Weak Comp Algebra in Secondary Hochschild Cohomology

• Weak comp algebra is a graded structure on secondary Hochschild complexes that organizes differentials, cup products, and higher operations via partial compositions and a distinguished 2-cochain.
• It arises from entwining structures over a commutative base, where an algebra, a coalgebra, and a twisting map interact to control cohomological operations and deformation theory.
• The introduction of an equivariant subcomplex refines the weak comp structure to a full comp algebra, yielding a Gerstenhaber algebra on cohomology with merged cup product operations.

Weak comp algebra, in the sense developed for secondary Hochschild cohomology of entwining structures, is a graded algebraic structure carried by the secondary Hochschild complex $C^*(A,B,C,\varphi)$ associated with an entwining structure $(A,B,C,\psi,\varphi)$ over a commutative $k$-algebra $B$. Its defining data are partial composition operations $\circ_i$ and a distinguished element $\alpha\in V^2$, and its role is to organize the differential, cup products, and higher algebraic operations on cochains. In this setting, the secondary Hochschild complex becomes a right weak comp algebra, admits two cup products on cohomology, and contains a subcomplex on which the weak structure strengthens to a comp algebra; the cohomology of that subcomplex forms a Gerstenhaber algebra (Balodi et al., 2019).

1. Entwining structures over a commutative base

An entwining structure over $k$ is a triple $(A,C,\psi)$ consisting of an associative unital $k$-algebra $A$, a coassociative counital $(A,B,C,\psi,\varphi)$0-coalgebra $(A,B,C,\psi,\varphi)$1, and a $(A,B,C,\psi,\varphi)$2-linear map

$(A,B,C,\psi,\varphi)$3

subject to four compatibility conditions. Writing $(A,B,C,\psi,\varphi)$4, these are: $(A,B,C,\psi,\varphi)$5

$(A,B,C,\psi,\varphi)$6

$(A,B,C,\psi,\varphi)$7

$(A,B,C,\psi,\varphi)$8

Over a commutative $(A,B,C,\psi,\varphi)$9-algebra $k$0, the relevant datum is a tuple

$k$1

where $k$2 is a $k$3-algebra map with central image $k$4, $k$5 is an entwining structure over $k$6, and

$k$7

The coefficient objects are $k$8-bimodules $k$9 satisfying the $B$0-centrality condition

$B$1

This framework combines an algebra $B$2, a coalgebra $B$3, and a twisting map $B$4 with a distinguished commutative base algebra $B$5. The condition involving $B$6 forces the $B$7-part of the structure to commute with the entwining in the strongest possible way (Balodi et al., 2019).

2. Secondary Hochschild complex

For $B$8, the secondary cochain group attached to $B$9 and coefficients $\circ_i$0 is

$\circ_i$1

The tensor factors are written as upper triangular tensor matrices. For $\circ_i$2, one writes

$\circ_i$3

with diagonal entries in $\circ_i$4 and upper triangular entries in $\circ_i$5.

The differential is defined by transporting Staic’s secondary Hochschild differential through the canonical isomorphism

$\circ_i$6

Here $\circ_i$7 is endowed with the $\circ_i$8-bimodule structure

$\circ_i$9

With this bimodule structure, $\alpha\in V^2$0 is a $\alpha\in V^2$1-central $\alpha\in V^2$2-bimodule, and the resulting operator $\alpha\in V^2$3 satisfies $\alpha\in V^2$4. The cohomology groups are denoted

$\alpha\in V^2$5

In this form, the secondary Hochschild complex generalizes ordinary Hochschild cohomology, Staic’s secondary Hochschild cohomology, and the Hochschild-type cohomology attached to entwining structures (Balodi et al., 2019).

3. Weak comp algebra structure

A right weak comp algebra is a graded $\alpha\in V^2$6-vector space $\alpha\in V^2$7, together with an element $\alpha\in V^2$8 and operations

$\alpha\in V^2$9

such that $k$0 for $k$1, the relation

$k$2

holds, the relation

$k$3

holds when either $k$4 or $k$5, and

$k$6

On the secondary Hochschild complex, the paper defines operations

$k$7

by explicit block-matrix formulas, and introduces the distinguished $k$8-cochain

$k$9

The resulting system satisfies the weak comp algebra axioms. Precisely,

$(A,C,\psi)$0

is a right weak comp algebra.

The differential is recovered from $(A,C,\psi)$1 and the insertion operations by the identity

$(A,C,\psi)$2

This is the secondary analogue of the classical expression of the Hochschild differential in terms of the multiplication cochain (Balodi et al., 2019).

A comp algebra is the stronger structure in which the insertion operations satisfy the full Gerstenhaber–Schack associativity pattern for all relative positions of iterated insertions. Every comp algebra is a weak comp algebra, but not conversely.

4. Cup products and cohomology

The weak comp structure yields two cochain-level cup products,

$(A,C,\psi)$3

on $(A,C,\psi)$4. Their explicit formulas are given in terms of block tensor matrices, products of the $(A,C,\psi)$5-entries through $(A,C,\psi)$6, and the two coproduct components $(A,C,\psi)$7 of the coalgebra element $(A,C,\psi)$8. The difference between them is in how the comultiplication of $(A,C,\psi)$9 is assigned to the two cochains and where the entwining $k$0 acts.

These products admit compact descriptions in terms of $k$1 and the insertion operations: $k$2

$k$3

for $k$4, $k$5.

Both products are graded associative at the cochain level, and the differential is a graded derivation for each: $k$6

$k$7

Hence both descend to cohomology,

$k$8

On cohomology they satisfy

$k$9

Thus the secondary Hochschild cohomology of the entwining structure carries two distinct cup product structures. Their coexistence is a characteristic feature of the weak comp algebra formalism in this setting (Balodi et al., 2019).

5. Equivariant subcomplex and the passage to comp algebras

The paper isolates an equivariant subcomplex

$A$0

defined by compatibility of cochains with the $A$1-bicomodule structure determined by the entwining maps $A$2 and $A$3. Concretely, a cochain belongs to $A$4 when a diagram involving $A$5, $A$6, $A$7, and the cochain itself commutes.

This subcomplex has several decisive properties. The distinguished element $A$8 belongs to $A$9, and $(A,B,C,\psi,\varphi)$00 is closed under all operations $(A,B,C,\psi,\varphi)$01. Consequently,

$(A,B,C,\psi,\varphi)$02

is a weak comp subalgebra of

$(A,B,C,\psi,\varphi)$03

and $(A,B,C,\psi,\varphi)$04 is a subcomplex.

The key simplification occurs at the level of cup products: for $(A,B,C,\psi,\varphi)$05 and $(A,B,C,\psi,\varphi)$06,

$(A,B,C,\psi,\varphi)$07

The paper then considers this subcomplex as the locus on which the two cup products coincide and on which the stronger comp algebra axioms hold. The abstract states that this subcomplex satisfies the axioms for being a comp algebra, and that the cohomology of this subcomplex forms a Gerstenhaber algebra (Balodi et al., 2019).

This passage from the full weak comp algebra to the equivariant comp algebra is structurally important. On the full complex, the two cup products remain distinct; on the equivariant subcomplex, they merge into a single product compatible with the standard Gerstenhaber package.

6. Deformations and mathematical position

The weak comp algebra formalism is not introduced only to organize cochain operations. It is also tied to deformation theory. The paper constructs a bicomplex that controls the deformations of the entwining structure over $(A,B,C,\psi,\varphi)$08. In this sense, the secondary Hochschild complex of $(A,B,C,\psi,\varphi)$09 plays the same role for entwining structures over a commutative base that ordinary Hochschild complexes play for associative algebras and that secondary Hochschild complexes play for $(A,B,C,\psi,\varphi)$10-parametrized algebra structures.

Within this framework, weak comp algebra occupies an intermediate position. It is weaker than a comp algebra, because the insertion identities are only imposed in full when one of the entries is the distinguished element $(A,B,C,\psi,\varphi)$11, yet it is strong enough to recover the differential from $(A,B,C,\psi,\varphi)$12, define two graded associative cup products, and make the differential a graded derivation for both. The equivariant subcomplex then restores the stronger comp algebra regime and yields a Gerstenhaber algebra on cohomology.

Accordingly, “weak comp algebra” denotes not merely a weakened operadic gadget, but the specific algebraic mechanism by which secondary Hochschild cochains of entwining structures over $(A,B,C,\psi,\varphi)$13 support insertion operations, multiple cup products, and deformation-theoretic control in a single formalism (Balodi et al., 2019).