99 lines
2.7 KiB
Idris
99 lines
2.7 KiB
Idris
||| This module defines tensor products, which are later used to define
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||| the concept of profunctor strength. The two primary tensor products
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||| in `Idr` are the product (`Pair`) and the coproduct (`Either`).
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module Data.Tensor
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%default total
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------------------------------------------------------------------------------
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-- Tensor products
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------------------------------------------------------------------------------
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||| A bifunctor that admits an *associator*, i.e. a bifunctor that is
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||| associative up to isomorphism.
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||| Laws:
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||| * `mapFst assocl . assocl . assocl = assocl . mapSnd assocl`
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public export
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interface Bifunctor ten => Associative ten where
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assocl : a `ten` (b `ten` c) -> (a `ten` b) `ten` c
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assocl = assoc.leftToRight
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assocr : (a `ten` b) `ten` c -> a `ten` (b `ten` c)
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assocr = assoc.rightToLeft
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assoc : a `ten` (b `ten` c) <=> (a `ten` b) `ten` c
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assoc = MkEquivalence assocl assocr
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||| A bifunctor that admits a swap map, i.e. a bifunctor that is
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||| symmetric up to isomorphism.
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||| The bifunctor `ten` is generally also associative.
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public export
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interface Bifunctor ten => Symmetric ten where
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swap' : a `ten` b -> b `ten` a
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swap' = symmetric.leftToRight
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symmetric : a `ten` b <=> b `ten` a
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symmetric = MkEquivalence swap' swap'
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||| A tensor product is an associative bifunctor that has an identity element
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||| up to isomorphism. Tensor products constitute the monoidal structure of a
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||| monoidal category.
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||| Laws:
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||| * `mapSnd unitl.leftToRight = mapFst unitr.leftToRight . assocl`
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public export
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interface Associative ten => Tensor ten i | ten where
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unitl : i `ten` a <=> a
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unitr : a `ten` i <=> a
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------------------------------------------------------------------------------
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-- Cartesian monoidal structure
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------------------------------------------------------------------------------
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public export
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Associative Pair where
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assocl (x,(y,z)) = ((x,y),z)
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assocr ((x,y),z) = (x,(y,z))
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public export
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Symmetric Pair where
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swap' = swap
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public export
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Tensor Pair () where
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unitl = MkEquivalence snd ((),)
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unitr = MkEquivalence fst (,())
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------------------------------------------------------------------------------
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-- Cocartesian monoidal structure
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------------------------------------------------------------------------------
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public export
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Associative Either where
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assocl (Left x) = Left (Left x)
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assocl (Right (Left x)) = Left (Right x)
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assocl (Right (Right x)) = Right x
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assocr (Left (Left x)) = Left x
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assocr (Left (Right x)) = Right (Left x)
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assocr (Right x) = Right (Right x)
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public export
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Symmetric Either where
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swap' = either Right Left
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public export
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Tensor Either Void where
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unitl = MkEquivalence (either absurd id) Right
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unitr = MkEquivalence (either id absurd) Left
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