Intending to introduce a method for the topological analysis of fields, we present a pipeline that takes as an input a weighted and based chain complex, produces a factored chain complex, and encodes it as a barcode of tagged intervals (briefly, a tagged barcode). We show how to apply this pipeline to the weighted and based chain complex of a gradient-like Morse-Smale vector field on a compact Riemannian manifold in both the smooth and discrete settings. Interestingly for computations, it turns out that there is an isometry between factored chain complexes endowed with the interleaving distance and their tagged barcodes endowed with the bottleneck distance. Concerning stability, we show that the map taking a generic enough gradient-like vector field to its barcode of tagged intervals is continuous. Finally, we prove that the tagged barcode of any such vector field can be approximated by the tagged barcode of a combinatorial version of it with arbitrary precision.

翻译：为引入一种针对向量场的拓扑分析方法，我们提出了一套处理流程：该流程以加权带基链复形为输入，生成分解链复形，并将其编码为带标记区间的条形码（简称标记条形码）。我们展示了如何将该流程应用于紧致黎曼流形上梯度类Morse-Smale向量场的加权带基链复形，涵盖光滑与离散两种情形。值得关注的计算特性是：赋予交错距离的分解链复形与赋予瓶颈距离的标记条形码之间存在等距关系。在稳定性方面，我们证明了将足够一般的梯度类向量场映射至其标记条形码的过程具有连续性。最后，我们论证了任意此类向量场的标记条形码均可通过其组合版本的标记条形码以任意精度逼近。