A depth-bounded naturality meta-operation in the Catt type theory constructs and machine-checks cylinder and cone composites in weak omega-categories.
Hom $\omega$-categories of a computad are free
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abstract
We provide a new description of the hom functor on weak $\omega$-categories, and we show that it admits a left adjoint that we call the suspension functor. We then show that the hom functor preserves the property of being free on a computad, in contrast to the hom functor for strict $\omega$-categories. Using the same technique, we define the opposite of an $\omega$-category with respect to a set of dimensions, and we show that this construction also preserves the property of being free on a computad. Finally, we show that the constructions of opposites and homs commute.
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Naturality for higher-dimensional path types
A depth-bounded naturality meta-operation in the Catt type theory constructs and machine-checks cylinder and cone composites in weak omega-categories.