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\(\DeclareMathOperator{\prj}{prj}\) \(\DeclareMathOperator{\aprj}{aprj}\) \(\DeclareMathOperator{\tprj}{tprj}\) \(\DeclareMathOperator{\gprj}{gprj}\)

+ 6.1 Arrays

An array \(A\) of length \(n\) is just a tuple of \(n\) components of the same type \(T\): \[ A = T^n = \underbrace{T \times T \times T \times \ldots \times T}_\text{n times} \] A special property of arrays is that all the projection functions have the same type: \[ A\rightarrow T \] This means we can have a generalised projection function called an array projection instead: \[ \aprj_A : {\Bbb Z}_n \rightarrow (A \rightarrow T) \] where \[ i \mapsto \prj_{A,i} \]

However, even more general is a generalised tuple projection shown here for a pair: \[ \tprj_{T_0\times T_1} : {\Bbb Z}_2 \rightarrow (T_0 \times T_1 \rightarrow T_0 + T_1) \] which suggests a more general form for an individual projection: \[ \gprj_{T_0\times T_1,i}:T_0 \times T_1 \rightarrow T_0 + T_1 \] for \(i \in \lbrace 0,1\rbrace\). In simple terms, the generalised projection returns the tuple index along with the value.

We also have the codiagonal function. For a sum: \[ S = \underbrace{T + T + T + \ldots + T}_\text{n times} \]

\[ \nabla_S : S \rightarrow T \] which simply forgets the case index.