Parenthesised tensor words and their evaluation functors

Fix symbols $x_1,\dots ,x_n$. A parenthesised tensor word on these letters is formed recursively by:

• each $x_i$ is a word;
• the unit symbol $\mathbf{1}$ is a word;
• if $u$ and $v$ are words, then $(u\otimes v)$ is a word;

subject to the condition that, after deleting every $\mathbf{1}$ and every pair of parentheses, the letters $x_1,\dots ,x_n$ appear exactly once and in that order.

If $\mathcal{C}$ is a monoidal category (Monoidal category), each such word determines an evaluation functor $E_u:{\mathcal{C}}^n\to \mathcal{C}$ recursively:

• $E_{x_i}$ is the $i$th projection;
• $E_{\mathbf{1}}$ is the constant functor at the unit object;
• $E_{(u\otimes v)}$ is the composite of $(E_u,E_v):{\mathcal{C}}^n\to \mathcal{C}\times \mathcal{C}$ with the tensor bifunctor.

These evaluation functors are the only tensor expressions regarded as defined on this page before coherence is proved.