Pushouts

Module for the categorical pushout of cellular sheaves over a fixed graph.

Given a span of sheaf morphisms

K --f--> F
|
g
|
v
G

the pushout P = F ⊕_K G is the unique sheaf (up to isomorphism) equipped with morphisms iF : F → P and iG : G → P such that iF ∘ f = iG ∘ g, universal among all such cones.

For network sheaves over a fixed graph the pushout is computed stalkwise as a cokernel:

• At each vertex v: P(v) = coker((f_v, -g_v) : K(v) → F(v) ⊕ G(v))
• At each edge e: P(e) = coker((f_e, -g_e) : K(e) → F(e) ⊕ G(e))
• Restriction maps of P are induced from those of F and G simultaneously via the naturality conditions for iF and iG.

This module provides SheafSpan and pushout_sheaf.

SheafSpan

A span of sheaf morphisms

F ← K → G

where left : K → F and right : K → G share the same domain apex = K.

The inner constructor validates that the morphism index maps have the correct lengths relative to the apex's vertex and edge counts, and that all Vmap entries are valid indices into the respective feet. An AssertionError is thrown if any check fails.

Fields

• left :: SheafMorphism — morphism K → F.
• right :: SheafMorphism — morphism K → G.
• apex :: EuclideanSheaf — the apex sheaf K.
• F :: EuclideanSheaf — left foot.
• G :: EuclideanSheaf — right foot.

pushout_sheaf(span::SheafSpan) -> (EuclideanSheaf, SheafMorphism, SheafMorphism)

Convenience overload: compute the pushout of a SheafSpan.

pushout_sheaf(f::SheafMorphism, g::SheafMorphism, K, F, G) -> (EuclideanSheaf, SheafMorphism, SheafMorphism)

Compute the pushout of the span K --f--> F, K --g--> G in the category of network sheaves over the shared underlying graph.

Returns (P, iF, iG) where:

• P is the pushout sheaf.
• iF : F -> P is the canonical injection from F.
• iG : G -> P is the canonical injection from G.

The universal property holds: iF ∘ f ≈ iG ∘ g (up to numerical tolerance).

All three sheaves K, F, G must share the same underlying graph.