The Periplectic Brauer Algebra III: The Deligne Category

We construct a faithful categorical representation of an infinite Temperley-Lieb algebra on the periplectic analogue of Deligne's universal monoidal category. We use the corresponding combinatorics to classify thick tensor ideals in this periplectic Deligne category. This allows us to determine the objects in the kernel of the monoidal functor going to the module category of the periplectic Lie supergroup. We use this to classify indecomposable direct summands in the tensor powers of the natural representation, determine which are projective and determine their simple top.