Dear Andree, I admire very much the pioneer work of Charles Ehresmann on structures, but I think faithful+amnestic is too weak. if S is contained in X the inclusion S --> X satisfies these two properties . Let X se the category of finite sets and S be the full subcategory of finite ordinals. Do you know any structure on finite sets such that 1 has that structure and some other singleton doesn't? Thus if an inclusion is a structure S should be a replete subcategory of X. Dear Steve, My question was NOT what is a structure on an object of a category, but what is a structure FUNCTOR p: S --> X. Topological spaces are the objects of many DIFFERENT categories S, e.g. with morphisms continuous, open, closed, proper maps, each of these categories is equipped with a functor with codomain Sets. All these functors coincide on the objects, but their properties AS FUNCTORS are completely different. The situation is the same with toposes with geometric, open, closed, and proper morphisms and the structure functorS with codomain in CAT. (As you see neither points nor whales are involved; but they could be with Fred's help ). Dear Thomas, I totally agree with your definition. which is confirmed by George Janelidze. I asked the question because I have seen the word structure used with various meanings; and sometimes without any meaning at all.In particular in the nLab. I hope the category community agrees with this definition. I shall wait a few days to see if there are any more comments, and then write a new mail to show how rich this definition is, and how much can be done with it. Meanwhile let me add a definition, and make a few remarks. DEFINITION.. A structure functor p: S --> X is algebraic if p reflects isos. REMARKS ON TERMINOLOGY Let me first introduce some notations adapted to the typographical requirements of this list. If p; S --> X is a functor. I denote by Iso(S) and Iso(X) the groupoids of or isos of S and X and by p/Iso the restriction of p to these groupoids. For each object x of X I denote by p*(x) the fiber of p over x . As I do not want to run into set-class distinctions, which are meaningless here, I shall assume S and X small. I think we should call transportable what is called uniquely transportable in the book on Algebraic Theories (AT), and weakly transportable what they call transportable. This would fit with Bourbaki's definition, given even before the birth of Category Theory,. It means that p/Iso is not merely a fibration but a DISCRETE fibration Thus we would say that p is a structure (resp. a weak structure) if it is faithful and transportable (resp. weakly transportable). In presence of faithfulness the weak case amounts to saying that p/Iso is not merely a fibration but what I call an elementary fibration, i.e. equivalent to a discrete fibration, which is then unique up to a unique isomorphism of fiibrations. What about amnesticity? I think it should be kept as a very secondary notion. It says nothing global about the functor p but only about its fibers; which isn't much unless p is a fibration. Let me make an analogy. If p is faithful all the fibers are posets. The converse is of course not true. Would anyone consider functors such that all the fibers are posets without any cohesion between these posets? Or again; p is amnestic iff its restriction to the subcategory Vert(p) of vertical maps of p is amnestic. It is only in presence of other strong axioms on p that amnesticity makes sense. (Look also at my answer to Andree). I'd of course appreciate any comments or objections, in particular about the last definition and the terminology I propose in the last section. Many thanks., Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]