Some monads on Set have algebras that can be thought of as data structures labeled by elements of a set. For example, the free monoid on a set X can be thought of as the collection of finite lists whose elements are in X. The list monad T has a tensorial strength, a transformation t_{X, Y}: X x TY -> T(X x Y), natural in X and Y. The strength pairs x with each element of the list M: t_{X, Y}(x, M) = T( y |-> (x, y) )(M). There are other transformations for which I'd like to know names and what coherence laws they have to satisfy. The first is a transformation u_{X, Y}: T(X x Y) -> (X -> TY), natural in Y and dinatural in X. In the case of lists, u_{X, Y}(M) = x |-> flatten( T( (x',y) |-> [y] if x = x', [] otherwise )(M) ). That is, it filters out those pairs in M that do not have x as the first element of the pair and lists the second elements of the pairs that do. The second is a transformation v_{X, Y}: T(X -> Y) -> (X -> TY) natural in X and Y. In the case of lists, v_{X, Y}(M) = x |-> T( f |-> f(x) )(M) ). That is, it applies each f in M to x. I've worked out a few of the coherence laws relating these three transformations, but I'd like to be sure I haven't missed any. -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]