Invariant Morphisms in Algebra and Geometry

• Invariant Morphisms are morphisms that preserve key structures and operations, ensuring compatibility under endomorphisms and automorphisms.
• They play a crucial role in module theory, operator algebras, and persistent homology by refining matching invariants and enabling classification theorems.
• Their application across algebra, topology, and logic facilitates transfer of properties between structures, driving advances in both theoretical and applied research.

An invariant morphism is a morphism characterized by its preservation or compatibility with distinguished structures—such as submodules, invariants, or operations—under relevant algebraic, categorical, or model-theoretic transformations. The concept spans several mathematical contexts, notably module theory, category theory, operator algebras, persistent homology, and many-sorted algebraic logic, where invariance properties of morphisms yield both classification results and connections between algebraic or categorical structures. The definition and study of invariant morphisms are central to understanding how algebraic, operator-theoretic, or categorical data behave under the action of endomorphism groups, automorphisms, or other internal transformations.

1. Invariant Morphisms in Module Theory

In the module-theoretic context, a submodule $K \subset M$ is called invariant under a class of morphisms (endomorphisms or automorphisms) of $M$ if it is preserved by those morphisms. This leads to the following structure:

• For a monomorphism $u: K \rightarrow M$, define two subrings:
  • $$\mathrm{End}_R^M(K) = \{f \in \mathrm{End}_R(K) \mid \exists\, g \in \mathrm{End}_R(M),\ g \circ u = u \circ f\}$$
  • $$\mathrm{End}_R^K(M) = \{g \in \mathrm{End}_R(M) \mid \exists\, f \in \mathrm{End}_R(K),\ g \circ u = u \circ f\}$$

The subring $\mathrm{End}_R^K(M)$ comprises endomorphisms of $M$ preserving $u(K)$, i.e., $K$ is invariant under these endomorphisms.

The main structural result is that if $u: K \rightarrow M$ is a left minimal monomorphism (no non-isomorphic endomorphism of $M$ fixes $u$), then there is a ring isomorphism

$$\mathrm{End}_R^M(K)/J(\mathrm{End}_R^M(K)) \cong \mathrm{End}_R^K(M)/J(\mathrm{End}_R^K(M))$$

where $J$ denotes the respective Jacobson radicals. This allows transfer of properties between endomorphism rings of $K$ and $M$ under invariance conditions.

Specifically, if $K$ is invariant under all endomorphisms of $M$ (i.e., fully invariant), or under all automorphisms (i.e., automorphism-invariant), one obtains isomorphisms of endomorphism rings modulo radicals with strong ring-theoretic consequences. For example, fully invariant submodules in injective or cotorsion envelopes inherit von Neumann regularity and idempotent lifting properties from the envelope to the submodule endomorphism ring (Cortés-Izurdiaga et al., 2020).

2. Invariance in Homological and Persistent Module Categories

In persistent homology and the theory of persistence modules, invariance of morphisms manifests in the construction of invariants that refine the correspondence between barcodes of modules under morphisms. Given a morphism $f: V \to U$ between persistence modules, the operator $M_f$ defines a partial matching between the barcodes $\mathrm{Bar}(V)$ and $\mathrm{Bar}(U)$, encoding in a functorial and additive manner how $f$ transforms interval decompositions.

The induced block-matrix $M_f$ is strictly finer than the information given by the image module $fV$ or the rank invariant, and is additive with respect to direct sums of morphisms. The construction ensures invariance of the matching under algebraically meaningful operations on the persistence modules (e.g., direct sum, interpretable as categorical invariance), and can even subsume previously established partial matchings in specific cases (e.g., Morse filtrations) (Gonzalez-Diaz et al., 2020).

3. Invariant Morphisms and Zero-Dimensionality in Operator Algebras

In the setting of $C^*$-algebra classification, invariant morphisms are realized as *-homomorphisms with prescribed nuclear dimension and compatibility with algebraic invariants (the "total invariant"). A *-homomorphism $\varphi: A \rightarrow B$ is called zero-dimensional (or has nuclear dimension zero) if it can be approximated arbitrarily well by maps factoring through a finite-dimensional $C^*$-algebra, or equivalently, through an AF-algebra.

The total invariant, denoted as

$$\mathrm{KTu}(A) = (K_*(A), K_*(A; \mathbb{Z}/n), T(A), K_1^{\mathrm{alg}}(A), \Pi_A, \Theta_A)$$

enables complete classification of zero-dimensional *-homomorphisms via the induced morphism on invariants. Under real rank zero and torsion-freeness conditions, nuclear dimension zero can be detected purely at the level of the total invariant, specifically by triviality of $K_1$-maps, evidencing a deep interaction between morphism properties and homological invariants (Castillejos et al., 2023).

4. Categorical and 2-Categorical Perspectives: Polyderivors, Transformations, and 2-Institutions

In categorical logic and many-sorted universal algebra, invariant morphisms are formalized through the notion of polyderivors—generalized signature morphisms assigning families of derived terms to sorts and operations. The structure extends to a 2-category $\mathbf{Sig}_{\mathfrak{pd}}$, where:

• Objects are many-sorted signatures.
• 1-cells are polyderivors.
• 2-cells are transformations between polyderivors (natural families of terms satisfying compatibility conditions).

These transformations preserve invariance of term realization across the induced translations, and the construction leads to strict 2-institutions (in the sense of Goguen and Burstall), generalizing the classical institution framework.

The 2-functoriality ensures that realization of terms in algebras is invariant under the action of polyderivors and compatible with their transformations. This cements invariance properties in the algebraic realization of formal specifications and supports equivalences between different classes of many-sorted algebraic theories (e.g., Hall and Bénabou specifications) (Vidal et al., 2012).

5. Invariant Tensors and Congruent Morphisms in Information Geometry

In the geometric theory of statistical models, invariant morphisms manifest in the classification of tensor fields that are invariant under congruent Markov morphisms (i.e., morphisms admitting a right inverse up to a statistic). On the space of measure roots, the canonical symmetric multilinear forms $L^n_\Omega$ serve as universal invariants, and any congruent family of $n$-tensors is algebraically generated by these canonical tensors. In particular, the only invariant $2$-tensor is the Fisher metric, and the only invariant $3$-tensor is the Amari-Chentsov tensor; for general $n$, all invariants are linear combinations of the pullbacks of these canonical forms (Schwachhöfer et al., 2017).

This identification of invariant tensors under congruent morphisms yields a structural understanding of information geometry, unifying classical and higher-order invariants.

6. Examples, Applications, and Future Directions

Several settings illustrate the depth and breadth of invariant morphisms:

• Invariant submodules in module categories lead to isomorphism theorems for endomorphism rings and lifting results for idempotents, with explicit applications in the theory of quasi-injective and pure-injective modules (Cortés-Izurdiaga et al., 2020).
• In persistent homology, the induced matching $M_f$ provides a strictly finer invariant than classical rank or image-based invariants, and is crucial for distinguishing morphisms in topological data analysis (Gonzalez-Diaz et al., 2020).
• In $C^*$-algebra theory, factorization through AF-algebras determined by invariant-theoretic data enables unified classification theorems for morphisms (Castillejos et al., 2023).
• The 2-categorical framework for polyderivors and term realization supports logical invariance and model-theoretic transfer in many-sorted algebraic logic (Vidal et al., 2012).
• Invariant tensors in information geometry encode the unique statistical structures preserved under congruent Markov morphisms (Schwachhöfer et al., 2017).

Open directions include the classification of higher-dimensional morphisms in operator algebras, categorical generalizations of invariance principles (notably for non-simple or non-unital structures), and further development of invariance concepts in higher category theory and logic. The transfer of ring-theoretic and homological properties via invariant morphisms remains a fruitful area for exploration in both algebra and geometry.