Sydney Reading Group: Classifying Category

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Fix a type theory $𝕋$ (no interesting type formers).

The classifying category $𝒞(𝕋)$:

has as

  • objects: contexts of the form $Φ \; ≝ \; (x_0 ∈ A_0, x_1 ∈ A_1(x_0), … , x_n ∈ An(x_0, …, x_{n-1}))$

  • morphisms are context morphisms $f \; ≝ \; (b_0, …, b_n): Φ ⟶ \underbrace{Ψ}{(y_0 ∈ B_0, …, \, y_n ∈ B_n(y_0, ⋯, y{n-1}))}$ consisting of

    b_0 ∈ B_0\\ b_1 ∈ B_1(b_0)\\ \vdots\\ b_n ∈ B_n(b_0, ⋯, b_{n-1})

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