(also nonabelian homological algebra)
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
Where a linear persistence module, as traditionally considered in persistent homology theory, is a diagram of vector space/modules of the directed form
in a zig-zag persistence module one allows the linear maps to go in either direction, such as in regular zigzags
or in more general zigzags such as
In the language of quiver theory, zigzag persistence modules are precisely the quiver representations of A-type quivers. But in quiver representation theory the classical Gabriel theorem says (see there) that for all possible zig-zag patterns as above the indecomposable quiver representations are still “interval modules”, ie. those for which the $V_i$ are zero except in some interval where they are 1-dimensional and connected by (zigzags of) identity maps.
This means that the key property of persistence modules – namely that they are entirely characterized by persistence barcodes, i.e. by multisets of such intervals – immediately generalizes to zigzag persistence modules.
A typical type of a zigzag persistence module appearing in the practice of topological data analysis consists of homology groups $H(X_i)$ of stages $\cdots \hookrightarrow X_{i - 1} \hookrightarrow X_i \hookrightarrow X_{i + 1} \hookrightarrow \cdots$ of a filtered topological space, as usual, but evaluating now on the zigzag of inclusions into the unions
of consecutive filter stages:
It is claimed (…) that such zigzag persistence modules still retain the same persistent information of interest, but are more robust.
As the level varies, the well groups of a continuous function naturally form a zigzag persistence module, called a well module (Edelsbrunner, Morozov & Patel 2011group#EMP11)).
The concept of zigzag persistence modules as a tool in topological data analysis and their relation to quiver representation theory (including a re-proof of Gabriel's theorem for the case of A-type quivers) is due to:
Application to level sets:
Gunnar Carlsson, Vin de Silva, Dmitriy Morozov, Zigzag persistent homology and real-valued functions, in: SCG ‘09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 247–256 $[$doi:10.1145/1542362.1542408$]$
Gunnar Carlsson, Vin de Silva, Sara Kališnik, Dmitriy Morozov, Parametrized Homology via Zigzag Persistence, Algebr. Geom. Topol. 19 (2019) 657-700 $[$arXiv:1604.03596, doi:10.2140/agt.2019.19.657$]$
Review:
The algebraic stability theorem for zigzag persistence modules:
Last revised on May 20, 2022 at 09:49:21. See the history of this page for a list of all contributions to it.