uconf.models.cocommutative

Cocommutative cooperad: linear dual of the commutative operad.

CoCommutative(n) is the arity-n component of Com^*, rank-one in each arity n >= 1 with zero differential. The unique basis element is () (empty tuple) and the infinitesimal cocomposition satisfies Δ^{i;m,n}(e_{m+n-1}) = e_m ⊗ e_n for all valid i, dualising Com.compose(e_m, i, e_n) = e_{m+n-1}.

Classes

CoCommutative(n, base_ring)

Cocommutative cooperad component in fixed arity.

class uconf.models.cocommutative.CoCommutative(n, base_ring)

Bases: Commutative

Cocommutative cooperad component in fixed arity.

This is the linear dual of uconf.commutative.Commutative. As a graded module it is identical to Commutative; the additional cooperad structure is provided by counit(), reduced(), and infinitesimal_cocompose().

Parameters:

n (int)

static counit(x)

Cooperadic counit: extracts the coefficient of the arity-1 generator.

The counit ε: CoCom(1) → k evaluates to 1 on the unique basis element () and to 0 for all other arities.

Parameters:

x (Element)

static reduced(x)

Project to the reduced part (kills the arity-1 unit component).

Parameters:

x (Element)

Return type:

Element

static infinitesimal_cocompose(x, i, m, n)

Partial cocomposition dual to Commutative.compose(·, i, ·).

Since Com.compose(e_m, i, e_n) = e_{m+n-1} for all i, the dual satisfies Δ^{i;m,n}(e_{m+n-1}) = e_m ⊗ e_n.

Parameters:

• x (Element)

• i (int)

• m (int)

• n (int)

factory

alias of CoCommutative

CoCommutative.Element

alias of Element