Symmetric monoidal closed comma categories

Symmetric monoidal closed comma categories

[F W Lawvere]

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)


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F. William Lawvere			Mathematics Dept. SUNY 
wlawvere@acsu.buffalo.edu               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA

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