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< simplex category


simplex category Δ 𝒰\Delta_\mathcal{U} of a universe 𝒰\mathcal{U} is the category of all inhabited finite totally ordered types in 𝒰\mathcal{U} whose hom types are the monotonic functions.

Δ 𝒰 n: A:Fin 𝒰(n)isTotallyOrdered(A)×[A]\Delta_\mathcal{U} \coloneqq \sum_{n:\mathbb{N}} \sum_{A:\mathrm{Fin}_\mathcal{U}(n)} \mathrm{isTotallyOrdered}(A) \times \left[A\right]
Hom Δ 𝒰(A,B) f:ABisMonotonic(f)Hom_{\Delta_\mathcal{U}}(A, B) \coloneqq \sum_{f:A \to B} \mathrm{isMonotonic}(f)

See also

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