This concerns the discussion by Mike Barr, Paul Taylor, and Masahito Hasegawa regarding symmetric monoidal closed comma categories: The construction (later called 'comma') in the category of categories was introduced in 1963 primarily for foundational simplification (though it was clear that certain particular cases, such as slice, were already in direct use). Besides the 2-categorical equational description of adjointness, one needs the description in terms of a bijection between arrows, but that does not require the complicated assumption that there exists a category of sets in which two given categories can be enriched. Namely, an adjunction between two given categories can be described by giving a third 'adjunction category', related by appropriate functors to them, which is isomorphic to two differently-constructed 'comma' categories. It seems that there are many cases in which this third category is of interest in itself, whether or not one of the two given categories is or is not monadic or comonadic over the other. Emilio Faro's notes from my Fall 1990 Buffalo course Categories of Space and of Quantity, mention essentially the result cited by Masahito Hasegawa. If an adjunction involves monoidal functors, then the adjunction category tends to be a monoidal closed category. This remark was essentially intended to supply semantically-based examples of closed categories which have one aspect which is linear (in the straightforward sense that coproducts equal products) and an opposite aspect which is cartesian (in the sense that the tensor is the categorical product). Of course, the most immediate subclass of examples, based on the data of a rig in a cartesian closed category, involve monadic adjunctions. On the other hand, several published papers on related matters axiomatically assume comonadic adjunctions. However, the simple algebraic stance, as Masahito Hasegawa points out, is that both aspects, as well as the relation between them, are all regarded as equally given. As part of the logic (= natural structure) of the resulting situation there will be unary (= modal?) operators reflected on each aspect by composing. A further step is to investigate to what extent the data can be approximated via data which is reconstructed on the basis of only one aspect or the other using this additional reflected structure. Both that step a la M. Stone, as well as the simple algebraic stance in the spirit of Chu and G. Mackey, are of course involved in the full study of any related pair of aspects (e.g. algebra and geometry). A problem from topology, where related considerations may help, concerns the operation of collapsing a connected subspace to a point (the effect of this operation on relative homology is part of the content of algebraic topology). In extending this operation to apply to not-necessarily-connected subspaces (and more generally, from inclusion maps to arbitrary maps), collapsing all these to a point would be an unnecessarily discontinuous functor. Rather, within the category whose objects are continuous maps, consider the subcategory wherein the domains of these structural maps are discrete (or zero-dimensional, if that is different in the model of continuity being considered). That subcategory is reflective (with the help of pushout) in case the model admits a left adjoint connected-components functor. In the case of a subspace, the reflector collapses each of its components to a distinct point in the new ambient space, and the lifted unit of the adjunction is epimorphic if the original one (to the connected components pi zero) is, even where the subspace is empty. I am wondering: under what conditions are these categories and functors cartesian monoidal closed? Indeed these things are probably folklore, but listed below are some references containing partial indications. Bill Functorial Semantics of Algebraic Theories Thesis Columbia University (1963) The Category of Categories as a Foundation for Mathematics, Proceedings of La Jolla Conference, Springer-Verlag (1966) 1 - 20 Categories of Space and of Quantity, Buffalo Course Notes by Emilio Faro (1990) ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA *******************************************************************************