Emily Riehl JohnsHopkinsUniversity Categorifying cardinal arithmetic MAA MathFest, August 4, 2018 by taking a tour of some deep ideas from category theory. • Step 1: categorification • Step 2: the Yoneda lemma • Step 3: representability • Step 4: the proof • Epilogue: what was the point of that? Plan Goal: prove 𝑎×(𝑏+𝑐) = (𝑎×𝑏)+(𝑎×𝑐) for any natural numbers 𝑎, 𝑏, and 𝑐. • Step 1: categorification • Step 2: the Yoneda lemma • Step 3: representability • Step 4: the proof • Epilogue: what was the point of that? Plan Goal: prove 𝑎×(𝑏+𝑐) = (𝑎×𝑏)+(𝑎×𝑐) for any natural numbers 𝑎, 𝑏, and 𝑐 by taking a tour of some deep ideas from category theory. • Epilogue: what was the point of that? Plan Goal: prove 𝑎×(𝑏+𝑐) = (𝑎×𝑏)+(𝑎×𝑐) for any natural numbers 𝑎, 𝑏, and 𝑐 by taking a tour of some deep ideas from category theory. • Step 1: categorification • Step 2: the Yoneda lemma • Step 3: representability • Step 4: the proof Plan Goal: prove 𝑎×(𝑏+𝑐) = (𝑎×𝑏)+(𝑎×𝑐) for any natural numbers 𝑎, 𝑏, and 𝑐 by taking a tour of some deep ideas from category theory. • Step 1: categorification • Step 2: the Yoneda lemma • Step 3: representability • Step 4: the proof • Epilogue: what was the point of that? 1 Step 1: categorification Q: What is the deeper meaning of the equation 𝑎×(𝑏+𝑐) = (𝑎×𝑏)+(𝑎×𝑐)? Q: What is the role of the natural numbers 𝑎, 𝑏, and 𝑐? The idea of categorification The first step is to understand the equation 𝑎×(𝑏+𝑐) = (𝑎×𝑏)+(𝑎×𝑐) as expressing some deeper truth about mathematical structures. Q: What is the role of the natural numbers 𝑎, 𝑏, and 𝑐? The idea of categorification The first step is to understand the equation 𝑎×(𝑏+𝑐) = (𝑎×𝑏)+(𝑎×𝑐) as expressing some deeper truth about mathematical structures. Q: What is the deeper meaning of the equation 𝑎×(𝑏+𝑐) = (𝑎×𝑏)+(𝑎×𝑐)? The idea of categorification The first step is to understand the equation 𝑎×(𝑏+𝑐) = (𝑎×𝑏)+(𝑎×𝑐) as expressing some deeper truth about mathematical structures. Q: What is the deeper meaning of the equation 𝑎×(𝑏+𝑐) = (𝑎×𝑏)+(𝑎×𝑐)? Q: What is the role of the natural numbers 𝑎, 𝑏, and 𝑐? A: Natural numbers define the cardinalities, or sizes, of finite sets. Natural numbers 𝑎, 𝑏, and 𝑐 encode the sizes of finite sets 𝐴, 𝐵, and 𝐶. 𝑎 ≔ |𝐴|, 𝑏 ≔ |𝐵|, 𝑐 ≔ |𝐶|. Categorifying natural numbers Q: What is the role of the natural numbers 𝑎, 𝑏, and 𝑐?