Faithful linear and relational representations of diagram categories and monoids

We study representations of diagram categories by binary relations and matrices over rings and semirings. Our main result is a faithful involutive tensor representation of the partition category $P$ (and consequently of each partition monoid $P_n$) by zero-one matrices over an arbitrary (additively) idempotent semiring. The dimensions of the matrices involved are powers of $2$, and we show that these are minimal with respect to faithful involutive tensor representations by matrices over any semiring. Intriguingly, these matrices encode the number of floating components formed when composing partitions, and can therefore be used to construct faithful representations of ($d$-)twisted partition categories $P^\Phi$ and $P^{\Phi,d}$ (and the respective twisted partition monoids $P_n^\Phi$ and $P_n^{\Phi, d}$) over rings of appropriate characteristic. We also give lower-dimensional involutive representations of the Brauer and Temperley--Lieb categories $B$ and $TL$. In the case of $TL$, the dimensions are given by Fibonacci numbers.