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Sets

GAT signature​

⦃ X : U ⦄

Algebras and Homomorphisms​

Set-⁡Alg  :=Set\mathfrak{Set}\operatorname-\mathsf{Alg}\;:= \mathsf{Set}
Set-⁡Hom  X0  X1  :=X0→X1\mathfrak{Set}\operatorname-\mathsf{Hom}\;X_0\;X_1\;:= X_0 \to X_1

Initial Algebra​

Tm‾  Set  (El  0)=∅\overline{\mathsf{Tm}}\;\mathfrak{Set}\;(\mathsf{El}\;0) = \emptyset

Displayed Algebras and Sections​

N-⁡DAlg  X  :=X→Set\mathfrak{N}\operatorname-\mathsf{DAlg}\;X\;:= X \to \mathsf{Set}
N-⁡Sect  X  XD  :=(x ⁣:X)→XD  x\mathfrak{N}\operatorname-\mathsf{Sect}\;X\;X^D\;:= (x \colon X) \to X^D\;x