertices at the bottom and top. Then the category TL is a generalization of this kind of representation without the last constraint, satisfying the same relations. Allowing, for example, a "cup" diagram as a valid diagram, instead of the usual generator, which consist of two opposite-sided cups.
The idempotent completion $Kar(TL)$ gives you more objects than the original category. It is an additive monoidal category which contains formal direct summands of the objects $n$. It includes the images of idempotents in $\operatorname{End}_{TL}(n)$. These objects can be expressed as direct sums of the Jones-Wenzl projectors $JW_n\in \operatorname{End}_{TL}(n)$.
This is the direct relationship between those three different objects.
A functor defined from $TL$ over an additive Karoubian category induces a functor from Kar(TL). It is induced by the universal property of the Karoubi envelope since it is the smallest additive Karoubian category, i.e. a category where every idempotent split. RepSL2C is a Karoubian category, then you should not having trouble extending it.
