  The first part of this dissertation defines "dependently typed algebraic
theories", which are a strict subclass of the generalised algebraic theories
(GATs) of Cartmell. We characterise dependently typed algebraic theories as
finitary monads on certain presheaf categories, generalising a well-known
result due to Lawvere, B\'enabou and Linton for ordinary multisorted algebraic
theories. We use this to recognise dependently typed algebraic theories for a
number of classes of algebraic structures, such as small categories,
n-categories, strict and weak omega-categories, planar coloured operads and
opetopic sets. We then show that every locally finitely presentable category is
the category of models of some dependently typed algebraic theory. Thus, with
respect to their Set-models, these theories are just as expressive as GATs,
essentially algebraic theories and finite limit sketches. However, dependently
typed algebraic theories admit a good definition of homotopy-models in spaces,
via a left Bousfield localisation of a global model structure on simplicial
presheaves. Some cases, such as certain "idempotent opetopic theories", have a
rigidification theorem relating homotopy-models and (strict) simplicial models.
  The second part of this dissertation concerns localisations of presentable
$(\infty,1)$-categories. We give a definition of "pre-modulator", and show that
every accessible orthogonal factorisation system on a presentable
$(\infty,1)$-category can be generated from a pre-modulator by iterating a
plus-construction resembling that of sheafification. We give definitions of
"modulator" and "left-exact modulator", and prove that they correspond to those
factorisation systems that are modalities and left-exact modalities
respectively. Thus every left-exact localisation of an $\infty$-topos is
obtained by iterating the plus-construction associated to a left-exact
modulator.
