uconf.constructions.bar_construction

Bar construction for connected dg-operads.

The bar construction B(P) of a connected augmented dg-operad P is the cofree conilpotent cooperad on the suspension of the augmentation ideal:

B(P) = (T^c(sP̄), d_1 + d_2)

where: - P̄ is the augmentation ideal (P̄(1) = 0 for connected operads) - sP̄ denotes the suspension used here (degree shift by +1 per internal vertex) - T^c denotes the cofree conilpotent cooperad (decorated rooted trees) - d_1 is the internal differential from P - d_2 is the structural differential from edge contractions

Note

This module requires a connected operad input. Connectedness means P(0) = 0 and P(1) = k (unit only), so every internal tree vertex has arity >= 2. For a tree with n leaves this bounds the number of internal vertices by n - 1, making the basis in every (arity, degree) finite.

Reference: Loday-Vallette “Algebraic Operads”, Chapter 6.

Classes

BarConstruction(operad_cls)

Factory for bar construction components of a connected dg-operad.

class uconf.constructions.bar_construction.BarConstruction(operad_cls)

Bases: UniqueRepresentation

Factory for bar construction components of a connected dg-operad.

Parameters:

operad_cls (OperadLike) – Base operad provider (class or wrapper instance). Must be a connected operad (P(0) = 0, P(1) = k·unit).

The bar construction B(P) is a dg-cooperad whose arity-n component has basis elements given by rooted trees with n leaves, where internal vertices are decorated by elements of P̄ (the augmentation ideal).

For connected operads, P̄(1) = 0, so all internal vertices have arity >= 2. This bounds the number of internal vertices in arity n by n - 1, making every (arity, degree) basis finite without requiring an external weight cap.

property connectivity: int

Connectivity of the bar construction.

For an operad P with connectivity k (degrees >= k*(n-1)), the minimum bar degree of a single-vertex tree in arity n is k*(n-1) + 1. The connectivity of B(P) as a cooperad is therefore k + 1 in the sense that B(P)(n) is concentrated in degrees >= (k+1)*(n-1) + 1 (for k >= 0). We store k here as a reference value derived from the underlying operad.

static counit(x)

Cooperadic counit at the factory level.

Parameters:

x (Element)

static reduced(x)

Reduced projection at the factory level.

Parameters:

x (Element)

Return type:

Element

static infinitesimal_cocompose(x, i, m, n)

Infinitesimal cocomposition at the factory level.

Parameters:

• x (Element)

• i (int)

• m (int)

• n (int)

counit_element(base_ring)

Return the counit element (single leaf tree in arity 1).

The bar construction B(P) has a canonical counit ε: B(P)(1) → k, which is non-trivial on the single-leaf tree with no internal vertices. This method returns that generator.

Return type:

Element

unit_key()

Return the basis key of the counit generator in arity 1.

In the bar construction, arity-1 elements are the single-leaf tree, whose basis key is the integer 1.

Return type:

int

BarConstruction.Component

alias of Component

BarConstruction.Element

alias of Element