How to Specify the Topology of Space?

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A modern approach goes back to the idea of the fundamental group a, but replaces it by a something called the fundamental infinity group oid. Owe recover now, every possible path that an ant could take between points in he space. And since i promised i would tell exa listeners what an infinity groupoit is, let me do it now. So this is the classical approach to saying, what is a space? Algebraically, what are some algebra stuff that tell you everything that you would want to know about the homotope type of th space. Depending on your point of view, either it's a theorem or it's a tautology really does capture the full homot

“A way that math can make the world a better place is by making it a more interesting place to be a conscious being.” So says mathematician Emily Riehl near the start of this episode, and it’s a good summary of what’s to come. Emily works in realms of topology and category theory that are far away from practical applications, or even to some non-practical areas of theoretical physics. But they help us think about what is possible and how everything fits together, and what’s more interesting than that? We talk about what topology is, the specific example of homotopy — how things deform into other things — and how thinking about that leads us into groups, rings, groupoids, and ultimately to category theory, the most abstract of them all.

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Emily Riehl received a Ph.D in mathematics from the University of Chicago. She is currently an associate professor of mathematics at Johns Hopkins University. Among her honors are the JHU President’s Frontier Award and the Joan & Joseph Birman Research Prize. She is author of Categorical Homotopy Theory, and co-author of the upcoming Elements of ∞-Category Theory. She competed on the United States women’s national Australian rules football team, where she served as vice-captain.

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