M Theory Lesson 190

Euler characteristic as an alternating sum is related to inclusion-exclusion

said Scott Carter at the n-Category Cafe today. He attributes the quote to Vassiliev. The $n$-simplices used to calculate ordinary Euler characteristics may be viewed as dual to $n$ intersecting sets. For example, the full intersection of three sets corresponds to the face of a triangle, whereas the three edges of the triangle come from the double intersections. The union of the three sets counts vertices once and edges twice, so one takes away the double intersections and then adds back on the single face of the triple intersection. This parity of simple intersections is what gives the terms in $\chi$ their sign. In M Theory, we like to think of set intersections (or the vector space analogue) as topos theory pullbacks, which turns the triangle into the three faces at the corner of a cube! For mass operators, it is important to look at tricategorical analogues. This is why we study ternary geometry!