looks like $$\xymatrix@!0{ &(1,0)\ar[dr] && (1,1)\ar[dr] && (1,2)\ar[dr] && (1,3)\ar[dr] && (1,4)\ar[dr] && \ldots \\
             \ldots \ar[ur]\ar[dr] && (2,0)\ar[ur]\ar[dr] && (2,1)\ar[ur]\ar[dr] && (2,2)\ar[ur]\ar[dr] && (2,3)\ar[ur]\ar[dr] && (2,4)\ar[ur]\ar[dr] & \\
              &(3,-1)\ar[ur]\ar[dr] && (3,0)\ar[ur]\ar[dr] && (3,1)\ar[ur]\ar[dr] && (3,2)\ar[ur]\ar[dr] && (3,3)\ar[ur]\ar[dr]  && \ldots &&&&\\
             \ldots \ar[ur] && (4,-1)\ar[ur] && (4,0)\ar[ur] && (4,1)\ar[ur] && (4,2)\ar[ur] && (4,3)\ar[ur]  & \\
   }$$

If $Q = \xymatrix{ 1\ar@<.3ex>[r] \ar@<-.3ex>[r]  & 2}$ is the Kronecker quiver, then ${\mathbb{Z}}Q$ looks like $$\xymatrix@!0{& (1,0)\ar@<.3ex>[dr] \ar@<-.3ex>[dr] && (1,1)\ar@<.3ex>[dr] \ar@<-.3ex>[dr] && (1,2)\ar@<.3ex>[dr] \ar@<-.3ex>[dr] && (1,3)\ar@<.3ex>[dr] \ar@<-.3ex>[dr] && \ldots \\
             \ldots \ar@<.3ex>[ur] \ar@<-.3ex>[ur] && (2,0) \ar@<.3ex>[ur] \ar@<-.3ex>[ur] && (2,1) \ar@<.3ex>[ur] \ar@<-.3ex>[ur] && (2,2)\ar@<.3ex>[ur] \ar@<-.3ex>[ur] && (2,3)\ar@<.3ex>[ur] \ar@<-.3ex>[ur] 
   }$$

Define the *mesh category* $k({\mathbb{Z}}Q)$ to be the category whose objects are the vertices of ${\mathbb{Z}}Q$ and whose morphisms are $k$-linear combinations of paths in ${\mathbb{Z}}Q$, modulo the *mesh relations*: whenever we have $$\xymatrix@!0{ & \bullet \ar[ddr]^{b_1} & \\
            & \bullet \ar[dr] & \\
            (i,n) \ar[uur]^{a_1}\ar[ur]\ar[dr]_{a_r} & \vdots & (i,n+1) \\
            & \bullet \ar[ur]_{b_r} & \\
 }$$ in ${\mathbb{Z}}Q$, where the $a_j$ are all arrows leaving $(i,n)$ and the $b_j$ are all arrows arriving in $(i,n+1)$, then $\sum_{j=1}^{r} b_ja_j = 0$.

For any category ${\mathcal{C}}$, let $ind({\mathcal{C}})$ be the full subcategory of indecomposable objects of 