  We introduce a notion of signature whose sorts form a direct category, and
study computads for such signatures. Algebras for such a signature are
presheaves with an interpretation of every function symbol of the signature,
and we describe how computads give rise to signatures. Generalising work of
Batanin, we show that computads with certain generator-preserving morphisms
form a presheaf category, and describe a forgetful functor from algebras to
computads. Algebras free on a computad turn out to be the cofibrant objects for
certain cofibrantly generated factorisation system, and the adjunction above
induces the universal cofibrant replacement, in the sense of Garner, for this
factorisation system. Finally, we conclude by explaining how many-sorted
structures, weak $\omega$-categories, and algebraic semi-simplicial Kan
complexes are algebras of such signatures, and we propose a notion of weak
multiple category.
