Algebraic interleavings of spaces over the classifying space of the circle

We bring spaces over the classifying space $BS^1$ of the circle group $S^1$ to persistence theory via the singular cohomology with coefficients in a field. Then, the {\it cohomology} interleaving distance (CohID) between spaces over $BS^1$ is introduced and considered in the category of persistent differential graded modules. In particular, we show that the distance coincides with the {\it interleaving distance in the homotopy category} in the sense of Lanari and Scoccola and the {\it homotopy interleaving distance} in the sense of Blumberg and Lesnick. Moreover, upper and lower bounds of the CohID are investigated with the cup-lengths of spaces over $BS^1$. As a computational example, we explicitly determine the CohID for complex projective spaces by utilizing the bottleneck distance of barcodes associated with the cohomology of the spaces.