Relating Brauer categories, Koszul complexes, and graph complexes

Abstract: The purpose of this paper is to investigate the relationship between hairy graph complexes associated to cyclic operads and their counterparts for operads (and, more generally, dioperads). This is based on the author's interpretation of these as Koszul complexes for the associated modules over the respective appropriate twisted downward (walled) Brauer category. The general question of relating such Koszul complexes is addressed by analysing the relationships between the respective twisted Brauer-type categories, proceeding through a direct analysis. The passage from the walled to unwalled context involves functors induced by the disjoint union of finite sets. As an application, for the cyclic operad associated to an operad, this leads to an explicit relation between the respective (hairy) graph homologies.

• The paper introduces a unified framework that relates twisted Brauer categories, Koszul complexes, and graph homologies, bridging diverse operadic structures.
• The paper establishes explicit functorial correspondences between cyclic operads, operads, and dioperads using twisted linearizations and Day convolution.
• The paper verifies isomorphisms and stabilization phenomena via explicit tensor identities and exact sequences in operadic homology.

Synthesis of Brauer Categories, Koszul Complexes, and Graph Complexes

Introduction

This work develops a unified framework relating several advanced objects in algebraic topology and representation theory through categorical and homological methods. Specifically, it elucidates the relationships between "hairy" graph complexes stemming from cyclic operads and dioperads, their Koszul complexes, and module categories constructed via twisted variants of the (walled and unwalled) Brauer categories. The construction enables explicit comparisons between graph homologies associated to cyclic operads, operads, and dioperads, synthesizing and extending structures from classical and modern perspectives: graph complexes à la Kontsevich, operadic and dioperadic constructions, and categorical approaches using Day convolution and twisted module categories.

Brauer Categories and Twisted Linearizations

The paper provides a precise presentation of both standard and twisted forms of the upward (ub), downward (db), and walled (uwb/dwb) Brauer categories, motivated by applications in representation stability and the configuration of graph complexes. Key to the construction is the systematic use of graded $k$-linearizations and twisted module structures, where $k$ is a field of characteristic zero. The twisted linearizations involve orientation data (sign representations vs trivial representations) both for the order of edges/pairs and, in the walled case, for the separation of objects.

The categories are constructed as homogeneous quadratic categories over suitable groupoid (often $k$-linearizations of symmetric groups), and the relationships between walled and unwalled, upward and downward, and ordered and unordered categories are clarified using functors induced by disjoint union and ordering data.

The explicit description of generating morphisms and structure functors (unwalling, walling, transfer, section, and restriction) establishes the basis for relating different diagram categories and their module categories.

Operadic Structures and Module Categories

The framework accommodates both cyclic operads and (walled) dioperads, unified via Day convolution. The composition structures in cyclic operads and dioperads (modulo units) are encoded through symmetric and exterior powers under convolution, with structure morphisms derived from those of the respective operads. The paper provides a categorical comparison between these structures by explicitly relating modules over the (twisted) Brauer categories to operadic data.

The passage from cyclic operad $C$ to dioperad $\amalg^* C$, and reciprocally from dioperad $D$ to cyclic operad $\amalg_* D$, is analyzed in detail, including how the module structures and associated Koszul complexes are transformed via these functors. Notably, these correspond to restriction and induction along the functor induced by disjoint union between categories of finite sets.

Koszul Complexes and Their Comparisons

A systematic account is given of the Koszul complexes for modules over the (twisted) Brauer categories. The paper distinguishes between the "first" and "second" Koszul complexes, defined via canonical bimodule resolutions and their duals, and connects these to classical interpretations (e.g., hairy graph complexes and Chevalley-Eilenberg complexes).

The main technical results show, by explicit construction, how the second Koszul complex of a cyclic operad or a dioperad can be described in terms of one another through natural isomorphisms or inclusions involving the functors $d_{(\pm;\mp)}$ and $d^{*}_{(\pm;\mp)}$ (that transfer between dioperadic and operadic/cyclic contexts). The comparison uses explicit combinatorics of disjoint unions and dualities in the setting of $k$-linear categories.

It is established that the second Koszul complexes for cyclic operads correspond to the usual hairy graph complexes, and their comparison with those from associated dioperads or underlying operads is given by explicit morphisms of complexes (isomorphisms in the dioperad-to-cyclic direction, inclusions in the cyclic-to-dioperad direction). The impact of units (augmentation properties) and positivity (vanishing on boundary cases) for operads and dioperads is analyzed to determine when these morphisms are isomorphisms or merely inclusions.

Strong Structural Results

A highlight is the explicit calculation (Corollary on p. 153 of the paper) for operads with unit $k$0: $k$1 with vanishing of all other homology groups. This asserts that, in this case, the (non-hairy) graph homologies computed via the operadic and cyclic operadic contexts are naturally isomorphic, a formalization of classical stabilization phenomena (Lie algebra derivations of free $k$2-algebras for symplectic vector spaces).

The paper also gives tensor identities and exact sequences relating homologies in the presence of units or positivity assumptions, using the full machinery of module categories over Brauer-type categories.

Theoretical and Practical Implications

The comparative framework developed has several theoretical implications:

• It provides a unified categorical mechanism to compute and relate hairy and non-hairy graph homologies in various operadic settings, capturing both classical and modern constructions in a single technical apparatus.
• The explicit functorial connections between cyclic operads, operads, and dioperads allow the transfer of homological computations and conceptual insights between these settings, which is particularly useful for studying representation stability, graph homologies, and deformation theory.
• The machinery of twisted Brauer categories and their bimodule Koszul complexes opens up generalizations to higher genus graph complexes, potential applications to modular operads, and further categorical abstraction (e.g., derived Koszul duality over operadic or modular settings).

Practically, the results provide concrete homological tools for calculating stable and unstable homologies in graph complexes and for relating algebraic invariants of various flavors of operads and their associated algebraic structures (e.g., derivations, Lie algebra homologies).

Conclusion

This work synthesizes categorical and homological aspects of the relationships between Brauer categories, operadic and dioperadic structures, and graph complexes. By formulating exact functorial transformations and explicit Koszul complex comparisons, it enables a precise transfer of graph homology computations between cyclic operads, operads, and dioperads. The technical apparatus built on twisted module categories and Day convolution is robust and flexible, suggesting further generalizations for deeper study in both representation theory and homological algebraic topology.