Many topics in unsupervised learning can be viewed as dealing with the relative geometry of data. In mathematics, topology and homotopy theory are the fields that deal with similar kinds of questions. Using ideas, techniques, and language from topology can prove fruitful for unsupervised learning. This talk will look at how topological approaches can be brought to bear upon unsupervised learning problems as diverse as dimension reduction, clustering, anomaly detection, word embedding, and metric learning. Through the lens and language of topology and category theory we can draw common threads through all these topics, pointing the way toward new approaches to these problems. I hope to convince you that topological approaches offer a rich and growing field of research for unsupervised learning.