Theorem. Background symmetries form a presheaf of groupoids [fdip-009F]
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  • November 3, 2026

Let a group \(G\) act on the left on a set \(X\). For each subgroup \(H\), the action groupoid \(H\ltimes X\) has the elements of \(X\) as objects, and an arrow \(h:x\to y\) for each \(h\in H\) with \(h\cdot x=y\). If \(H\) is contained in a subgroup \(K\), sending each arrow to itself is a functor \(H\ltimes X\to K\ltimes X\), identical on objects. These functors preserve identities and composition, and they compose with the inclusions of subgroups. The assignment is therefore a functor from the poset of subgroups of \(G\) to groupoids. Equivalently, it is a groupoid-valued presheaf on the opposite of that poset.

The isomorphism classes of \(H\ltimes X\) are the orbits of \(H\). Thus the objects modulo the symmetries in \(H\) are the isomorphism classes of that groupoid. The enriched four-cycle backgrounds carry an action of the dihedral group constructed there. The action groupoid in that note is the value of this presheaf at that whole group. That note is transcluded by enriched backgrounds. The reader guide places that comparison in Part V, with the Clifford algebras and the Dirac operators. The chapter on backgrounds and symmetry does not transclude the four-cycle note or the enriched-backgrounds note. It says that its examples prepare those comparisons. The arrow pattern is the action groupoid. This note does not claim that every background in that chapter arises from the four-cycle.

No stack is proved. A stack would require a topology and a descent equivalence for compatible families. No cover is defined here, and no descent map is shown to be an equivalence. The finite tests of where the topos layer begins record that the action-groupoid example does not require a quotient stack, and that those tests have supplied no site, descent theorem, or physical correspondence. Whether this presheaf organises a stack of backgrounds remains open.

Proof. 

  • November 3, 2026

Composition in \(H\ltimes X\) sends \(g:x\to y\) and \(k:y\to z\) to \(kg:x\to z\). The action law gives \((kg)\cdot x=k\cdot (g\cdot x)\), so the composite is an arrow. The identity arrow is \(1:x\to x\). The inverse of \(h:x\to h\cdot x\) is \(h^{-1}:h\cdot x\to x\), and \(h^{-1}\) lies in \(H\) because \(H\) is a subgroup. The two inverse identities and associativity are the corresponding group laws. Every arrow is therefore invertible.

Suppose \(H\subset K\). An arrow labelled by an element of \(H\) is an arrow of \(K\ltimes X\) with the same source and target. The identity element and the product \(kg\) are unchanged by this inclusion, so the map on arrows preserves identities and composition. If \(H\subset K\subset L\), the composite of the two inclusions of arrows is the inclusion of \(H\) into \(L\). The inclusion of a subgroup into itself is the identity functor. This is a functor from the subgroup poset to groupoids. That poset has one morphism from \(H\) to \(K\) exactly when \(H\) is contained in \(K\), and the functor sends it to the inclusion of action groupoids. A groupoid-valued presheaf on a category is a functor from the opposite of that category. The opposite of the opposite of the subgroup poset is the subgroup poset, so the functor just constructed is a groupoid-valued presheaf on the opposite poset.

Two objects are isomorphic exactly when some element of \(H\) carries one to the other, which is the definition of an orbit. For the dihedral action, the groupoid at the whole group is the one whose arrows are already identified in the enriched four-cycle note. No topology on the subgroup poset is defined, so the presheaf is not given the data of a stack.