String diagram

Definition

Let $(\mathcal{C}, \otimes, I)$ be a strict monoidal category. A string diagram in $\mathcal{C}$ is a picture in the strip $\R \times [0, 1]$ with the following parts.

1. Finitely many disjoint boxes, each labelled by a morphism $f \colon A_1 \otimes \cdots \otimes A_m \to B_1 \otimes \cdots \otimes B_n$ of $\mathcal{C}$.
2. Finitely many disjoint wires, each an arc on which the height coordinate is strictly increasing, each labelled by an object of $\mathcal{C}$.
3. Each wire starts at the bottom edge of the strip or at the top of a box. It ends at the top edge of the strip or at the bottom of a box.
4. The wires entering a box labelled $f$, read from left to right, carry $A_1, \ldots, A_m$, and the wires leaving it carry $B_1, \ldots, B_n$.

The value of the diagram is a morphism from the tensor product of the bottom labels to the tensor product of the top labels. To compute it, cut the strip by horizontal lines so that each slice contains at most one box. Each slice is then the tensor product of identities and one box. The value is the composite of the slices from bottom to top.