Embedding tensors and their associated tensor hierarchies provide a nice and efficient way to the construction of supergravity theories and further on to higher gauge theories. A sharp and beautiful observation by Kotov and Strobl demonstrates the mathematical nature behind the various calculations from embedding tensors to their associated tensor hierarchies in the physics literature: An embedding tensor gives arise to a Leibniz algebra, which further gives arise to an L∞ algebra. Both procedures are functorial. In fact we conjecture that the second one is even functorial in an ∞-category sense. Namely, KS functor should be stable with respect to homotopy.
We will provide in this talk a rich math tool box for embedding tensors, which seem not yet existing in mathematics literature since as a sort of algebra (or operad), embedding tensors involve not only binary but also einary operations. We develop the theory of controlling algebra, thus further the theory of cohomology and homotopy of embedding tensors and their Lie-Leibniz triples.