Functorial Framework for Persistent Homology

• The paper establishes that persistent homology exhibits local functoriality through a categorical framework that leverages geometric symmetries, yet global functoriality is obstructed by non-geometric transformations.
• It defines enriched data-set categories and Grothendieck graphs to systematically encode the structure and symmetries of data, enabling decomposition into canonical blocks.
• The work demonstrates that set-equivariant operators facilitate the formation of natural transformations, offering practical tools to analyze symmetry-induced features in persistence modules.

A functorial framework for persistent homology provides a categorical language for encoding, transferring, and enriching the algebraic and topological invariants derived from data via persistent homology. Central to this approach is the identification and exploitation of functoriality: the property that persistence invariants behave coherently under morphisms of data, often via the structure of categories, functors, and natural transformations. Functorial frameworks allow for an explicit accounting of symmetries, equivariances, decompositions, and the precise scope and limitations of persistent homology as a categorical invariant. Recent advances formalize this perspective by introducing categories of enriched data sets, colored directed graphs, and set-equivariant operators; providing precise statements and counterexamples regarding the global functoriality of persistent homology; and establishing foundational theorems for decomposing such structures into canonical blocks, all as detailed in "Landscapes of Data Sets and Functoriality of Persistent Homology" (Chacholski et al., 2020).

1. Enriched Data-Set Categories and Morphisms

An enriched data set—termed an "incarnation"—is a pair $(\Phi, M)$, where $\Phi=\{\varphi_1,\ldots,\varphi_m\}$ is a finite set of real-valued functions (measurements) on a finite domain $X$, and $M \subseteq \operatorname{End}_{\Phi}(X)$ is a chosen set of allowed domain endomorphisms that preserve the set $\Phi$ under composition, i.e., for every $g \in M$, $\varphi \circ g \in \Phi$ for all $\varphi \in \Phi$. The collection $\operatorname{End}_{\Phi}(X)$ forms the structure monoid of $\Phi$, and the pair $(\Phi, M)$ is referred to as a monoid incarnation if $M$ is a submonoid, or group incarnation if $M$ is a subgroup.

Morphisms between incarnations are set-equivariant operators (SEOs) $(\alpha, T):(\Phi, M) \to (\Psi, N)$, where $\alpha:\Phi \to \Psi$ and $T:M \to N$ satisfy the condition: $$\alpha(\varphi \cdot g) = \alpha(\varphi) \cdot T(g)$$ for all $\varphi \in \Phi, g \in M$. If $T$ is a homomorphism of monoids/groups, the SEO is said to be monoid-equivariant (MEO) or group-equivariant (GEO), respectively.

Operations such as product and coproduct in the base category of data sets lift to incarnations, and a decomposition theorem (Proposition 6.1) asserts that every incarnation $(\Phi, M)$ canonically splits (up to SEO-isomorphism) as a coproduct of block-incarnations according to the orbits of $M$ on $\Phi$ (Chacholski et al., 2020).

2. Grothendieck Graph Data Structure

Each incarnation $(\Phi, M)$ is encoded by a Grothendieck graph $G=(V=\Phi, M, E)$, where the edge set $$E = \{ (\varphi, g, \psi)\mid \psi = \varphi \circ g, g \in M \}$$ is colored by elements of $M$. From every vertex $\varphi$ and color $g$, there is a unique outgoing $g$-edge, satisfying a bijectivity condition on projections. This structure is formalized as the category $\operatorname{Gr}_M V$, whose morphisms are composable triples with group or monoid multiplication in the color. Morphisms of Grothendieck graphs correspond to pairs $(\alpha, T)$ as above that preserve the edge structure.

The assignment $(\Phi, M) \mapsto (\Phi, M, E)$, together with SEOs as morphisms, defines a fully faithful embedding of the category of incarnations ("Nirvana") into the category of colored directed graphs ("GGraph") (Chacholski et al., 2020).

3. Decomposition Theorems and Morphism Formation

The categorical structure supports fine decompositional and morphism-forming properties:

• Bases and Dimension: A basis for $(\Phi, M)$ is an independent set $\Omega \subset \Phi$ generating $\Phi$ under $M$-action, with all bases sharing the same cardinality ("dimension"). In the transitive group case, dimension is $1$.
• Morphism Extension: Every SEO is determined by its action on a basis, subject to a relation-preserving condition: for compatible relations in the source, their images under $\alpha$ and $T$ must maintain those relations in the target. This yields unique SEO extensions (Proposition 4.8).
• Block Diagonalization: Every incarnation $(\Phi, M)$ decomposes as a coproduct of its $M$-orbit blocks, with the corresponding SEO given by the identity on blocks and the diagonal action of $M$ across blocks (Proposition 6.1).

These properties encode the intrinsic symmetry and orchestrate the classification of morphisms between incarnations (Chacholski et al., 2020).

4. Global vs. Local Functoriality of Persistent Homology

Persistent homology, defined as $$PH_d^{\Phi}:(\Phi, M, E)^{op}\to Tame([0,\infty)\times \mathbb{R},Vect)$$, assigns to each measurement the persistent homology module and to each domain action the induced homology map. However, this assignment fails to extend to a global functor $Nirvana \to Fun(-, Tame)$: non-geometric SEOs lack canonical maps between Vietoris–Rips complexes, as demonstrated by a sign-flip counterexample that destroys comparability between the persistent modules of the transformed data (Chacholski et al., 2020). Thus, persistent homology is not a functor on the entire category of incarnations; its global functoriality is obstructed except for the subcategory of geometric SEOs.

5. Local Functoriality via Grothendieck Graphs

Despite failures at the global level, for each fixed incarnation $(\Phi, M)$, the assignment $\varphi \mapsto PH_d(\varphi)$ and $(\varphi, g, \varphi g) \mapsto PH_d^{-g}(\varphi)$ (i.e., functorially pulling back along domain symmetries) yields a genuine contravariant functor from the Grothendieck graph $(\Phi, M, E)^{op}$ to $Tame([0,\infty)\times \mathbb{R}, Vect)$. The local functoriality theorem asserts that such confined functoriality is always respected, and the composition rules hold by the functorial properties of domain maps (Chacholski et al., 2020).

6. Role of Equivariant Operators and Symmetry Extension

Set-equivariant operators allow for the comparison of structures not visible to persistent homology. Geometric SEOs (i.e., those corresponding to actual maps between domains) enable the definition of natural transformations between persistent homology functors, preserving structure. Non-geometric SEOs—such as those corresponding to unit changes or domain modifications—encode symmetries acting beyond the scope of persistent homology functoriality but organize complementary invariants. By composing persistent homology on incarnations with SEOs, both functorial and equivariant analytical tools are available, broadening the invariant landscape associated with a data set (Chacholski et al., 2020).

7. Synthesis, Examples, and Implications

The functorial framework subsumes the classical pipeline of persistent homology: starting with a data set of functions, endowing it with symmetries $M$ to form an incarnation, and then forming the Grothendieck graph to encode those symmetries. Persistent homology then acts functorially on this object at the local (fixed-incarnation) level, even as global functoriality is precluded by non-geometric transformations. Canonical decomposition theorems guarantee that every incarnation splits into $M$-orbits; morphisms are uniquely determined by their effects on bases.

A central example is the case where $X \subset S^1$, and $M$ is generated by rotation, so the persistent homology functor becomes $S^1$-equivariant and detects $H_1$-classes invisible in the absence of symmetry. Consequently, the persistent homology of data sets with internal symmetries is categorically positioned: local functoriality is preserved along geometric symmetries, while the broader organizational role of SEOs structures all measurement-based operators, revealing and situating the extent of the functorial reach of persistent homology within a rigorous categorical framework (Chacholski et al., 2020).

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References:

• "Landscapes of Data Sets and Functoriality of Persistent Homology" (Chacholski et al., 2020)