Are Schur functors and categorification somehow related ?


If yes, probably looking on Schur functors (which I know) one can illustrate on this example why "categorification" (which I do not know) is so important/popular now ? 
(I am intersted to learn somehting about "categorification", but I would prefer to have
some "good entering point", meaning to relate it to what I know).


Schur functors - are some functors from category of vector spaces to itself.
<http://en.wikipedia.org/wiki/Schur_functor>
For example take vector space and send it to $S^n(V)$ n-th symmetric power (can be antisymmetric).


"Categorification" - briefly looking at MO discussions about it and abstracts of some papers, I got an impression that it is about realizing certain algebras as functors of some categories.
For example abstact of Khovanov's lectures <http://arxiv.org/abs/1008.5084> contains the
following sentence " diagrammatic categorification of positive halves of quantum groups".
 Is my understanding correct ?


PS


[What precisely Is "Categorification"?](https://mathoverflow.net/questions/4841/what-precisely-is-categorification)


[Algebraic relations between Schur functors](https://mathoverflow.net/questions/89001/algebraic-relations-between-schur-functors)



Categorification and Schur functors 
Categorification can be thought of as the process of replacing equalities with isomorphisms (in some category). A basic example is replacing a numerical combinatorial identity such as
$$2^n = \sum\_{k=0}^n {n \choose k}$$


with a bijection between two sets (an isomorphism in $\text{Set}$); in this case, between the set of subsets of $\{ 1, 2, ... n \}$ and the disjoint union of the sets of subsets of size $k$ over all $k$. 


As another example, the category $\text{FinSet}$ of finite sets itself categorifies the rng $\mathbb{N}\_{\ge 0}$ of non-negative integers; the coproduct categorifies addition and the product categorifies multiplication. (Also taking Hom-sets categorifies exponentiation.) In this way we replace equalities such as $1 + 1 = 2$ with isomorphisms $1 \sqcup 1 \cong 2$ (where $1$ is the set with one element, $2$ is the set with two elements, and $\sqcup$ is the coproduct). 


Schur functors categorify the theory of symmetric functions (specifically of [Schur functions](http://en.wikipedia.org/wiki/Schur_polynomial)). Taking the direct sum of Schur functors categorifies addition of symmetric functions, taking the tensor product categorifies multiplication of symmetric functions, and composing categorifies [plethysm](http://ncatlab.org/nlab/show/plethysm). So here we replace equalities between expressions involving symmetric functions with natural isomorphisms between expressions involving Schur functors. 




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As far as the linked arXiv paper goes, the idea is to replace a ring $A$ by a category $C$ with direct sums and tensor products such that the latter distributes over the former, such as the category of $(R, R)$-bimodules for some ring $R$. Equality of elements in $A$ is then replaced by isomorphism between objects in $C$. Taking the Grothendieck group of $C$ (a form of decategorification) should then recover the original ring.


One reason to attempt to do this is that the category $C$ may have a distinguished set of objects that provides a distinguished set of generators for $A$. For example, the Hecke algebra $H\_n$ of $S\_n$ turns out to be the Grothendieck group of the category of [Soergel bimodules](http://sbseminar.wordpress.com/2007/07/23/soergel-bimodules/). Among these we can distinguish the indecomposable modules, and these turn out to give the Kazhdan-Lusztig basis of $H\_n$. 


