Collage

definition example

Let $\Phi :\mathcal{X}\nrightarrow \mathcal{Y}$ be a $\mathcal{V}$-Profunctor. Its collage $\mathrm{Col}(\Phi )$ (DaoFP: cograph) is the $\mathcal{V}$-category with $\mathrm{Ob}(\mathrm{Col}(\Phi )):=\mathrm{Ob}(\mathcal{X})\bigsqcup \mathrm{Ob}(\mathcal{Y})$ and

$$\mathrm{Col}(\Phi )(a,b):=\{\begin{array}{ll}\mathcal{X}(a,b)& a,b\in \mathcal{X}\\ \Phi (a,b)& a\in \mathcal{X},b\in \mathcal{Y}\\ \varnothing (=\bigvee \varnothing )& a\in \mathcal{Y},b\in \mathcal{X}\\ \mathcal{Y}(a,b)& a,b\in \mathcal{Y},\end{array}$$

with collage inclusions $i_{\mathcal{X}}:\mathcal{X}\to \mathrm{Col}(\Phi )$, $i_{\mathcal{Y}}:\mathcal{Y}\to \mathrm{Col}(\Phi )$. Its Hasse diagram is the union of the two Hasse diagrams plus the bridges as arrows — “put a box around the whole picture and see a new preorder”.

Sources: 7 Sketches §4.3.3 (Definition 4.42, Example 4.43, Exercise 4.44), Example 4.11; DaoFP §17.1 (“Collages”), Exercises 17.1.1–17.1.2.

Example 4.43 ($\mathbf{Cost}$): $\mathcal{X}=A\stackrel{2}{\to }B$, $\mathcal{Y}=x\rightleftarrows y$ (weights $3,4$), bridge $A\stackrel{5}{\to }x$. Collage matrix (rows/cols $A,B,x,y$): $A$: $0,2,5,8$; $B$: $\infty ,0,\infty ,\infty$; $x$: $\infty ,\infty ,0,3$; $y$: $\infty ,\infty ,4,0$ (7 Sketches prints the empty hom-object as $0=\bigvee \varnothing$, which in $\mathbf{Cost}$ is $\infty$).

DaoFP: the new morphisms across the collage are heteromorphisms, going only from $\mathcal{C}$ to $\mathcal{D}$; composition with them is by lifting along the profunctor. A profunctor ${\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}\to \mathbf{Set}$ “should really be called an endo-profunctor” — it defines a collage of $\mathcal{C}$ with itself. There is a functor from any collage to the Walking Arrow (DaoFP Exercise 17.1.1), and conversely any category with a functor to the walking arrow splits as a collage (DaoFP Exercise 17.1.2): profunctors $\mathcal{C}\nrightarrow \mathcal{D}$ are the same as categories over $2$ with fibres $\mathcal{C}$ and $\mathcal{D}$.

JuliaHaskell

Docs: plain Julia — Catlab has no dedicated API for this; related: Catlab v0.16 docs · GATlab standard library

Builds on: Cost (CostPre), Enriched Category (VCategory), Profunctor (VProfunctor) — run those notes’ Julia code first.

# collage of a V-profunctor between finite V-categories: block matrix [X Φ; 0 Y]
function collage(P::VProfunctor)
  V = P.X.base; zero = join(V, [])                    # ⋁∅
  hom = [P.X.hom P.Φ; fill(zero, size(P.Φ, 2), size(P.Φ, 1)) P.Y.hom]
  VCategory(V, vcat(P.X.objects, P.Y.objects), hom)
end
# Example 4.43
X = VCategory(CostPre(), [:A, :B], [0.0 2; Inf 0]); Y = VCategory(CostPre(), [:x, :y], [0.0 3; 4 0])
Φ = VProfunctor(X, Y, [5.0 8; Inf Inf])
is_vcategory(collage(Φ))     # true

-- the collage of two categories along a profunctor: objects are a sum, heteromorphisms are p a b
data ColObj x y = InX x | InY y
data ColHom p x y a b where
  HomX :: (a -> b) -> ColHom p x y (InX a) (InX b)   -- schematic: homs within X
  Het  :: p a b -> ColHom p x y (InX a) (InY b)      -- heteromorphisms from X to Y (never back)