Maths - Categorical Programming Languages

Here are some computer languages that may be suitable for coding category theory constructs:

Charity

http://pll.cpsc.ucalgary.ca/charity1/www/home.html

Squiggol

Bird–Meertens formalism

Hagino

http://web.sfc.keio.ac.jp/~hagino/thesis.pdf

OBJ language family

Haskell Code

Code from here.

Catagory Theory in Idris

Scala Code

Code from here. package scalaz

package scalaz


trait Category[F[_, _]] {
  val id: Id[F]
  val compose: Compose[F]

  def i[A]: F[A, A] =
    id.id[A]

  def comp[A, B, C](f: F[B, C], g: F[A, B]): F[A, C] =
    compose.compose(f, g)
}


object Category extends Categorys


trait Categorys {
  def category[F[_, _]](implicit ii: Id[F], c: Compose[F]): Category[F] = new Category[F] {
    val id = ii
    val compose = c
  }

  implicit val Function1Category: Category[Function1] =
    category[Function1]

  implicit val PartialFunctionCategory: Category[PartialFunction] =
    category[PartialFunction]

  implicit val `<:<_Category` : Category[<:<] =
    category[<:<]

  implicit val `=:=_Category` : Category[=:=] =
    category[=:=]
}

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see also:

Catsters youtube videos -

catsters youtube video - Pullbacks and pushouts 1 - Definition of pullback, example: pullbacks in Set
catsters youtube video - Pullbacks and pushouts 2 - Definition of pushouts, pushouts in Set, and the pullback/pushout example of intersection/union of sets
catsters youtube video - Slice categories - Definition of slice categories C/X and X/C, products in C/X as pullbacks in C
catsters youtube video - Comma categories - Definition of comma categories D over and under F for a functor F, and F over and under G for functors F and G with the same target category

Adjunctions from Morphisms

catsters youtube video - Adjunctions from Morphisms 1 - Motivation for the construction of adjoint functors for bundles over sets
catsters youtube video - Adjunctions from Morphisms 2 - The category of bundles on a set as a slice category and as a functor category into sets.
catsters youtube video - Adjunctions from Morphisms 3 - The definition of the pull-back and its right adjoint for bundles over sets.
catsters youtube video - Adjunctions from Morphisms 4 - A proof that the push-forward is right adjont to pull-back.
catsters youtube video - Adjunctions from Morphisms 5 - Description of the left adjoint to the pull-back

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.

Terminology and Notation

Specific to this page here:

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