data-category-0.10: Category theory

Data.Category.RepresentableFunctor

Description

Synopsis

# Documentation

data Representable f repObj Source #

Constructors

 Representable FieldsrepresentedFunctor :: f representingObject :: Obj (Dom f) repObj represent :: forall k z. (Dom f ~ k, Cod f ~ (->)) => Obj k z -> (f :% z) -> k repObj z universalElement :: forall k. (Dom f ~ k, Cod f ~ (->)) => f :% repObj

unrepresent :: (Functor f, Dom f ~ k, Cod f ~ (->)) => Representable f repObj -> k repObj z -> f :% z Source #

type InitialUniversal x u a = Representable ((x :*-: Cod u) :.: u) a Source #

initialUniversal :: Functor u => u -> Obj (Dom u) a -> Cod u x (u :% a) -> (forall y. Obj (Dom u) y -> Cod u x (u :% y) -> Dom u a y) -> InitialUniversal x u a Source #

An initial universal property, a universal morphism from x to u.

terminalUniversal :: Functor u => u -> Obj (Dom u) a -> Cod u (u :% a) x -> (forall y. Obj (Dom u) y -> Cod u (u :% y) x -> Dom u y a) -> TerminalUniversal x u a Source #

A terminal universal property, a universal morphism from u to x.

adjunctionInitialProp :: Adjunction c d f g -> Obj d y -> InitialUniversal y g (f :% y) Source #

For an adjunction F -| G, each pair (FY, unit_Y) is an initial morphism from Y to G.

adjunctionTerminalProp :: Adjunction c d f g -> Obj c x -> TerminalUniversal x f (g :% x) Source #

For an adjunction F -| G, each pair (GX, counit_X) is a terminal morphism from F to X.

initialPropAdjunction :: forall f g c d. (Functor f, Functor g, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d) => f -> g -> (forall y. InitialUniversal y g (f :% y)) -> Adjunction c d f g Source #

terminalPropAdjunction :: forall f g c d. (Functor f, Functor g, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d) => f -> g -> (forall x. TerminalUniversal x f (g :% x)) -> Adjunction c d f g Source #