  This article concerns Ehresmann structures in the partition monoid $P_X$.
Since $P_X$ contains the symmetric and dual symmetric inverse monoids on the
same base set $X$, it naturally contains the semilattices of idempotents of
both submonoids. We show that one of these semilattices leads to an Ehresmann
structure on $P_X$ while the other does not. We explore some consequences of
this (structural/combinatorial and representation theoretic), and in particular
characterise the largest left-, right- and two-sided restriction submonoids.
The new results are contrasted with known results concerning relation monoids,
and a number of interesting dualities arise, stemming from the traditional
philosophies of inverse semigroups as models of partial symmetries (Vagner and
Preston) or block symmetries (FitzGerald and Leech): "surjections between
subsets" for relations become "injections between quotients" for partitions. We
also consider some related diagram monoids, including rook partition monoids,
and state several open problems.
