Can anyone explain the persistent absolute cohomology bar codes? how are the indices defined in absolute persistent cohomology?
For example, 
corresponds to the filtration $X_1 \subset ... \subset X_6$
Recall we have the persistent module: $H^*(X_1) \leftarrow ... \leftarrow H^*(X_{5}) \leftarrow H^*(X_6)$
there should thus be absolute cohomology barcodes: $\{[1,\infty)_0, [2,3)_0, [4,5)_1, [6,\infty)_2\}$ where the subscript refers to dimension of the generating cocycle.
How do we explain the barcodes for this example?
The existence of a barcode of the form $[a,b)_n$ establishes that there is an $n$-cocycle with non-trivial cohomology class arising at time $a$ that persists up to time $b$.
For instance, the barcode $[2,3)_0$ corresponds to the $0$-cocycle associated to the point $2$. Note that $H^0(X_2)=\langle 1,2\rangle \simeq \Bbb{Z}^2$, where $1$ and $2$ denote the corresponding $0$-cocycles. The cochain $2$ is no longer a cocyle in $H^2(X_3)$, since its differential is not zero.