I am currently reading the following paper by Gunnar Carlsson: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/BB0DA0F0EBD79809C563AF80B555A23C/S0962492914000051a.pdf/topological_pattern_recognition_for_point_cloud_data.pdf.
However, I am facing a certain issue with trying to reconcile the homology groups of persistent homology with the barcode diagrams. I have been stuck on it for weeks but I can't seem to make any progress. I know that as we cover a finite set of points with balls of increasing radius we get the following commutative diagram of chains:
where $VK(\sum_{n}X,R_l)$ is the free vector space generated by the $n$-simplices formed by the balls of radius $R_l$. Since the diagram is commutative we have the following persistent vector space of homology groups:
In Gunnar Carlsson's notes(Pg:316-312), he describes the correspondence between finitely presented persistence vector spaces and barcodes. However, I can't seem to link the persistent vector space as shown above to some finitely presented persistent vector space. Could some please explain that portion?