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Exercise 58 - Chapter 2

Recall the monoidal preorder \( \textbf{NMY} := (P, \le, yes, min) \) from Exercise 2.31. Interpret what a category enriched in \( \textbf{NMY} \) would be.

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Comments

  • 1.

    I think it would look like a preorder, except that some edges (necessarily between different objects) would be labelled as "maybe" and all paths that include a "maybe" edge should be counted as "maybe", too. For example, if we are dealing with a set of propositions and we interpret edges (or lack thereof) as "there is no proof/there may be a proof/there is a proof", then there would be a proof between two propositions linked by a path of "yes" edges, there may be a proof between two propositions linked only by a path of "yes" and (at least one) "maybe" edges, and there is no proof otherwise (i.e., "no" means "no edges").

    Comment Source:I think it would look like a preorder, except that some edges (necessarily between different objects) would be labelled as "maybe" and all paths that include a "maybe" edge should be counted as "maybe", too. For example, if we are dealing with a set of propositions and we interpret edges (or lack thereof) as "there is no proof/there may be a proof/there is a proof", then there would be a proof between two propositions linked by a path of "yes" edges, there may be a proof between two propositions linked only by a path of "yes" and (at least one) "maybe" edges, and there is no proof otherwise (i.e., "no" means "no edges").
  • 2.

    It provides a sort of fuzzy logic, answers to "is it possible?" where "maybe" is an allowed answer.

    Comment Source:It provides a sort of fuzzy logic, answers to "is it possible?" where "maybe" is an allowed answer.
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