ONE MORE HISTORICAL CITATION The Pumplun paper cited by Wyler as well as the Auderset paper cited by Mme Ehresmann illustrate that the study of generic structures in 2-categories has been going on for some time. My own paper ORDINAL SUMS AND EQUATIONAL DOCTRINES, SLNM 80 (1969) 141-155 shows that the augmented simplicial category Delta serves as the generic monad, but moreover goes on to actually apply this to show that the Kleisli construction is a tensor product left-adjoint to the Eilenberg- Moore construction which is an enriched Hom. The Hom/tensor formalism appropriate to the case of strict monoid objects is all that is required here, as I will explain below. AN EXTENSION AND A RESTRICTION The important special case of FROBENIUS monads is explicitly characterized in three ways in my paper. Concerning the IDEMPOTENT case discussed a few days ago by Grandis and Johnstone, note that the publication of Schanuel and Street proves among other things that the monoid Delta in Cat has very few quotients (see below for significance of the monoid structure). THE GENERAL HOM/TENSOR FORMALISM AND A VERY PARTICULAR MONOID In any cartesian-closed category with finite limits and co-limits, a non-linear version of the Cartan-Eilenberg Hom/tensor formalism applies to actions and biactions of monoid objects. In Cat, Delta is a (strict) monoid and its actions are precisely monads on arbitrary categories. A crucial part of the formalism is that categories of actions are automatically enriched in the basic cartesian-closed category, which in this case is Cat. There is a particular biaction of Delta, which I called Delta plus, with the property that the enriched Hom of it into an arbitrary Delta-action is exactly the Eilenberg-Moore category of "algebras", automatically equipped with its structure as a Delta^op action (co-monad). The left-adjoint tensor assigns to any category equipped with a co-monad its Kleisli category, as a category with monad. Not only are the calculations in this particular case quite explicit, but the enriched Hom tensor formalism has a lot of content which is still under-exploited. SKETCHES VERSUS PLATONISM The often repeated slander that mathematicians think "as if" they were "platonists" needs to be combatted rather than swallowed. What mathematicians and other scientists use is the objectively developed human instrument of general concepts. (The plan to misleadingly use that fact as a support for philosophical idealism may have been an honest mistake by Plato, or it may have been part of his job as disinformation officer for the Athenian CIA organization; it probably would not have survived until now had it not been for the special efforts of Cosimo de' Medici.) It seems that a general concept has two related aspects, as I began to realize more explicitly in connection with my paper Adjointness in foundations, Dialectica vol. 23, 1969 281-296; I later learned that some philosophers refer to these two aspects as "abstract general vs. concrete general". For example, there is the algebraic theory of rings vs. the category of all rings, or a particular abstract group vs. the category of all permutation representations of the group. While it is "obvious" that, at least in mathematics, a concrete general should have the structure of a category, because all the instances embody the same abstract general and hence any two instances can be compared in preferred ways, by contrast it was not until the late fifties that one realized that an abstract general can also be construed as a category in its own right. That realization essentially made explicit the fact that substitution is a logical operation and indeed is the most fundamental logical operation. Thus an abstract general is essentially a special algebraic structure indeed a category with additional structure such as finite limits or still richer doctrines. As with other algebraic structures there are again two aspects, the structures themselves and their presentations which are closely related, yet quite distinct; for example, more than one presentation may be needed for efficient calculations determining features of the same algebraic structure. What is meant by a presentation depends on the doctrine: for example Delta as a mere category has an infinite presentation used in topology, but as a strict monoidal category it has a finite presentation. The notion of SKETCH is the most efficient scheme yet devised for the general construction of PRESENTATIONS OF ABSTRACT GENERALS. The fact that particular abstract generals and the idea of sketches exist within the historically developed objective science does not mean that they somehow always existed; to call them "platonic" seems to detract from the honor of their actual discoverers. Bill Lawvere ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************