[Paper Review] Additive versus abelian 2-representations of fiat 2-categories

Volodymyr Mazorchuk, Vanessa Miemietz|arXiv (Cornell University)|Dec 21, 2011

Algebraic structures and combinatorial models12 references4 citations

TL;DR

This paper establishes a bridge between additive and abelian 2-representations of fiat 2-categories by introducing an abelianization 2-functor, enabling the construction of new 2-representations through additive closures in abelian settings. The key contribution is a characterization of cell 2-representations via quotients of fiat 2-categories by 2-ideals, proving that the image of a fiat 2-category on a cell 2-representation is 'J-simple'—a 2-categorical analogue of simplicity in representation theory.

ABSTRACT

We study connections between additive and abelian 2-representations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell 2-representation. We also describe, in detail, examples of fiat 2-categories associated to $\mathfrak{sl}_2$-categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur algebras.

Motivation & Objective

• To unify additive and abelian 2-representation theories of fiat 2-categories by introducing a systematic abelianization 2-functor.
• To clarify the relationship between additive 2-representations and their abelianizations, especially in the context of constructing new 2-representations.
• To characterize the annihilator of a cell 2-representation and establish its role in the structure of fiat 2-categories.
• To extend the theory of cell 2-representations to the additive setting and show equivalence with abelian counterparts via projective objects.
• To provide a new construction of cell 2-representations as quotients of fiat 2-categories by 2-ideals associated to two-sided cells.

Proposed method

• Use multisemigroups to encode the combinatorics of 2-categories, with isomorphism classes of 1-morphisms forming a multisemigroup under horizontal composition.
• Define Green's relations (left, right, two-sided cells) on multisemigroups, showing they correspond to the classical cell structures in 2-categories.
• Construct the abelianization 2-functor that maps additive 2-representations to abelian ones, preserving categorical structure.
• Apply the abelianization process to principal additive 2-representations and extract additive subrepresentations via additive closure of stable object sets.
• Prove that every abelian 2-representation is equivalent to the abelianization of the additive 2-subrepresentation generated by projective objects.
• Use the notion of 2-ideals and quotients to define J-simplicity of the image of a fiat 2-category on a cell 2-representation, establishing a 2-categorical analogue of simplicity.

Experimental results

Research questions

• RQ1 How can additive and abelian 2-representations of fiat 2-categories be systematically related via a 2-functorial construction?
• RQ2 What is the role of the abelianization 2-functor in generating new 2-representations from additive ones?
• RQ3 How do the annihilators of cell 2-representations relate to the structure of two-sided cells in fiat 2-categories?
• RQ4 To what extent do cell 2-representations in the additive setting mirror their abelian counterparts, particularly in terms of simplicity and 2-ideals?
• RQ5 Can cell 2-representations be reconstructed as quotients of fiat 2-categories by 2-ideals associated to two-sided cells?

Key findings

• The abelianization 2-functor allows the construction of new 2-representations by taking additive closures of sets of objects stable under the action of the 2-category in the abelianized setting.
• Every abelian 2-representation of a fiat 2-category is equivalent to the abelianization of the additive 2-subrepresentation generated by its projective objects.
• The image of a fiat 2-category under a cell 2-representation is J-simple, meaning any nontrivial 2-ideal necessarily annihilates the representation.
• Cell 2-representations can be reconstructed as quotients of the fiat 2-category modulo the 2-ideal corresponding to the two-sided cell, providing an alternative construction.
• For the 2-category D_{n,r} associated to Schur algebras, the left and right cells are classified by the row and column entries of semistandard Young tableaux, respectively.
• The decategorification of D_{n,r} is isomorphic to the Schur algebra S(n,r), with indecomposable projective functors corresponding to Du’s canonical basis via the Robinson-Schensted-Knuth correspondence.