  We introduce the notion of pie algebra for a 2-monad, these bearing the same
relationship to the flexible and semiflexible algebras as pie limits do to
flexible and semiflexible ones. We see that in many cases, the pie algebras are
precisely those "free at the level of objects" in a suitable sense; so that,
for instance, a strict monoidal category is pie just when its underlying monoid
of objects is free. Pie algebras are contrasted with flexible and semiflexible
algebras via a series of characterisations of each class; particular attention
is paid to the case of pie, flexible and semiflexible weights, these being
characterised in terms of the behaviour of the corresponding weighted limit
functors.
