Below I describe an infinite (but locally finite) quiver with relations. My question is whether anyone recognizes it and can provide appropriate pointers to the literature. I'm mainly interested in computing its Hochschild homology. The path category of the quiver (with relations) is related to the Temperley-Lieb category at non-generic (non-semisimple) values of the parameter. (At least I think it is -- I haven't double-checked the calculations.)

(This is not my area of expertise, so I apologize in advance if I'm not using standard terminology.)

**The quiver**

- A vertex for each natural number $i=0, 1, 2, \ldots$.
- An arrow $u_i : i \to i+1$ for each $i\ge 0$.
- An arrow $d_i : i \to i-1$ for each $i \ge 1$.

**The relations**

- $u_i u_{i+1} = 0$ for all $i\ge 0$.
- $d_i d_{i-1} = 0$ for all $i \ge 2$.
- $u_i d_{i+1} = d_i u_{i-1}$ for all $i\ge 1$.
- $u_0 d_1 = 0$.

Here's a more geometric description of the quiver. Start with a 2-dimensional mesh: A vertex for each pair of integers $(j,i)$ such that $j+i$ is even; "up" arrows connecting $(j, i)$ to $(j+1, i+1)$; "down" arrows connecting $(j, i)$ to $(j+1, i-1)$. Impose a commutativity relation for the boundary of each square of the mesh. Impose another relation that the composition of two "up" arrows is zero, and similarly for two "down" arrows. Declare that objects $(j, i)$ with $j<0$ are zero (i.e. any path which factors through such an object (vertex) is zero. Finally, mod out by the horizontal translation $(j, i) \mapsto (j+2, i)$.