Product of Enriched Categories

definition example proof

Let $\mathcal{V}$ be a Symmetric Monoidal Preorder and $\mathcal{X},\mathcal{Y}$ be $\mathcal{V}$-categories. Their $\mathcal{V}$-product $\mathcal{X}\times \mathcal{Y}$ has

(i) $\mathrm{Ob}(\mathcal{X}\times \mathcal{Y}):=\mathrm{Ob}(\mathcal{X})\times \mathrm{Ob}(\mathcal{Y})$; (ii) $(\mathcal{X}\times \mathcal{Y})((x,y),(x^{\prime },y^{\prime })):=\mathcal{X}(x,x^{\prime })\otimes \mathcal{Y}(y,y^{\prime })$.

Sources: 7 Sketches Definition 2.74, Example 2.76, Exercises 2.75, 2.78; DaoFP §20.1 (“tensor product of $\mathcal{V}$-categories”); ordinary case: Product Category, preorder case: Product Preorder.

It is a $\mathcal{V}$-category (7S Exercise 2.75): $I=I\otimes I\le \mathcal{X}(x,x)\otimes \mathcal{Y}(y,y)$; and

$$\mathcal{X}(x_1,x_2)\otimes \mathcal{Y}(y_1,y_2)\otimes \mathcal{X}(x_2,x_3)\otimes \mathcal{Y}(y_2,y_3)\cong \mathcal{X}(x_1,x_2)\otimes \mathcal{X}(x_2,x_3)\otimes \mathcal{Y}(y_1,y_2)\otimes \mathcal{Y}(y_2,y_3)\le \mathcal{X}(x_1,x_3)\otimes \mathcal{Y}(y_1,y_3),$$

where symmetry is used exactly to swap the middle two factors — which is why $\mathcal{V}$ must be symmetric (DaoFP makes the same point for the tensor product of $\mathcal{V}$-categories, whose identity is $j_c\otimes j_d:I\otimes I\to \mathcal{C}(c,c)\otimes \mathcal{D}(d,d)$).

Examples. For $\mathcal{V}=\mathbf{Bool}$ the AND in ”$(p_1,q_1)\le (p_2,q_2)$ iff $p_1\le p_2$ and $q_1\le q_2$” is $\otimes$: the Product Preorder. For $\mathcal{V}=\mathbf{Cost}$ distances add: $d_{X\times Y}((x,y),(x^{\prime },y^{\prime }))=d_X(x,x^{\prime })+d_Y(y,y^{\prime })$ (Example 2.76: the product of the path $A\stackrel{2}{\to }B\stackrel{3}{\to }C$ with $p\rightleftarrows q$ (weights $5,8$) is a $2\times 3$ grid; 7S Exercise 2.78: in $\mathbb{R}\times \mathbb{R}$, $d((5,6),(-1,4))=6+2=8$). In matrix terms the product is the Kronecker product of hom-matrices with $\otimes$ in place of multiplication — a block matrix with one $X$-shaped block, shifted by an entry of $Y$, for every entry of $Y$.

JuliaLeanHaskell

Docs: Vignette: monoidal preorders & SMCs

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

function vproduct(X::VCategory, Y::VCategory)
  V = X.base
  objs = [Symbol(x, "_", y) for y in Y.objects for x in X.objects]
  hom = [otimes(V, X.hom[i, i′], Y.hom[j, j′])
         for (j, i) in Iterators.product(eachindex(Y.objects), eachindex(X.objects)) |> collect |> vec,
             (j′, i′) in Iterators.product(eachindex(Y.objects), eachindex(X.objects)) |> collect |> vec]
  VCategory(V, objs, hom)
end
# For Cost this is `kron`-like with + in place of *: kron(X, Y)[(i,j),(i′,j′)] = X[i,i′] + Y[j,j′]

-- Mathlib has products of ordinary categories (`CategoryTheory.prod`) and of preorders;
-- the enriched tensor product of V-categories is the componentwise ⊗ on Hom (by hand).

vproduct :: MonoidalPreorder v => VCat v o -> VCat v o' -> VCat v (o, o')
vproduct (VCat os h) (VCat os' h') =
  VCat [ (x, y) | x <- os, y <- os' ] (\(x, y) (x', y') -> h x x' <> h' y y')