Let $\Gamma$ be the category of finite pointed sets. The abelian category $\mathrm{Mod-}\Gamma$ is the category of functors $\Gamma^{\mathrm{op}} \to \mathrm{Vect}\_k$, where $k$ is a field (see [Pirashvili's paper](http://www.numdam.org/item?id=ASENS_2000_4_33_2_151_0)).


* Is every object of $\mathrm{Mod-}\Gamma$ the direct sum of simple objects? (If necessary, add finiteness assumptions to the modules.)
* If yes: Is every simple object a direct summand of the "regular representation" $\displaystyle\bigoplus\_{n \geq 0} K\bigl[\mathrm{Hom}(-,[n])\bigr]$?


I am aware that Peter Webb and others have studied [representation theory of categories](http://www.math.umn.edu/~webb/Publications/CategoryAlgebras.pdf), but I couldn't locate such a result so far.


Actually I am not so much interested in $\Gamma$, but rather the category of finite sets (i.e. without base points), but I thought it would be wise to ask the question for $\Gamma$ first since this category seems to appear more frequently in the literature, especially in algebraic topology.


I highly appreciate references to the literature, because I am not really not sure where to look for more properties of these module categories. Google almost only gives me results on $(\phi,\Gamma)$-modules in algebraic number theory.



Decomposition of $\Gamma$-modules into simple objects
Neither Finite sets or pointed finite sets have this property. If they did, their endomorphism monoids at each object would have it being essentially corners in the category algebra (one has to be a little careful here since your categories have infinitely many objects but it is ok). These monoid are well known not to be semisimple. See my forthcoming book on monoid representation theory or see Chapter 5 of Clifford and Preston's semigroup book. 


For the monoid of all self maps on an n-element set the natural n-dimensional module is indecomposable but not simple. The augmentation is the unique proper submodule. 


