@buchholtz @trebor In fact this question about representing the cup product in terms of torsors and gerbes was what originally led me to think about torsors and gerbes in HoTT. Eventually I became convinced that one should work with EM spaces representation-independently, just using their defining properties. So I became less interested in the original question. The paper linked above mentions the cup product but perhaps doesn't answer Trebor's question.
I think one answer would be as follows. Say we have groups A, B, C with C abelian, and a bilinear map μ : A → B → C. Say X is an A-torsor and Y is a B-torsor. Let's define the cup product of X and Y by its defining property, i.e. given a gerbe Z banded by C we say what it means for Z to be the cup product. It means we have a map h : X * Y → Z from the join of X and Y with the following property. Given points x₀ x₁ : X and y₀ y₁ : Y, we get a loop in X * Y that looks like glue(x₀,y₀) · glue(x₁,y₀)⁻¹ · glue(x₁,y₁) · glue(x₀,y₁)⁻¹. Now h sends this to a loop in Z. Since Z is banded by C, this loops corresponds to an element of C. The condition is that this element of C equals μ(x₁ - x₀, y₁ - y₀). But I don't have a proof that this works.
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