## 1 Introduction
### Quest from Single-Cell Resolution Data  
With the increasingly popular usage of single-cell genomic techniques, it might be possible to infer such cell-cell interaction (or cellular ecology) in a fine resolution. However, as far as we are aware, there are only a few works in the literature exploring the cellular ecology from single-cell RNA sequencing data. For instance, the authors of [21; 22] applied the ecology and multi-agent models to model single-cell systems. We wish to complement this line of work by connecting it to topological data analysis, where the focus is to model the shape or manifold of the data from the similarity of the data points. For instance, simplicial complexes are high-dimensional objects or generalizations of neighboring graphs that represent the cliques of data points or, in other words, a notion of _ecology_. The ecology does not have to be the organisms within a physical system. In the field of data science where we represent biological cells by their measurements (e.g., gene expression profiles) as data points residing in high-dimensional feature spaces, the ecology can be how these data points are connected to one another in the feature space. If we adopt an ecology research point of view, in order to characterize the dynamic systems of a community, one needs to have knowledge or prior knowledge regarding the causal relationships between the agents (e.g., how prey and predators interact and in what ways). In order to parse out causal relationships, the temporal sequence of these events matters. Thus, the property of the synchrony and asynchrony of the events is key to translating the feature space (represented by a similarity graph) to an ecology, which has directed (e.g., causal) relationships among the agents. This is why a temporal take on topological data analysis can potentially unlock the first step, from finding a static representation of the overall shape of the data points to discovering the event-directed representations (i.e., a temporal skeleton) of the data points.  
One challenge of this hybrid direction is to conceptually understand the biological meaning behind the dissimilarity of the omic data. For instance, what does it mean if two cells have similar gene expression profiles to each other? Does that indicate homogeneity if the two cells are from the same tissue, or is it an artifact from the manual labeling or classifications not being perfect? Can we measure the \"complexity\" of the cell populations based on the heterogeneity or diversity within populations? If we can, how do we evaluate and interpret lower-order versus higher-order \"complexity\"? These are some open questions we wish to engage the field to discuss and investigate together instead of answering them directly in this first work.  
The other challenge is the scalability and compariablity of the single-cell data. With the advancements in multi-channel, high-throughput data collection techniques in biological fields, how do we compute the pairwise distances of the point clouds efficiently? In different trials of single-cell experiments, how do we make sure that the persistent modules are comparable to one another?