[Paper Review] (Co)-Induced Two Crossed Modules

Ummahan Ege Arslan, Z. Arvasi+1 more|arXiv (Cornell University)|Jul 21, 2011

Organometallic Complex Synthesis and Catalysis6 references3 citations

TL;DR

This paper introduces the concept of (co)-induced 2-crossed modules, generalizing Brown and Higgins' induced crossed modules to higher dimensions. It constructs induced and co-induced 2-crossed modules via pullbacks and pushouts in the category of 2-crossed modules, providing a framework for applying the 3-dimensional Van Kampen Theorem with explicit algebraic descriptions of the resulting structures and their relations to pre-crossed modules and Peiffer subgroups.

ABSTRACT

We introduce the notion of an induced 2-crossed module, which extends the notion of an induced crossed module (Brown and Higgins).

Motivation & Objective

• To extend the theory of induced crossed modules to 2-crossed modules, enabling applications in 3-dimensional algebraic topology.
• To define and construct induced and co-induced 2-crossed modules using pullbacks and pushouts in the category of 2-crossed modules.
• To provide an algebraic framework for computing non-abelian homotopy 3-types via the 3-dimensional Van Kampen Theorem.
• To generalize the universal properties of induced and co-induced structures from crossed modules to 2-crossed modules.
• To describe the structure of pushouts of 2-crossed modules in terms of normal closures of specific relations in group constructions.

Proposed method

• Constructs the induced 2-crossed module φ∗(M) as a quotient of the free Q-group QM by the Peiffer subgroup and the action compatibility relation (q, p·m) = (qφ(p), m).
• Defines the co-induced (pullback) 2-crossed module φ∗(N) as the subgroup {(n, p) ∈ N × P | v(n) = φ(p)} with induced P-action and structure maps.
• Uses the pushout construction in X2Mod/(M,P) to build the 2-crossed module L from induced (M→P)-2-crossed modules Bi = (θi)∗(Li).
• Describes the resulting 2-crossed module L as B/S, where B is the pushout of groups and S is the normal closure of specific elements involving Peiffer brackets and actions.
• Applies the universal property of pullbacks and pushouts to ensure functoriality and compatibility with morphisms of pre-crossed modules.
• Verifies the axioms of 2-crossed modules (e.g., Peiffer relations, action compatibility) through explicit computation in the group-theoretic construction.

Experimental results

Research questions

• RQ1 How can the notion of induced crossed modules be generalized to 2-crossed modules in dimension 3?
• RQ2 What is the algebraic structure of the induced 2-crossed module along a group homomorphism φ: P → Q?
• RQ3 How can co-induced (pullback) 2-crossed modules be constructed from a given pre-crossed module and a group homomorphism?
• RQ4 What is the pushout of 2-crossed modules in terms of group presentations and normal subgroups?
• RQ5 How do the induced and co-induced 2-crossed modules satisfy the universal properties analogous to those in the crossed module case?

Key findings

• The induced 2-crossed module φ∗(M) is constructed as a quotient of the free Q-group QM by the Peiffer subgroup and the action compatibility relation, yielding a well-defined 2-crossed module over Q.
• The co-induced (pullback) 2-crossed module φ∗(N) is realized as the subgroup {(n, p) ∈ N × P | v(n) = φ(p)} with induced P-action and structure maps, satisfying the universal property of pullbacks.
• The pushout of 2-crossed modules is described as L = B/S, where B is the pushout of the underlying groups and S is the normal closure of elements involving Peiffer brackets and action relations.
• The construction ensures that the resulting 2-crossed module satisfies the axioms of a 2-crossed module, including the Peiffer identity and compatibility of the action with the boundary map.
• The induced 2-crossed module φ∗(M) satisfies a dual universal property: any morphism from M to a Q-module factors uniquely through φ∗(M).
• The paper establishes that the functor φ∗: X2Mod/P → X2Mod/Q is left adjoint to the pullback functor φ∗, extending the classical adjunction to the 2-crossed module setting.