Part : Cat → Cat
Part C = C^{● → ○}
● represents the total space
○ represents the set of indices
For our purposes C := Set.
⊔ : Part C → C
⊔ = composition with {●} → {● → ○}
D ⊣ ⊔ ⊣ ⊤
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The refinement of two partitions ● → ○₁ and ● → ○₂
with the same total space is ● → ○₁ × ○₂.
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