kan-extensions-4.2.2: Kan extensions, Kan lifts, various forms of the Yoneda lemma, and (co)density (co)monads

Copyright (C) 2013-2014 Edward Kmett, Gershom Bazerman and Derek Elkins BSD-style (see the file LICENSE) Edward Kmett provisional portable Trustworthy Haskell98

Data.Functor.Contravariant.Day

Description

The Day convolution of two contravariant functors is a contravariant functor.

http://ncatlab.org/nlab/show/Day+convolution

Synopsis

# Documentation

data Day f g a Source

The Day convolution of two contravariant functors.

Constructors

 forall b c . Day (f b) (g c) (a -> (b, c))

Instances

 Contravariant (Day f g) (Representable f, Representable g) => Representable (Day f g) Typeable ((* -> *) -> (* -> *) -> * -> *) Day type Rep (Day f g) = (Rep f, Rep g)

day :: f a -> g b -> Day f g (a, b) Source

Construct the Day convolution

````day1` (`day` f g) = f
`day2` (`day` f g) = g
```

runDay :: (Contravariant f, Contravariant g) => Day f g a -> (f a, g a) Source

Break apart the Day convolution of two contravariant functors.

assoc :: Day f (Day g h) a -> Day (Day f g) h a Source

Day convolution provides a monoidal product. The associativity of this monoid is witnessed by `assoc` and `disassoc`.

````assoc` . `disassoc` = `id`
`disassoc` . `assoc` = `id`
`contramap` f `.` `assoc` = `assoc` `.` `contramap` f
```

disassoc :: Day (Day f g) h a -> Day f (Day g h) a Source

Day convolution provides a monoidal product. The associativity of this monoid is witnessed by `assoc` and `disassoc`.

````assoc` . `disassoc` = `id`
`disassoc` . `assoc` = `id`
`contramap` f `.` `disassoc` = `disassoc` `.` `contramap` f
```

swapped :: Day f g a -> Day g f a Source

The monoid for Day convolution in Haskell is symmetric.

````contramap` f `.` `swapped` = `swapped` `.` `contramap` f
```

intro1 :: f a -> Day Proxy f a Source

Proxy serves as the unit of Day convolution.

````day1` `.` `intro1` = `id`
`contramap` f `.` `intro1` = `intro1` `.` `contramap` f
```

intro2 :: f a -> Day f Proxy a Source

Proxy serves as the unit of Day convolution.

````day2` `.` `intro2` = `id`
`contramap` f `.` `intro2` = `intro2` `.` `contramap` f
```

day1 :: Contravariant f => Day f g a -> f a Source

In Haskell we can do general purpose elimination, but in a more general setting it is only possible to eliminate the unit.

````day1` `.` `intro1` = `id`
`day1` = `fst` `.` `runDay`
`contramap` f `.` `day1` = `day1` `.` `contramap` f
```

day2 :: Contravariant g => Day f g a -> g a Source

In Haskell we can do general purpose elimination, but in a more general setting it is only possible to eliminate the unit. ``` day2 . intro2 = id day2 = snd . runDay contramap f . day2 = day2 . contramap f ```

diag :: f a -> Day f f a Source

Diagonalize the Day convolution:

````day1` `.` `diag` = `id`
`day2` `.` `diag` = `id`
`runDay` `.` `diag` = a -> (a,a)
`contramap` f . `diag` = `diag` . `contramap` f
```

trans1 :: (forall x. f x -> g x) -> Day f h a -> Day g h a Source

Apply a natural transformation to the left-hand side of a Day convolution.

This respects the naturality of the natural transformation you supplied:

````contramap` f `.` `trans1` fg = `trans1` fg `.` `contramap` f
```

trans2 :: (forall x. g x -> h x) -> Day f g a -> Day f h a Source

Apply a natural transformation to the right-hand side of a Day convolution.

This respects the naturality of the natural transformation you supplied:

````contramap` f `.` `trans2` fg = `trans2` fg `.` `contramap` f
```