Hi, just curious. Sorry I am trying to get a handle on this , will try to make it more precise: I am trying to see if the following has a categorical parallel/counterpart. Consider the case of measure spaces (X,S,m) : X any space, S a sigma algebra, m a measure and that of metric spaces (Y,d) with ( I would say) continuous maps. Given a metric space (Y,d) , there always exists a measure space associated with it, given by the sigma algebra generated by the open sets ( themselves generated by open metric balls ), and measure is n-dimensional volume. But, a given measure triple (X,S,m) does not necessarily correspond to a metric space (X,d), i.e, (X,S,m) is not necessarily a measure triple associated to (X,d). (Phew!) Is there a way of expressing/describing the above using Category Theory, e.g., can we describe the above in terms of the non-existence of functors between the categories (Metric spaces, Cont. Maps) and (measure spaces with measurable maps)? Thanks, sorry for the rambling.