On Computads for Strict Omega-Categories

Diagrams allow us to leverage our geometric intuition by abstracting pastability: every way we may compose the arrows will yield the same result. This essay provides a self-contained discussion illustrating some non-geometric behaviour exhibited by diagrams in higher dimensions, which are formalised by the notion of computads. This pathological behaviour is intimately linked with the failure of computads to form a presheaf category. This is ultimately caused by strictness, the requirement that cancellation of units or associativity hold up to strict equality, as opposed to weakly, or only up to isomorphism. In ordinary, 1-dimensional category theory, such constraints are baked into the definition of what it even means to be a category; we analyse in depth how this innocuous assumption can cause unexpected trouble when passing to higher dimensions.

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