We give a new description of computads for weak globular ω-categories by giving an explicit inductive definition of the free words. This yields a new understanding of computads, and allows a new definition of ω-category that avoids the technology of globular operads. Our framework permits direct proofs of important results via structural induction, and we use this to give new proofs that every ω-category is equivalent to a free one, and that the category of computads with variable-to-variable maps is a presheaf topos, giving a direct description of the index category. We prove that our resulting definition of ω-category agrees with that of Batanin and Leinster and that the induced notion of cofibrant replacement for ω-categories coincides with that of Garner.
We introduce various notions of globular multicategory with homomorphism types. We develop a higher dimensional modules construction that constructs globular multicategories with strict homomorphism types. We illustrate how this construction is related to iterated enrichment. We show how various collections of “higher category-like” objects give rise to globular multicategories with homomorphism types. We show how these structures suggest a new globular approach to the semantics of (directed) homotopy type theory.