Computads for weak $ω$-categories as an inductive type
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 generator-preserving 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.