Monoidal Functor

definition example

Let $(\mathcal{C},I_{\mathcal{C}},{\otimes }_{\mathcal{C}})$ and $(\mathcal{D},I_{\mathcal{D}},{\otimes }_{\mathcal{D}})$ be symmetric monoidal categories. A (lax) symmetric monoidal functor $(F,\phi )$ consists of

(i) a Functor $F:\mathcal{C}\to \mathcal{D}$; (ii) a morphism ${\phi }_I:I_{\mathcal{D}}\to F(I_{\mathcal{C}})$; (iii) morphisms ${\phi }_{c_1,c_2}:F(c_1){\otimes }_{\mathcal{D}}F(c_2)\to F(c_1{\otimes }_{\mathcal{C}}{c}_2)$, natural in $c_1,c_2$ (the coherence maps),

obeying bookkeeping axioms compatible with associators, unitors and symmetries. It is strong if the $\phi$ are isomorphisms and strict if identities. The preorder version is a Monoidal Monotone Map; reversing the $\phi$ gives oplax monoidal functors.

Sources: 7 Sketches §6.4.1 (Rough Definition 6.68, Example 6.69, Exercise 6.70), Definition 2.41, Exercise 5.69; DaoFP §14.9 (“Monoidal Functors”: lax monoidal functors, functorial strength, applicative functors, closed functors), §17.7 (applicatives as monoids under Day Convolution), §20.1 (change of enriching category).

Examples

• Power set $\mathcal{P}:(\mathbf{Set},1,\times )\to (\mathbf{Set},1,\times )$ with $\mathcal{P}(f)=$ direct image, ${\phi }_I(1)=\{1\}$, ${\phi }_{S,T}(A,B)=A\times B$ (Example 6.69; naturality is 7S Exercise 6.70).
• Decoration functors $(F,\phi ):(\mathbf{FinSet},\varnothing ,+)\to (\mathbf{Set},1,\times )$, e.g. $\mathbf{Circ}$ sending a finite set of nodes to the set of circuits on it, with ${\psi }_{V,V^{\prime }}$ the disjoint union of circuits (Eq. 6.81); these build decorated cospan categories (Theorem 6.77). The constant functor $F(c)=\{\ast \}$ recovers plain cospans (7S Exercise 6.78).
• $U:\mathbf{Mat}(R)\to \mathbf{Set}$, $n\mapsto R^n$, strong monoidal since $R^m\times R^n\cong R^{m+n}$ (7S Exercise 5.69); monoidal functors carry monoid objects to monoid objects.
• Applicative functors (DaoFP §14.9): a lax monoidal endofunctor of $\mathbf{Hask}$, unit :: () -> f (), (>*<) :: (f a, f b) -> f (a, b), equivalently pure and <*>; every Monad is applicative. Lax monoidal functors $[\mathcal{C},\mathbf{Set}]$ are monoids for Day Convolution.
• The semantics functor $S:{\mathbf{SFG}}_R\to \mathbf{Mat}(R)$ is a strict monoidal (prop) functor; Change of Base uses a monoidal functor between enriching categories; operad functors are the operadic analogue.
• Functorial strength ${\sigma }_{a,b}:a\otimes Fb\to F(a\otimes b)$ (DaoFP §14.9, §20.2) is a related notion: in a closed category strong = enriched.

Laxness as an approximation

In probabilistic learning, lax monoidality measures what an approximation throws away: Bayesian Inversion is only a lax monoidal functor on open models, because the parallel composite of two inversions sees only the marginals of a correlated prior — the defect is the mutual information of the branches (Lax Functor, Bayesian Lens).

JuliaLeanHaskell

Docs: Theories (Catlab)

# Example 6.69: the power set as a lax monoidal functor (Set, 1, ×) → (Set, 1, ×)
powerset(S) = [Set(c) for c in Iterators.map(collect, Iterators.filter(_ -> true, subsets(collect(S))))]
image(f, A::Set) = Set(f(a) for a in A)                         # P on morphisms
φ(A::Set, B::Set) = Set((a, b) for a in A, b in B)               # φ_{S,T}(A, B) = A × B
# naturality (Exercise 6.70): φ(image(f, A), image(g, B)) == image(((a,b),) -> (f(a), g(b)), φ(A, B))

#check CategoryTheory.LaxMonoidalFunctor      -- ε : 𝟙_ D ⟶ F.obj (𝟙_ C), μ : F.obj X ⊗ F.obj Y ⟶ F.obj (X ⊗ Y), coherence
#check CategoryTheory.MonoidalFunctor         -- strong: ε and μ isomorphisms
#check CategoryTheory.LaxBraidedFunctor       -- compatible with braidings/symmetry

-- DaoFP §14.9: a lax monoidal endofunctor of Hask (= Applicative)
class Functor f => Monoidal f where
  unit  :: () -> f ()
  (>*<) :: (f a, f b) -> f (a, b)
-- equivalent to Applicative: pure x = fmap (const x) (unit ()); ff <*> fa = fmap (uncurry ($)) (ff >*< fa)
 
-- functorial strength (free in Hask)
strength :: Functor f => (a, f b) -> f (a, b)
strength (a, fb) = fmap (\b -> (a, b)) fb

Lax monoidal functors in programming (DaoFP §14.9)

Monoidal functors map monoids to monoids, and only the lax data ${\phi }_I:j\to Fi$, ${\phi }_{ab}:Fa\oplus Fb\to F(a\otimes b)$ is needed: a Monoid Object $(m,\mu ,\eta )$ goes to $(Fm,F\mu \circ {\phi }_{mm},F\eta \circ {\phi }_I)$. For endofunctors preserving the cartesian product this is the Haskell class class Monoidal f where unit :: f (); (>*<) :: f a -> f b -> f (a, b) (DaoFP Exercise 14.9.1). In a Cartesian Closed Category lax monoidal endofunctors coincide with lax closed ones ($F(b^a)\to (Fb{)}^{Fa}$), i.e. applicative functors; every strong Monad is one. The category $\mathbf{MonCat}$ of monoidal categories and monoidal functors is a 2-Category.