Categories of elements (Pat Donaly)

[Jpdonaly@aol.com]

To all category theorists:

In various textbooks, I see reference to the common comma category Elts(G),
which is called the "category of elements of functor G". This category seems to
be drastically misnamed. Does anyone agree? Here is my side of the story,
beginning with a review of the nature of Elts(G) and some of its significance.

G is a functor from a small category C into the category F of small
functions. Denoting the singleton {0} of the void set 0 by 1 (as usual), Elts(G)
consists of all triples (g,a,f) with a in the domain category C and
g:1--->G(codomain a), f:1--->G(domain a) such that g = G(a) o f, where "o" denotes function
composition. (Warning: By my conventions, a.domain a = a; codomain a.a = a.) The
composition of Elts(G) is defined by (h,b,g)(g,a,f) = (h,ba,f). The objects
are the Elts(G)-morphisms of the form (f,u,f), where u is a C-object, and the
map (f,u,f)-->f(0) identifies each of these with an element of the set G(u), so
that the convention of naming categories after their objects (to the extent
possible) is what presumably leads to calling Elts(G) "the category of elements
of the values of G at objects" or, for short, "the category of elements of G".

Several basic features of Elts(G) are exposed by treating it as a subcategory
of a product BxC of a transition category B with the domain category C of G.
To define B, let X be the set of functions f:1--->G(u) as u varies over the
objects of C; B is then the full transition category or groupoid of X, that is,
the self-product XxX with the transition composition (h,g)(g,f) = (h,f). But
rather than taking the ordinary cartesian product for BxC, one uses the
attachment product consisting of those triples (g,a,f) with (g,f) in B and a in C, so
that C-morphism a is viewed as being attached on its left to g and on its
right to f. Then Elts(G) inherits by restriction the projection functor
(g,a,f)-->a, which is reasonably called the detaching functor from Elts(G) into
C---this functor will be generically denoted by "det".  There is also the transition
projection (g,a,f)-->(g,f) which maps Elts(G) functorially onto a transitive
relation on X, and the rule (g,a,f)-->g defines the entwining function of the
canonical natural transformation which entwines the constant functor
(g,a,f)-->1 on Elts(G) with the function composite functor G o det: Elts(G)--->F. (A
constant functor with value 1 will be denoted generically by "delta(1)".)

There are many important examples: If C is a group with object e so that G is
group action with action set Y=G(e), then, to within the identification
(g,a,f)-->(g(0),a,f(0)), Elts(G) is the traditional idea of a G-action as a
function from CxY into Y after correction to remove the categorically problematic
product CxY. If C is actually the group RxR, R being the additive group of real
numbers and G the action of C on the real affine plane by translation, then
Elts(G) is essentially the category of attached planar vectors as used in
Engineering Statics 101, which is why it seems appropriate to continue to use the
word "attach" in the context of a more general function-valued functor G:C--->F.
If C is a certain type of monoid, then Elts(G) is a semiautomaton. Among its
theoretical services is the fact that Elts(G) plays a role in the construction
of Kan extensions along inclusion functors, thus in particular in the theory
of induced group actions. It plays an analogous part in sheafification relative
to a Grothendieck site, and it is used to show that representable functors
are dense in the set of function-valued functors on C, providing, according to
Mac Lane and Moerdijk, "a plethora of tensor products". Even more basically, if
the domain C of G is discrete, then Elts(G) is a coproduct a.k.a. a disjoint
union of (the object values) of G. As will be noticed in a moment, the set of
functors A:C--->Elts(G) which are right inverse to the detaching functor det
on Elts(G)---that is, the "attaching functors" into Elts(G)---constitute a
(small) limit object of G, thus, in the case of discrete C, a product of the sets
G(u). From these examples it appears that Elts(G) is sufficiently important to
require a unique and unambiguous nomenclature, but, unfortunately, the things
in Elts(G) are really not the elements of G.

The (global) elements of an object u in a category are generally agreed to be
the morphisms from a given terminal object t to u. This convention
terminologically extends the observation that the functions f from the terminal object 1
into a (small) set X can be identified with the elements of X by the mapping
f-->f(0). The general definition has the virtue that each terminal object has
only one element, as should surely be the case, and the representable functor
of t provides a plausible (but not necessarily effective) attempt to convert a
given category into a category of functions between sets of elements. In
fact, this language seems to have found broad acceptance. But then the functor G
is an object in the morphismwise (i.e. "vertical") composition category F^C of
natural transformations whose (fully extended) entwining functions map from C
into F, and, because 1 is terminal in F, the constant functor delta(1) on C is
terminal in F^C. So G already has a set of elements, namely, those natural
transformations which entwine delta(1) with G. Such elements of G are not in
Elts(G) in any sense.

At first sight this terminological conflict might seem to be innocuous, since
Elts(G) and global elements of G occur in somewhat disparate contexts, but
the apparent separation does not hold up well when one considers how close the
set of global elements of G is to being a limit object of G. The only problem
with it is that it is not small; that is, it is not in the codomain category F
of G, and the only reason for this defect is that F, the common codomain of
the entwining functions of the things in F^C, is not small. Mac Lane in CWM
gives an ad hoc workaround which replaces, for a given G, the category F with a
small, G-dependent category of small functions, but this approach effectively
isolates G by artificially depriving it of morphisms into functors which do not
happen to map into Mac Lane's ad hoc replacement category; so one needs a more
perspicuous method of eliminating F and its untoward largeness.

F. W. Lawvere was apparently motivated by such considerations to introduce
comma categories in his thesis, an approach which works very well in addressing
the present awkwardnesses. One defines a category Law(C) whose objects are the
categories Elts(G) as G ranges through the functors in F^C and whose
morphisms are the cocompatible functors S:Elts(G)--->Elts(H) between such objects. The
composition is function composition of functors, and "cocompatible" means
that S does not disturb middle components of attached C-morphisms or,
alternatively put, det o S = det, where "det" continues to be the generic symbol for a
detaching functor. Then, if s in F^C entwines functor G with functor H, there is
a cocompatible functor S:Elts(G)--->Elts(H) which is evaluated at an attached
C-morphism (y*,a,x*) by

S(y*,a,x*)=(s(codomain a)(y)*, a, s(domain a)(x)*),

where I use y*, for example, to denote that function f:1--->G(codomain a)
whose value is y. Then the assignments s-->S define a functorial
isomorphism---which I call the Lawvere isomorphism (but should this be attributed to someone
else?) from F^C onto Law(C). Moreover, the objects Elts(G) of Law(C) are small.
This implies, of course, that the homset of cocompatible functors from
Elts(G) into Elts(H) is also small.

Elts(delta(1)) evidently consists of triples of the form (0*,a,0*) and can
thus be identified with C by the detaching functor (0*,a,0*)-->a on
Elts(delta(1)). With this identification, a cocompatible functor from Elts(delta(1)) into
Elts(G) becomes an attaching functor into Elts(G), so that, in the Lawvere
picture, the global elements of G are the attaching functors into the category
of...uh...elements of G. The set of such global elements is plainly small and
therefore must be what God intends to be the standard limit object of G, except
that it is difficult to believe that God would use such a verbal collision to
say what a global element is. These are my grounds for believing that Elts(G)
has to be renamed and redenoted.

As a related suggestion, I might recommend dropping the habit of referring to
categories by the names of their objects. This illogicality immediately
inhibits use of the subcategory concept (I still don't know what categorists use to
refer to the subcategory formed by the monomorphisms in an abstractly given
category), and then it just goes looking for the sort of trouble which has
turned up as "the category of elements of G". Besides this, the terminology "comma
category" is disrespectful of category theory, itself, due to the
inappropriateness of naming a fundamental, overarching categorical concept after a
punctuation mark. ("Slice category" doesn't seem to be any better.) Given the
precedent of attached vectors, which are used in a rough sense even by sophisticated
diagrammaticists, the category Elts(G) is obviously some kind of attachment
category, and since the transition components of any of its morphisms all have
domain 1, it is a based attachment category with base 1 or just a basement
category---or even just a basement denoted by something like G/1, if you're used
to placing domains on the right. Anyway, this is approximately what I use in
my study notes, and so far it works fine.

At the same time, I would be interested in seeing sharp, well reasoned
criticisms of this note provided that they are written at about the same technical
level so that I can understand them. I would like to emphasize that, aside from
what may be terminologically or notationally novel here, I am not making a
substantial research proposal or claiming priority for any discoveries. I have
no reason at all to doubt that all of the mathematics here is well known.

Pat Donaly