Tensor Algebra

The $n$-th tensor algebra of an $(R,R)$ Bimodule $M$ is defined as the $n$th-iterated Tensor Product of Bimodules

• $T^{n}(M)=\bigotimes_{k=0}^{n}M$

When Ring Bimodules are viewed as Profunctors (or, more accurately, Discrete Two-Sided Fibrations), then the tensor algebra acts a bit like a Reflexive-Transitive Closure: the elements of $T^{n}(M)$ are essentially $n$-chains of composable vectors.