Combinatorial Approach to Categorical Möbius Inversion and Magnitude

Explore a 54-minute lecture on the combinatorial approach to categorical Möbius inversion and magnitude, presented by Juan Pablo Vigneaux for the Applied Algebraic Topology Network. Delve into the numerical invariant of enriched categories known as magnitude, introduced by T. Leinster in 2008, and its definition using "Möbius coefficients" in the spirit of Rota's combinatorial theory. Discover a novel combinatorial interpretation of Möbius coefficients, expressed as quotients of sums indexed by specific path collections within categories. Examine how this result encompasses previous theorems by P. Hall for posets and T. Leinster for categories "without nontrivial cycles." Consider potential variations of magnitude homology suggested by this approach, opening avenues for further research and development in the field of applied algebraic topology.