Time-travelling visualization answers how the predictions of a deep classifier are formed during the training. It visualizes in two or three dimensional space how the classification boundaries and sample embeddings are evolved during training. In this work, we propose TimeVis, a novel time-travelling visualization solution for deep classifiers. Comparing to the state-of-the-art solution DeepVisualInsight (DVI), TimeVis can significantly (1) reduce visualization errors for rendering samples' travel across different training epochs, and (2) improve the visualization efficiency. To this end, we design a technique called temporality spatialization, which unifies the spatial relation (e.g., neighbouring samples in single epoch) and temporal relation (e.g., one identical sample in neighbouring training epochs) into one high-dimensional topological complex. Such spatio-temporal complex can be used to efficiently train one visualization model to accurately project and inverse-project any high and low dimensional data across epochs. Our extensive experiment shows that, in comparison to DVI, TimeVis not only is more accurate to preserve the visualized time-travelling semantics, but also 15X faster in visualization efficiency, achieving a new state-of-the-art in time-travelling visualization.