cartesian

The Cartesian coordinate system. Coordinates are (x,y,z).

Quite a few categories (monoidal categories) will permit "products" of objects as objects again – in the Haskell sense those are tuples – allowing for "dyadic morphisms" (x,y) ~> r. Together with a unique UnitObject, this makes for a monoidal structure, with a few natural isomorphisms. Ordinary tuples may not always be powerful enough to express the product objects; we avoid making a dedicated associated type for the sake of simplicity, but allow for an extra constraint to be imposed on objects prior to consideration of pair-building. The name Cartesian is disputable: in category theory that would rather Imply cartesian closed category (which we represent with Curry). Monoidal would make sense, but we reserve that to Functors.

Not on Stackage, so not searched. Coordinate systems

A Category equipped with a Tensor t where the Tensor unit i is the terminal object in cat and thus every object a is equipped with a morphism <math>, which we call kill.

Laws

fwd unitl . glmap kill . split ≡ id
fwd unitr . grmap kill . split ≡ id

Calculate the Cartesian product of two sets.

cartesianProduct xs ys = fromList $ liftA2 (,) (toList xs) (toList ys)

Example:

cartesianProduct (fromList [1,2]) (fromList ['a','b']) =
fromList [(1,'a'), (1,'b'), (2,'a'), (2,'b')]

Compute the Cartesian product of two sets, applying a function to each resulting pair.

Slightly unfair 2-way Cartesian product: given two (possibly infinite) lists, produce a single list such that whenever v and w have finite indices in the input lists, (v,w) has finite index in the output list. Lower indices occur as the fst part of the tuple more frequently, but not exponentially so.

Calculate the Cartesian product of two sets.

cartesianProduct xs ys = fromList $ liftA2 (,) (toList xs) (toList ys)

Example:

cartesianProduct (fromList (1:|[2])) (fromList ('a':|['b'])) =
fromList ((1,'a') :| [(1,'b'), (2,'a'), (2,'b')])

Returns the three Cartesian coordinates as a triple from a position.

Cartesian product of two sets. The elements are all pairs (x, y) for each x from s and for each y from t.