Cotruss: Superalgebraic Dual Structures

• Cotruss is a coordinate-superalgebra structure with binary and ternary comultiplications that represent truss operations in a functorial superalgebra context.
• It satisfies coassociativity, abelian quantum-heap axioms, and codistributivity, providing a dual framework analogous to a Hopf superalgebra for affine supergroups.
• The structure underlies the formal passage to affine superbraces and supports Yang–Baxter map constructions by encoding both multiplicative and heap operations on superschemes.

Cotruss is the coordinate-superalgebra structure introduced in the study of affine supertrusses: it is the representing object of a truss-valued functor on superalgebras, in the same role that a Hopf superalgebra plays for an affine supergroup. In the formal setting of "Affine Supertrusses and Superbraces" (Bruce, 24 Apr 2026), the operative notion is the supercotruss, namely a superalgebra $X$ endowed with a binary comultiplication $\Delta^{(2)}:X\to X\otimes X$ and a ternary comultiplication $\Delta^{(3)}:X\to X\otimes X\otimes X$ satisfying coassociativity, abelian quantum-heap identities, and codistributivity. The concept is introduced to represent affine supertrusses functorially, and it underlies the subsequent passage to affine superbraces and, indirectly, to the proposed Yang--Baxter constructions on affine superschemes (Bruce, 24 Apr 2026).

1. Representable-functor origin

The paper defines an affine supertruss as a representable functor

$T:{SAlg}_{\mathbb K}\to {Truss},$

where ${SAlg}_{\mathbb K}$ is the category of unital associative supercommutative superalgebras over $\mathbb K$, and ${Truss}$ is the category of trusses. If $T$ is represented by a superalgebra $X$, then

$$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$

The cotruss structure is extracted from the requirement that $\Delta^{(2)}:X\to X\otimes X$0 take values in trusses rather than in sets (Bruce, 24 Apr 2026).

The motivation is explicitly tied to the supergeometric method. A direct $\Delta^{(2)}:X\to X\otimes X$1-graded version of a bare truss is not adopted, because a truss replaces addition by an abelian heap operation, so there is no additive group structure from which an even/odd decomposition analogous to $\Delta^{(2)}:X\to X\otimes X$2 could be read off. The paper therefore defines the super-object functorially: a supertruss is not a graded set with extra operations, but a representable truss-valued functor on superalgebras (Bruce, 24 Apr 2026).

Yoneda then forces the representing superalgebra to carry dual operations. A natural binary operation on $\Delta^{(2)}:X\to X\otimes X$3 corresponds to a map $\Delta^{(2)}:X\to X\otimes X$4, while a natural ternary operation corresponds to a map $\Delta^{(2)}:X\to X\otimes X$5. These are exactly the binary and ternary comultiplications

$\Delta^{(2)}:X\to X\otimes X$6

For $\Delta^{(2)}:X\to X\otimes X$7, the induced operations are

$\Delta^{(2)}:X\to X\otimes X$8

$\Delta^{(2)}:X\to X\otimes X$9

Thus $\Delta^{(3)}:X\to X\otimes X\otimes X$0 represents truss multiplication, and $\Delta^{(3)}:X\to X\otimes X\otimes X$1 represents the heap operation.

2. Formal definition and axioms

The paper’s formal definition is concise: a supercotruss is a triple $\Delta^{(3)}:X\to X\otimes X\otimes X$2 satisfying the conditions extracted in §2, namely the quantum-heap axioms for $\Delta^{(3)}:X\to X\otimes X\otimes X$3, coassociativity for $\Delta^{(3)}:X\to X\otimes X\otimes X$4, and left/right codistributivity between them (Bruce, 24 Apr 2026).

The ternary part requires $\Delta^{(3)}:X\to X\otimes X\otimes X$5 to be an abelian quantum heap. The axioms are

$\Delta^{(3)}:X\to X\otimes X\otimes X$6

$\Delta^{(3)}:X\to X\otimes X\otimes X$7

$\Delta^{(3)}:X\to X\otimes X\otimes X$8

$\Delta^{(3)}:X\to X\otimes X\otimes X$9

Here $T:{SAlg}_{\mathbb K}\to {Truss},$0 is multiplication in $T:{SAlg}_{\mathbb K}\to {Truss},$1, and $T:{SAlg}_{\mathbb K}\to {Truss},$2 flips the first and third tensor factors. The last identity is the abelian condition, dualizing commutativity of the heap operation.

The binary part requires $T:{SAlg}_{\mathbb K}\to {Truss},$3 to be a non-counital coassociative coalgebra: $T:{SAlg}_{\mathbb K}\to {Truss},$4

The compatibility between the two comultiplications is expressed by left and right codistributivity: $T:{SAlg}_{\mathbb K}\to {Truss},$5

$T:{SAlg}_{\mathbb K}\to {Truss},$6

On a homogeneous tensor

$T:{SAlg}_{\mathbb K}\to {Truss},$7

the maps $T:{SAlg}_{\mathbb K}\to {Truss},$8 and $T:{SAlg}_{\mathbb K}\to {Truss},$9 are

${SAlg}_{\mathbb K}$0

${SAlg}_{\mathbb K}$1

with

${SAlg}_{\mathbb K}$2

Taken together, these conditions make the cotruss the explicit dualization of the defining truss data: the heap operation becomes a ternary comultiplication, multiplication becomes a binary comultiplication, and distributivity becomes codistributivity.

3. Superalgebraic setting and morphisms

The grading enters through the ambient category and the tensor calculus. A test object ${SAlg}_{\mathbb K}$3 is a unital associative supercommutative superalgebra

${SAlg}_{\mathbb K}$4

with

${SAlg}_{\mathbb K}$5

for homogeneous ${SAlg}_{\mathbb K}$6. All tensor products are ${SAlg}_{\mathbb K}$7-graded tensor products, and both ${SAlg}_{\mathbb K}$8 and ${SAlg}_{\mathbb K}$9 are required to be superalgebra homomorphisms (Bruce, 24 Apr 2026).

The sign in the codistributivity maps is the only fully explicit sign formula in the cotruss axioms. It is the Koszul sign needed to bring the factors $\mathbb K$0, or $\mathbb K$1, together in the graded tensor product. The paper also notes that supercommutativity of the test algebra $\mathbb K$2 is essential in the proof that the codistributivity equations yield distributivity for the induced truss operations on $\mathbb K$3.

A morphism of supercotrusses $\mathbb K$4 is a superalgebra morphism satisfying

$\mathbb K$5

$\mathbb K$6

If counits or cozeros are present, one also requires preservation of them. This gives the category of supercotrusses that appears in the representation theorem.

4. Duality with trusses and categorical role

The paper presents cotrusses as a dual notion to trusses through representability and Yoneda, rather than through an abstract duality doctrine. An ordinary truss carries an abelian heap operation

$\mathbb K$7

and an associative multiplication

$\mathbb K$8

satisfying distributivity. In a cotruss, these become

$\mathbb K$9

together with the codistributivity identities. Heap associativity and identity become the quantum-heap axioms; abelianness becomes ${Truss}$0; semigroup associativity becomes coassociativity of ${Truss}$1 (Bruce, 24 Apr 2026).

“The philosophy is that a cotruss is to a truss what a Hopf algebra is to a group.”

This slogan is conceptual rather than theorematic, but it captures the intended role exactly. The principal structural result is the equivalence between affine supertrusses and their representing cotruss-bearing superalgebras: every affine supertruss has a representing triple ${Truss}$2 satisfying the cotruss axioms, and there is an equivalence between the opposite category of affine supertrusses and the category of supercotrusses (Bruce, 24 Apr 2026).

The paper also defines extra structure dual to unit and zero. A counit is a superalgebra morphism

${Truss}$3

satisfying

${Truss}$4

A cozero is a superalgebra morphism

${Truss}$5

satisfying the dual absorber identities. If these exist, they induce on each ${Truss}$6 a multiplicative unit and a zero element, respectively.

5. Examples and standard sources of cotrusses

The paper gives three explicit examples. The minimal one is ${Truss}$7, with the only possible structure maps

${Truss}$8

Then for any ${Truss}$9, $T$0 is a singleton, and the induced truss is trivial (Bruce, 24 Apr 2026).

A more informative example uses

$T$1

The cotruss maps are

$T$2

$T$3

$T$4

For $T$5, write

$T$6

Then the induced multiplication and heap law are

$T$7

$T$8

This is the clearest explicit cotruss in the note: the binary coproduct encodes a super-analogue of multiplication on pairs, while the ternary coproduct is affine and heap-like on both even and odd coordinates. The same example also carries a counit and a cozero, yielding distinguished points $T$9 and $X$0.

A third source is canonical rather than ad hoc. If $X$1 is the Hopf superalgebra of an affine abelian supergroup, then

$X$2

defines a cotruss structure. The paper gives the explicit supergroup example

$X$3

with $X$4, $X$5, and $X$6. This shows that cotrusses arise canonically from Hopf superalgebras as well as from abstract representability.

6. Passage to affine superbraces and Yang--Baxter theory

The cotruss is not an auxiliary device: it is the mechanism by which the paper constructs affine superbraces. If the supercotruss $X$7 has a counit $X$8, then each represented truss $X$9 has a distinguished multiplicative unit

$$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$0

Using Brzeziński’s pointwise construction, the paper defines on $$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$1 the abelian group law

$$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$2

With this choice, each $$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$3 becomes a two-sided semi-brace, and the construction is natural in $$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$4; hence one obtains an affine superbrace

$$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$5

from the cotruss-represented affine supertruss (Bruce, 24 Apr 2026).

The role of cotrusses in the Yang--Baxter discussion is indirect but foundational. The paper defines a Yang--Baxter map as a natural transformation

$$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$6

whose components

$$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$7

satisfy the set-theoretic Yang--Baxter equation. Since $$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$8 and $$T(-)\cong h^X(-):=\operatorname{Hom}_{SAlg_{\mathbb K}}(X,-).$$9 are built from brace operations, and brace operations come from the represented truss structure, the cotruss underlies the entire construction even though the Yang--Baxter equation is not reformulated directly in cotruss language. A plausible implication is that cotrusses function as the coordinate-level infrastructure for extending brace-theoretic Yang--Baxter constructions from sets to affine superschemes.

In the paper’s own conceptual summary, the supercotruss is the actual coordinate-algebra object underlying the theory: it carries the binary coproduct encoding truss multiplication, the ternary coproduct encoding the heap operation, the quantum-heap and coassociativity axioms, and the codistributivity identities dualizing truss distributivity. Within that framework, cotrusses form the structural bridge between trusses, supergeometry, superbraces, and the proposed Yang--Baxter theory (Bruce, 24 Apr 2026).