uconf.constructions.cobar_construction¶
Cobar construction for connected dg-cooperads.
The cobar construction Ω(C) of a connected coaugmented dg-cooperad C is the free operad on the desuspension of the coaugmentation coideal:
Ω(C) = (T(s⁻¹C̄), d_1 + d_2)
where: - C̄ is the coaugmentation coideal (C̄(1) = 0 for connected cooperads) - s⁻¹C̄ denotes the desuspension used here (degree shift by -1 per internal vertex) - T denotes the free operad (decorated rooted trees) - d_1 is the internal differential from C - d_2 is the structural differential from vertex expansions
Note
This module requires a connected cooperad input. Connectedness means C(0) = 0 and C(1) = k (counit 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
|
Factory for cobar construction components of a connected dg-cooperad. |
- class uconf.constructions.cobar_construction.CobarConstruction(cooperad_cls)[source]¶
Bases:
UniqueRepresentationFactory for cobar construction components of a connected dg-cooperad.
- Parameters:
cooperad_cls (CooperadLike) – Base cooperad provider (class or wrapper instance). Must be a connected cooperad (C(0) = 0, C(1) = k·counit).
The cobar construction Ω(C) is a dg-operad whose arity-n component has basis elements given by rooted trees with n leaves, where internal vertices are decorated by elements of C̄ (the coaugmentation coideal).
For connected cooperads, C̄(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.
Note
Trees are automatically normalized to shuffle form via
to_shuffle_tree_cobarin the element constructor, analogous to the bar construction.- property connectivity: int¶
Connectivity inherited from the underlying cooperad.
- unit(base_ring)[source]¶
Return the unit element (identity in arity 1).
For the free operad, the unit is represented by a single leaf.
- Return type:
- unit_key()[source]¶
Return the basis key of the unit element in arity
1.In the cobar construction, the arity-1 unit is the single-leaf tree, whose basis key is the integer
1.- Return type:
int
- compose(x, i, y)[source]¶
Free operad composition: graft y onto leaf i of x.
In the free operad
T(s⁻¹C̄), composition is tree grafting with a Koszul sign from inserting y’s vertex factors into x’s DFS linearisation:x ∘_i y = (-1)^{|y| · A(x, i)} · graft(x, i, y)where
A(x, i)is the total cobar degree of internal vertices of x that come after leaf i in DFS order.