  We define a simple kind of higher inductive type generalising dependent
$W$-types, which we refer to as $W$-types with reductions. Just as dependent
$W$-types can be characterised as initial algebras of certain endofunctors
(referred to as polynomial endofunctors), we will define our generalisation as
initial algebras of certain pointed endofunctors, which we will refer to as
pointed polynomial endofunctors.
  We will show that $W$-types with reductions exist in all $\Pi W$-pretoposes
that satisfy a weak choice axiom, known as weakly initial set of covers (WISC).
This includes all Grothendieck toposes and realizability toposes as long as
WISC holds in the background universe.
  We will show that a large class of $W$-types with reductions in internal
presheaf categories can be constructed without using WISC.
  We will show that $W$-types with reductions suffice to construct some
interesting examples of algebraic weak factorisation systems (awfs's).
Specifically, we will see how to construct awfs's that are cofibrantly
generated with respect to a codomain fibration, as defined in a previous paper
by the author.
