Abstract: Consider \(m(A\cup B) = m(A) + m(B) - m(A\cap B)\) and \(m(A\times B) = m(A)\times m(B)\). Here "\(m\)" could be the cardinality of a finite set or a probability measure or (replacing \(\cup\) with \(+\) and the first \(\times\) with \(\otimes\) ) the dimension of a finite dimensional vector space or the Euler characteristic of a compact simplicial complex. Categorically speaking, the equations above indicate well-behavior with respect to certain (co)limits like push-outs and products. About 20 years ago Tom Leinster added \(m =\) the magnitude of a finite (enriched) category to this list. We may regard metric spaces as enriched categories (which will be explained in the talk). Finite metric spaces come up in Graph Theory, Error Correcting Codes and Topological Data Analysis (as data clouds). The central, motivating example for "\(m\)" is the Euler characteristic (topological analog of cardinality). Euler characteristic is also a homotopy invariant, another desirable property. Betti numbers refine Euler characteristic and (co)homology categorifies Betti numbers. This shifts the emphasis from spaces to maps and yields proofs of fundamental results like the Brouwer and Lefschetz fixed point theorems, the Borsuk-Ulam theorem, etc. We'll chat a little about "categorification" and "de-categorification". After defining (enriched) categories, the magnitude of finite and compact subsets of Euclidean space and the magnitude function, we'll work out a few examples and mention some properties. We may get to magnitude (co)homology. Time permitting, the concept of path homology of a di(rected )graph, generalizing simplicial homology (Grigoryan-Lin-Muranov-Yau, 2012) and its recently established relation to magnitude homology will be mentioned. While some familiarity with algebraic topology (homology, homotopy) may help to appreciate a few of the results, these concepts will not be used in the talk. Matrices and Euclidean space are assumed to be known, other relevant notions will be defined.