Enriched Functor

definition example

Let $\mathcal{X}$ and $\mathcal{Y}$ be $\mathcal{V}$-categories. A $\mathcal{V}$-functor $F:\mathcal{X}\to \mathcal{Y}$ consists of a function $F:\mathrm{Ob}(\mathcal{X})\to \mathrm{Ob}(\mathcal{Y})$ such that

$$\mathcal{X}(x_1,x_2)\le \mathcal{Y}(F(x_1),F(x_2))\text{for all }{x}_1,x_2\in \mathrm{Ob}(\mathcal{X}).$$

Sources: 7 Sketches Definition 2.69, Examples 2.70, 2.72, Exercise 2.73; DaoFP §20.2; the ordinary case is Functor.

The analogy: preorder is to $\mathbf{Bool}$-category as Monotone Map is to $\mathbf{Bool}$-functor (Example 2.70). Likewise a Cost-functor between Lawvere metric spaces is a function with $d_X(x_1,x_2)\ge d_Y(F{x}_1,F{x}_2)$: a 1-Lipschitz (distance non-increasing) map (Example 2.72) — “a friend from the theory of metric spaces”. A dagger $\mathcal{V}$-category is one where the identity is a $\mathcal{V}$-functor $\mathcal{X}\to {\mathcal{X}}^{\mathrm{op}}$ (7S Exercise 2.73).

In a monoidal category (DaoFP §20.2)

When $\mathcal{V}$ is a Monoidal Category, a $\mathcal{V}$-functor maps objects to objects and hom-objects to hom-objects via $\mathcal{V}$-morphisms $F_{ab}:\mathcal{C}(a,b)\to \mathcal{D}(Fa,Fb)$ that commute with composition and identity (diagrams in $\mathcal{V}$). Both categories must be enriched over the same $\mathcal{V}$. Examples:

• The Hom Functor $\mathrm{Hom}:{\mathcal{C}}^{\mathrm{op}}\otimes \mathcal{C}\to \mathcal{V}$ of a category enriched in a closed $\mathcal{V}$ (treating $\mathcal{V}$ as self-enriched); its action on hom-objects $\mathcal{C}(b,a)\otimes \mathcal{C}(a^{\prime },b^{\prime })\to [\mathcal{C}(a,a^{\prime }),\mathcal{C}(b,b^{\prime })]$ is built by currying and composing twice. Lifting a global element $f:I\to \mathcal{C}(a,b)$ gives $\mathcal{C}(c,f):\mathcal{C}(c,a)\to \mathcal{C}(c,b)$ as ${\lambda }^{-1}$, then $f\otimes \mathrm{id}$, then $\circ$.
• Enriched co-presheaves $\mathcal{C}\to \mathcal{V}$ with $F_{ab}:\mathcal{C}(a,b)\to [Fa,Fb]$; enriched profunctors ${\mathcal{C}}^{\mathrm{op}}\otimes \mathcal{D}\to \mathcal{V}$.
• $\otimes :\mathcal{V}\otimes \mathcal{V}\to \mathcal{V}$ is a $\mathcal{V}$-functor when $\mathcal{V}$ is closed (DaoFP Exercise 20.2.2).
• A Haskell Functor is an enriched endofunctor of $\mathbf{Hask}$ (self-enriched): fmap :: (a -> b) -> (f a -> f b) maps internal homs. In a self-enriched category, enriched endofunctors are exactly strong endofunctors: strength gives $[a,b]\otimes Fa\stackrel{\sigma }{\to }F([a,b]\otimes a)\stackrel{F\epsilon }{\to }Fb$, and conversely enrichment gives strength via the coevaluation ${\eta }_{ab}:a\to [b,a\otimes b]$ — in Haskell strength (a, bs) = fmap (a,) bs.

JuliaLeanHaskell

Docs: Vignette: monoidal preorders & SMCs

Builds on: Enriched Category (VCategory) — run that note’s Julia code first.

# check a V-functor between finite V-categories given as an object map (Dict)
function is_vfunctor(X::VCategory, Y::VCategory, F::Dict)
  ix = Dict(o => i for (i, o) in enumerate(X.objects)); iy = Dict(o => i for (i, o) in enumerate(Y.objects))
  all(leq(X.base, X.hom[ix[a], ix[b]], Y.hom[iy[F[a]], iy[F[b]]]) for a in X.objects, b in X.objects)
end

#check CategoryTheory.EnrichedFunctor      -- EnrichedFunctor V C D
-- Mathlib: `Monotone.functor` is the Bool-enriched case; `LipschitzWith 1` the Cost case
#check LipschitzWith

-- a V-functor on finite V-categories: an object map satisfying hom x y <= hom (F x) (F y)
isVFunctor :: (MonoidalPreorder v, Eq o, Eq o') => VCat v o -> VCat v o' -> (o -> o') -> Bool
isVFunctor (VCat os h) (VCat _ h') f = and [ leq (h x y) (h' (f x) (f y)) | x <- os, y <- os ]
 
-- Hask-enriched endofunctor = Functor; enrichment gives strength for free
strength :: Functor f => (a, f b) -> f (a, b)
strength (a, bs) = fmap (\b -> (a, b)) bs