Geometric partition categories: On short Brauer algebras and their blob subalgebras

The main result here gives an algebra(/linear category) isomorphism between a geometrically defined subcategory $J_0^1$ of a short Brauer category $J_0$ and a certain one-parameter specialisation of the blob category $b$. That is, we prove the Conjecture in Remark 6.7 of [14]. We also define a sequence of generalisations $J_{i-1}^i$ of the category $J_0^1$. The connection of $J_0$ with the blob category inspires a search for connections also with its beautiful representation theory. Here we obtain formulae determining the non-semisimplicity condition (generalising the classical `root-of-unity’ condition).