A group presentation consists of a bunch of symbols and some equations between strings of some of the symbols and some of their formal inverses. The symbols are the arrows of the graph, and the diagrams exhibit the equations. The type of theory (instead of FP theory, FL theory, coherent theory or whatever) is that of categories in which all arrows have inverses (groupoids) to which if you insist you may add any two arrows compose (groups). A model of a sketch in a category takes each arrow to an isomorphism in such a way that the diagrams commute. The presentation generates a group, and that group is the theory. The category of actions of the theory on category is clearly equivalent to the category of models of the presentation as sketch. This brings up a point. An FP theory (for example) is typically thought of as having models in an FP category, but in fact it can have models in any category. The necessary finite products must exist, but others can be missing. Nevertheless, most of the examples that have actually been considered are in categories that do have all finite products (in other words are within the doctrine). In the case of a group presentation, it is _normal_ to consider models in categories in which most arrows don't have isomorphisms (they are not within the doctrine). This is only a psychological difference, but it is interesting. -- Charles Wells professional website: http://www.cwru.edu/artsci/math/wells/home.html blog: http://www.gyregimble.blogspot.com/ abstract math website: http://www.abstractmath.org/MM//MMIntro.htm personal website: http://www.abstractmath.org/Personal/index.html