How to make in R matrix of intersections and unions over categories?

Maybe something like this?

days <- levels(allt$day)

f <- function(x, y) {
  xids <- allt$id[allt$day == x]
  yids <- allt$id[allt$day == y]
  length(intersect(xids, yids)) / length(union(xids, yids))
}
f <- Vectorize(f)

outer(days, days, f)

#      [,1] [,2] [,3]
# [1,]  1.0  0.5  0.2
# [2,]  0.5  1.0  0.5
# [3,]  0.2  0.5  1.0

optionally pipe that into set_colnames(days) and set_rownames(days)