Morita Traces: Invariance in Algebra & Topology

• Morita traces are invariant trace maps that remain unchanged under Morita equivalence, unifying distinct settings in algebra, topology, and quantum theory.
• They are constructed via bicategorical shadows, ensuring functoriality and adherence to trace-associativity and unit coherence properties.
• Practical applications include invariants in mapping class groups, homology cobordisms, and fusion categories, leading to robust quantum topology measures.

Morita traces are a generalization of classical trace maps in algebra and topology, characterizing the invariance and functorial properties of trace-like operations under Morita equivalence in a bicategorical framework. They unify and extend classical invariants (notably Hochschild homology traces and categorical dimensions) across algebraic, topological, and quantum settings. The notion of Morita trace arises naturally in contexts ranging from mapping class and cobordism groups to higher representation theory and topological quantum field theory, and is tightly linked with the theory of bicategorical shadows, cyclic homology, and equivalence invariance (Hess et al., 2021, Klanderman et al., 2023, Fuchs et al., 13 Jul 2024).

1. Traces and Morita Invariance: Historical and Conceptual Origins

The classical concept of trace arises in several paradigms, such as the Hattori–Stallings trace in algebraic $K$-theory, or the Dennis trace from algebraic $K$-theory to Hochschild homology. Central to these constructions is Morita invariance: the property that the trace is unchanged under passage to Morita equivalent rings or categories—in other words, that it is an invariant of the equivalence class of (bi)modules rather than their specific model (Hess et al., 2021, Klanderman et al., 2023).

In low-dimensional topology, Morita introduced trace-like invariants encoding the behavior of Johnson homomorphisms and their generalizations, and later extended these to 1-cocycles on groups of homology cobordisms, unifying the trace data of the mapping class group with the structure of infinite-dimensional representation spaces (Massuyeau et al., 2016).

2. Morita Traces via Bicategorical Shadows

The modern categorical framework for Morita traces is provided by the theory of bicategorical shadows. Given a bicategory $B$ (objects, 1-cells, and 2-cells), a shadow is the data of a family of functors $\{\cdot\}_X: B(X,X) \to D$ (one for each 0-cell $X$) together with natural isomorphisms satisfying "trace-associativity" and unit coherence conditions. The paradigm ensures that trace-like constructions are functorial under horizontal composition and, crucially, invariant under adjoint equivalences (known as Morita equivalences in classical settings) (Hess et al., 2021).

Universal Property and Examples

The category of shadows on $B$ with values in $D$ is equivalent to the category of functors out of the Hochschild homology object (universal shadow) $\mathrm{HH}(B)$, defined as the appropriate colimit of the bar resolution (Hess et al., 2021). This generalizes cyclic invariants (e.g., the free-loop groupoid for a bigroupoid $B$) and underpins the universality of Morita invariance for trace maps.

3. Explicit Constructions: Algebraic and Topological Contexts

Mapping Class Groups and Homology Cobordisms

Morita’s original trace maps on the Johnson homomorphism are equivariant maps $ITr_k: \mathfrak{h}_k^\to S^k(H^)$, vanishing on the image of the $k$th Johnson homomorphism and surjective for odd $k\geq3$. The traces are constructed via derived Fox calculus and symplectic representation theory, and satisfy vanishing/surjectivity properties indexed by degree parity (Massuyeau et al., 2016).

A new 1-cocycle $$Ab^\theta: H \to \widehat{H_1}(\widehat{\mathfrak{h}}^+)$$ is defined on the group $H$ of homology cobordisms, with strict functorial (group cocycle) and invariance properties, extending the Johnson filtration to the full group. The leading homogeneous part recaptures the classical Morita trace, while composition with $ITr$ recovers the classical trace structure and Johnson filtration.

Notably, $Ab^\theta$ encodes Magnus representation data and tree-level (LMO) quantum invariants. Applications include the construction of finite-type invariants that detect nontriviality in the rational abelianization of the group $H$ (Massuyeau et al., 2016).

Trace Methods for coHochschild Homology

In coalgebraic settings, the Hattori–Stallings trace and its colinear analog for finitely generated comodules can be formulated via shadows in the bicategory of bicomodules (Klanderman et al., 2023). Both the coHochschild homology $\mathrm{coHH}$ and the corresponding traces are Morita–Takeuchi invariant: under Morita–Takeuchi equivalence, $\mathrm{coHH}(C)\simeq \mathrm{coHH}(D)$ and the traces coincide (Klanderman et al., 2023).

4. Morita Traces in Fusion Categories and Quantum Topology

In higher representation theory, Nakayama-twisted traces provide the fundamental Morita-invariant trace functional on bimodule categories. For pivotal (and unimodular) tensor categories, these traces satisfy cyclicity, nondegeneracy, and compatibility with partial traces. When pivotal structure and unimodularity are imposed, left and right traces coincide (“trace-sphericality”) and yield Calabi–Yau structures, enabling the construction of 3-manifold invariants (such as Turaev–Viro state sums) that are manifestly invariant under pivotal Morita equivalence (Fuchs et al., 13 Jul 2024).

Evaluative Example

For the group algebra $H = \Bbbk[C_2]$ in characteristic two, the ordinary quantum trace may vanish identically, whereas the Nakayama-twisted trace is manifestly nondegenerate—demonstrating genuine Morita-invariant sensitivity beyond classical centers (Fuchs et al., 13 Jul 2024).

5. Key Properties and Theorems

Universal Morita-Invariance

The theory of bicategorical shadows guarantees that any shadow (i.e., trace) construction sends Morita (adjoint) equivalences to isomorphisms. The free adjunction bicategory formalizes universal Morita-invariance via Hochschild homology functoriality, and all shadow-like invariants (including traces in $K$-theory, Hochschild, and cyclic homology) emanate as functors out of the universal shadow (Hess et al., 2021, Klanderman et al., 2023).

Finite-Type and Representation-Theoretic Aspects

Morita traces encode the Johnson filtration and provide explicit finite-type invariants distinguishing cosets beyond the mapping class subgroup. The structure of invariants (e.g., in the rational abelianization of homology cobordism groups) is governed by infinite-dimensional representation theory developed from Morita’s original traces (Massuyeau et al., 2016).

In automorphism groups of free groups, the trace cocycles and associated filtrations (Andreadakis, lower central series) have concrete algebraic consequences, ensuring surjectivity properties in the graded Lie algebra structure after rationalization (Massuyeau et al., 2016).

6. Significance and Broader Applications

Morita traces unify a spectrum of invariants across algebra, topology, and quantum field theory, mechanizing trace properties that are robust under categorical and derived equivalence. They provide the categorical underpinnings for quantum invariants of 3-manifolds, ensure structure preservation for higher traces (Dennis, Hattori–Stallings), and anchor the notion of shadow/invariant in modern higher category theory.

These constructions bridge the gap between classical algebraic invariants and their categorified and derived analogs, laying the groundwork for systematic generalizations in $(\infty,2)$-categorical frameworks, and supporting the development of homotopically Morita-invariant topological Hochschild and cyclic homologies (Hess et al., 2021, Klanderman et al., 2023).