Representation of algebraic theories and non-expanding maps

Finite product preserving full faithful functors of monosorted finitary algebraic theories into categories with finite products are investigated. One consequence of a general method presented here is the result that for every triple m≤n≤p of natural numbers, there exist metric spaces X1=(P,d1) and X2=(P,d2) such that the categories Nk(Xi) formed by all non-expanding maps between finite powers Xi0, Xi1, …, Xik−1 of Xi with i=1,2 have the following properties: