The multi-filtration and the multi-persistence on a total object

Table of Contents

Discussion and Verdict

I have observed a tendency to conflate, or at least structurally identify, (multi-) filtration and (multi-) persistence within Topological Data Analysis (TDA) literature¹. This occurs even though they are distinct concepts that arguably should remain theoretically separate.

Here, I discuss multi-filtration on a total object $C\in\mathcal{A}$ . This can be formulated as a family of filtrations $F^\alpha_\ast \in \mathcal{A}^I, \alpha=1,2,\ldots$ on a fixed total object $C$ , where $\mathcal{A}$ is an abelian category and $I$ is a filtered category. This notion gives rise to a “spectral decomposition” of $C$ from multiple perspectives—where each filtration is responsible for “slicing” along a specific parameter—while behaving coherently along the indices $\alpha$ .

Although this naive, intrinsic construction of “multi-filtration” generally fails to be a $k$ -filtration ( $k\geq 2$ ) under the standard definition², the spirit is similar: slicing a complex object into smaller, comprehensible pieces from multiple perspectives. In many practical applications, we might assume $C$ is an algebraic object (e.g., a homology or homotopy group) and $F^\alpha\to C$ imposes an exhaustive filtration of subobjects. Through this, the total object is recovered by completion; namely $\displaystyle{\varprojlim_i C/F^\alpha_i \approx C}$ up to quasi-isomorphism for each $\alpha$ . The difficulty in extending this to multi-dimensional filtration lies in coherence, specifically where the resulting $k$ -filtration structure forces the internal filtrations to commute.

In this framework, multi-persistence on a total object $C$ (prototypically a global, ill-posed point cloud) is defined differently. It captures higher-dimensional (i.e., multiple) features (generally in $\mathbb{R}^n$ ) that admit a sense of “robustness” for measuring the topological invariants of $C$ , alongside a complete track of feature growth. In this context, we assume $C$ is a point cloud or cluttered object, often devoid of inherent algebraic structure.

Here is a comparison summary (partially generated by Gemini 3.0 Pro):

Feature	Multi-Filtration	Multi-Persistence
Primary Context	Algebraic Topology / Homotopy Theory	Data Analysis / Applied Topology
Input Object $C$	Algebraic (Complex, Module, Group)	Geometric (Point Cloud, Image)
Primary Tool	Spectral Sequences (Iterative homological slicing)	Representation Theory of Quivers (Decomposition)
Key Output	Convergence (or say, limit. Recovering $C$ from slices)	Invariants (Barcodes, Betti numbers)
Concept of Indices	Filtered category, giving a sense of distinct, often orthogonal perspectives (e.g., “Weight” vs “Hodge”)	Ordered dimensions of a single parameter space (e.g., $\mathbb{R}^n$ )
Wildness	Generally avoided by imposing “Strictness” conditions (Deligne).³ For more concrete case, see Rips Bifiltrations in An Introduction to Multiparameter Persistence.	Unavoidable or the solutions are not known; requires new invariants (Rank, Sheaves)

Top-Down view

Several approaches interpret concrete objects, such as (multi-)persistence modules, through the lens of abstract theory, specifically spectral sequences.

In the single-parameter setting, the dimensional equivalence (i.e., Betti numbers)⁴ between the spectral sequence associated with a filtration and the persistence module is established in Spectral Sequences, Exact Couples and Persistent Homology of Filtrations by Saugata Basu et al.

However, for the multi-persistence case (with two or more parameters), I have not found extensive results beyond dimensional equivalence, which is canonically deduced from the single-parameter results (c.f., Computing Invariants for Multipersistence via Spectral Systems by A. Guidolin et al.).

(e.g., The Theory of Multidimensional Persistence by Gunnar Carlsson and Afra Zomorodian, An Introduction to Multiparameter Persistence by Magnus Bakke Botnan and Michael Lesnick)
For the consistence of k-derived objects, refer to Cartan-Eilenberg Resolution and similar discussions.
As of 2025, the core taming result of Deligne — Splitting Theorem — has been generalized by the works of Bondarko et al. in multiple ways, especially to the Triangulated Category under some conditions.
along with mild relationships fitting specific forms of exact sequences