Combinatorics and algebra of tensor calculus

TL;DR

This paper explores the combinatorial and algebraic structures underlying tensor calculus in strict tensor categories, introducing a monad that captures these properties through graph coarse-graining.

Contribution

It reveals the combinatorial nature of tensor calculus and introduces a monad based on graph coarse-graining to characterize its algebraic structure.

Findings

Identification of a monad describing tensor calculus

Connection between operads, PROPs, and tensor calculus

Framework for algebraic and combinatorial analysis of tensor categories

Abstract

In this paper, motivated by the theory of operads and PROPs we reveal the combinatorial nature of tensor calculus for strict tensor categories and show that there exists a monad which is described by the coarse-graining of graphs and characterizes the algebraic nature of tensor calculus.