Homotopy cardinality via extrapolation of Morava–Euler characteristics

We answer a question of John Baez, on the relationship between the classical Euler characteristic and the Baez–Dolan homotopy cardinality, by constructing a unique additive common generalization after restriction to an odd prime p. This is achieved by ℓ -adically extrapolating to height n= - 1 the sequence of Euler characteristics associated with the Morava K(n) cohomology theories for (any) ℓ∣ p- 1 . We compute this sequence explicitly in several cases and incorporate in the theory some folklore heuristic comparisons between the Euler characteristic and the homotopy cardinality involving summation of divergent series.