----- Begin Included Message ----- Can somebody give me references to literature on Burroni's graphical algebras? The only paper I know of is: "Algebres graphiques" (A. Burroni), Cahiers de Topologie et Geometrie Differentielle, XXII, 1981. Many thanks, Frank Piessens. ----- End Included Message ----- The subject of graphical algebras is highly interesting (to my taste !) but rather controversial : there have been many mathematical misunderstandings and many quarrels about who did what and when. Here are the main lines of my account of the subject. In my opinion the fundamental notion is that of an algebraic functor in the sense of Coppey : this definition of algebricity is of a very intuitive and elementary nature and is for me THE definition of algebricity. The corresponding framework was introduced in [Coppey72] and is also described in [Coppey-Lair85]. An algebraic functor need not have a left adjoint. In fact U is monadic iff U is algebraic and has a left adjoint ! However an algebraic functor "with rank" is monadic. Now a category of graphical algebras in the sense of Burroni is exactly a category C with an algebraic functor C -> Graph. A controversial question (between Coppey and Burroni !) is : are graphical algebras of any special importance ? The only partial answer I can give is about the use of categories in computer science : I have never seen any computer scientist make use of the fact that a given type of structured categories (say c.c.c.'s) is algebraic over graphs. What is really used is the algebraicity over "graphs with formal constructors" (some kind of generalized sketches). The problem is that many sources that use that kind of algebricity refer to Burroni's graphical algebras which is completely misleading : for example the book [Lambek-Scott86] and the paper [Coquand-Ehrard87]. Categories with specified (co)limits of a given kind are algebraic over categories : this is a classical result by Lair ([Lair75] or [Lair79]). In some cases ((co)products, (co)equalizers) they are also algebraic over graphs : see [Burroni81] and [Coppey-Lair85]. Cartesian closed categories as well as elementary topoi are also algebraic over graphs : the history of these results is not simple and involves at least [Burroni81], [Dubuc-Kelly83], [MacDonald-Stone84] and [Ageron91]. An interesting example is that of Peano-Lawvere categories. It follows from general reasons that they are algebraic over categories; Burroni raised the question : are they algebraic over graphs - it was solved negatively in [Coppey-Lair85]. Much more could be said on that subject, e.g. the link between algebricity and sketchability but this message is already too long. I refer to [Ageron91] for a synthetic introduction and to [Coppey-Lair85] for more. Pierre Ageron References. [Coppey72] Theories algebriques et extensions de prefaisceaux, Cahiers 13 (72) 3-40 [Coppey-Lair85] Algebricite, monadicite, esquissabilite et non-algebricite, Diagrammes 13 (1985) 1-112 [Lambek-Scott86] Introduction to higher order categorical logic (Cambridge, 1986) [Coquand-Ehrard87] An equational presentation of higher order logic, in LNCS 283 (Springer,1987) [Lair75] Esquissabilite et triplabilite, Cahiers 16 (1975) 274-279 [Lair79] Condition syntaxique de triplabilite d'un foncteur algebrique esquisse, Diagrammes 1 (1979) 1-16 [Dubuc-Kelly83] A presentation of topoi as algebraic relative to categories or graphs, J. of A. 81 (1983) 420-433 [MacDonald-Stone84] Topoi over graphs, Cahiers 25 (1984) 51-62 [Ageron91] Structure des logiques et logique des structures, these, Universite Paris 7, 1991, 196 pages