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| ===Cartesian Closed Category=== | | ===Cartesian Closed Category=== |
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− | <pre> | + | {| align="center" cellpadding="8" width="90%" <!--QUOTE--> |
− | | A 'cartesian closed category' is a cartesian category $A$ with
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− | | additional structure R4 satisfying the additional equations:
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− | | E4a. !e!_A,B <h* p1_C,B, p2_C,B> = h,
| + | <p>A ''cartesian closed category'' is a cartesian category <math>\mathcal{A}</math> with additional structure <math>\text{R4}\!</math> satisfying the additional equations:</p> |
| + | |- |
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− | | E4b. (!e!_A,B <k p1_C,B, p2_C,B>)* = k,
| + | <p><math>\begin{array}{ll} |
| + | \text{E4a.} & \varepsilon^{}_{A,B} \langle h^* \pi^{}_{C,B}, \pi^\prime_{C,B} \rangle = h, |
| + | \\[8pt] |
| + | \text{E4b.} & (\varepsilon^{}_{A,B} \langle k \pi^{}_{C,B}, \pi^\prime_{C,B} \rangle)^* = k, |
| + | \\[8pt] |
| + | & \text{for all}~ h : C \land B \to A \quad \text{and} \quad k : C \to (A \Leftarrow B). |
| + | \end{array}</math></p> |
| + | |- |
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− | | for all h : C & B -> A, k : C -> (A <= B).
| + | <p>Thus, a cartesian closed category is a positive intuitionistic propositional calculus satisfying the equations <math>\text{E1}\!</math> to <math>\text{E4}.\!</math> This illustrates the general principle that one may obtain interesting categories from deductive systems by imposing an appropriate equivalence relation on proofs.</p> |
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| + | |
− | | Thus, a cartesian closed category is
| + | <p>(Lambek & Scott, 53).</p> |
− | | a positive intuitionistic propositional
| + | |} |
− | | calculus satisfying the equations E1 to E4.
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− | | This illustrates the general principle that
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− | | one may obtain interesting categories from
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− | | deductive systems by imposing an appropriate
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− | | equivalence relation on proofs.
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− | </pre> | |
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| ==Document History== | | ==Document History== |