Categories for Types

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Additionally, you can add a custom label to replace the "Ticket Type Category" text that will appear when registrants select from the category drop-down. Next, click on the Registration or Ticket Types section of your event. Then, add your Registration of Ticket Types.

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For instance, if your event is a mobile app developer conference, your various Registration Types might look something like this:. In this case, the categories would be:. When finished adding your ticket categories, you can view them as a registrant will see them by viewing the registration form. Then some perhaps different programmer implements the interface by defining those functions for a concrete type.

This is in a sense necessary: a structure should be able to implement the requirements for many different signatures, and a signature should be able to have many different structures implement it. On the other hand, not having any way to think of a structure as a type at all would be completely useless. A functor in ML is a procedure which accepts as input a structure and produces as output another structure.

That is, we can create a structure that defines the functions laid out in a signature and perhaps more.

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Here is an example of a structure for a mag object of integers:. If it fails, it raises a compiler error. A more detailed description of this process can be found here. We can then use the structure as follows:.

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  • In any case, the important part about signatures and structures is that one can write functions which accept as input structures with a given signature and produce structures with other signatures. In any case, we can do the same thing with a category. A general category will be represented by a signature, and a particular instance of a category is a structure with the implemented pieces. This list looks quite imbalanced, but one might argue that the cons are relatively minor compared to the pros. In particular and this is what this author hopes to be the case , being able to explicitly construct proofs to theorems in category theory will give one a much deeper understanding both of category theory and of programming.

    This is the ultimate prize. Like Like. I find it weird how it appears you assume knowledge of ML, throw it out the window to explain signatures, then throw functors into the mix like nothing. I also said that all that stuff about signatures and functors could be safely skipped.

    Categories as Values

    The point of this post is not to learn advanced features of ML but to see how categories can be represented as types in a computer. Safely skipped, then what is the point of including it? In any programming language there are many ways to accomplish equivalent tasks and each gives a different perspective on the same problem, and perspective is valuable.

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    • What Do We Want? Composition is associative. Every object has an identity morphism. Even so, this should be enough to define a category.

      Categories as Values In order to see this type in action, we defined and included in the source code archive for this post a type for homogeneous sets. To be completely clear, the type of the Poset category defined above is 'a Set, 'a Set PosetArrow Category and so we can define a shortcut for this type.

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      Pros: We can prove results by explicit construction more on this next time. Different-looking categories are structurally similar in code. We can faithfully represent the idea of using objects and morphisms of categories as parameters to construct other categories.

      What Do We Want?

      Writing things in code gives a fuller understanding of how they work. Cons: All computations are finite, requiring us to think much harder than a mathematician about the definition of an object. The type system is too weak. As such, the programmer becomes responsible for any failure that occur from bad definitions. The type system is too strong. We cannot ensure the ability to check equality on objects.

      Categories and Types in Logic, Language, and Physics

      This showed up in our example of diagram categories. The functions used in defining morphisms, e. For example, nothing about category theory requires functions to be computable. Moreover, nothing about our implementation requires the functions to have any outputs at all they may loop infinitely!

      Moreover, it is not possible to ensure that any given function terminates on a given input set this is the Halting problem. Until then! Like this: Like Loading One question: How do you write code in a gray box?

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