uconf.algebraic.configuration

Euclidean configuration model via operadic bar construction.

This module implements the configuration model for computing the real homotopy type of the ordered/unordered configuration spaces of a smooth closed manifold, following the construction in article.tex.

Overview

The construction builds a chain complex whose homology computes the factorization homology of a manifold model. The pipeline is:

Layer 0: Basic operads
  sLie = ShiftedOperad(Lie, -1)    # s⁻¹Lie, degree shift -(n-1) at arity n
  XsLie = HadamardProduct(sLie, Surjection)   # H = s⁻¹Lie ⊙ Surjection

Layer 1: Bar/cobar cooperad/operad
  BXsLie = BarConstruction(XsLie)   # cooperad C = B(H), connectivity -1
  OBXsLie = CobarConstruction(BXsLie)  # operad P = Ω(C), quasi-free resolution

Layer 2: Manifold model and free algebra
  manifold_model = SurjectionSphereCochainAlgebra(dim, R)   # Surjection-algebra
  coefficients = TrivialModule(dim, R)     # k[dim], rank-1 module
  free_alg = FreeOperadAlgebra(OBXsLie, coefficients)  # free P-algebra

Layer 3: Tensor and pullback
  tensor_alg = HadamardTensorAlgebra(manifold_model, free_alg)
     # (Surj ⊙ P)-algebra on (cochains ⊗ free_alg)
  comodule_morphism: P → Surj ⊙ P   # via E-comodule + table reduction
  pulled_back = PullbackAlgebra(comodule_morphism, tensor_alg)
     # P-algebra via pullback along comodule_morphism

Layer 4: Final bar algebra
  pi = canonical_projection(P)   # twisting morphism π: B(P) → P
  bar = BarAlgebra(pi, pulled_back)   # B_π(pulled_back)

The result bar is a C-coalgebra whose underlying dg-module bar.module can be truncated by weight and degree to compute homology via compute_chain_complex().

Key classes used

Functions

euclidean_unordered_configuration_model(...)

Build the configuration model for Euclidean space R^d.

labelled_configuration_model(manifold_model, ...)

Build the labelled configuration model for a manifold model.

unordered_configuration_model(...)

Build the unordered configuration model with a trivial coefficient module.

Classes

ConfigurationLayers(manifold_model, ...)

Typed container for intermediate configuration-model layers.

TrivialModule(dimension, base_ring)

Rank-1 graded module concentrated in a single degree.
class uconf.algebraic.configuration.ConfigurationLayers(manifold_model, coefficients, sLie, XsLie, BXsLie, OBXsLie, free_alg, tensor_alg, comodule_morphism, pulled_back, pi, bar)[source]

Bases: object

Typed container for intermediate configuration-model layers.

Parameters:
uconf.algebraic.configuration.labelled_configuration_model(manifold_model, coefficients)[source]

Build the labelled configuration model for a manifold model.

Given a Surjection-algebra manifold_model (encoding the manifold’s real homotopy type) and a coefficient module, this constructs the bar algebra whose homology computes the factorization homology.

Parameters

manifold_model:

A Surjection-algebra, e.g. from SurjectionSphereCochainAlgebra.

coefficients:

A graded module with boundary and connectivity, used as the label space for the free algebra.

Returns

BarAlgebra

The bar construction B_π(pulled_back). Its .module gives the underlying dg-module for homology computation.

Parameters:

  • manifold_model (OperadAlgebra)

  • coefficients (CombinatorialFreeModule)

class uconf.algebraic.configuration.TrivialModule(dimension, base_ring)[source]

Bases: CombinatorialFreeModule

Rank-1 graded module concentrated in a single degree.

The module has a single basis element β with degree = dimension and trivial (zero) boundary. Used as the coefficient module for unordered configuration models.

Parameters:

dimension (int)

TrivialModule.Element[source]

alias of Element

uconf.algebraic.configuration.unordered_configuration_model(manifold_model, dimension)[source]

Build the unordered configuration model with a trivial coefficient module.

Parameters

manifold_model:

A Surjection-algebra encoding the manifold’s real homotopy type.

dimension:

Dimension of the manifold. Used to set the degree of the trivial coefficient module k[dimension].

Returns

BarAlgebra

The bar construction B_π(pulled_back) with coefficients in k[d].

Parameters:
uconf.algebraic.configuration.euclidean_unordered_configuration_model(base_ring, dimension)[source]

Build the configuration model for Euclidean space R^d.

Uses the sphere cochain algebra as the manifold model (since S^d is the one-point compactification of R^d).

Parameters

base_ring:

Coefficient ring (e.g. QQ, GF(2)).

dimension:

Dimension d ≥ 0.

Returns

BarAlgebra

The configuration model. Use model.module for the underlying dg-module and compute_chain_complex() to build a chain complex for homology computation.

Parameters:

dimension (int)