  There are many category-theoretic notions of algebraic theory, including
Lawvere theories, monads, PROPs and operads. The first central notion of this
thesis is a common generalisation of these, which we call a proto-theory. In
order to define models of a proto-theory in a category, we need a way of
relating the arities of the proto-theory with the objects of the category. This
leads to our second central notion, that of an interpretation of arities, or
aritation for short. We show that every aritation gives rise to a semantics
functor sending proto-theories to models. In fact this functor always has an
adjoint, giving a structure-semantics adjunction. Furthermore, we show that the
semantics of proto-theories generalises the classical semantics of many
existing notions of algebraic theory.
  Another aim of this thesis is to find a convenient category of monads in the
following sense. Every right adjoint gives rise to a monad on its codomain, and
more generally so does any functor that admits a codensity monad. However, not
all functors have codensity monads. This means that the semantics functor for
monads on a category, viewed as a functor into the category of all functors
into the base category, does not have a left adjoint. We seek a generalisation
of monads with a semantics functor that does have a left adjoint.
  This can be accomplished with proto-theories, but at a cost. The classical
semantics functor for monads is full and faithful; this is a kind of
completeness theorem, and is highly desirable in a notion of algebraic theory.
However, the semantics functor for general proto-theories is not full and
faithful. Thus we seek a generalisation of monads and their semantics for which
the semantics functor is both full and faithful and has a left adjoint. We show
that such a generalisation is given, for suitable base categories, by a certain
kind of topological proto-theory.
