Models of the Theory of Categories
largepai...@gmail.com
Long ago, John Baez wrote this very interesting sentence in one of
his unimaginably instructive "Week" n's.
>For example, a category is a set of "objects" and a set of "morphisms"
It seems innocuous, however, it grates on me a bit.
Let us suppose for a moment that we believe in a thing called a
"theory". Theories take on many different forms. One form is that
which it takes in my head, but that is another matter. To form a
concrete example, we can mutually establish the meaning of a set of
symbols (after long periods of debate) and then form axioms, or
sentences, from those symbols. From those axioms, using something
called inference, we develop more and more sentences and then start to
wonder about ALL the sentences which we may eventually develop. At
this point we have come to the realization of a theory of X.
What do we have REALLY, though? Well, we have a SET of statements.
Inclusion in the set is equivalent to a notion of TRUTH. Now, as Godel
did in his paper on incompleteness, I could perform a transformation of
this set to another set, say the natural numbers, and this too should
form a viable model of the theory, however, interpreting the theory in
this setting should prove difficult.
The category of sets which embody the theory is, in my mind, the
model of the theory. If we start within the theory of categories, we
just simply say that there exists some category of the theory X. In
short, there is a category (not a set) which embodies the theory of X.
The objects in this category are NOT sets. Along with this category,
is a functor from this category to the category of sets, and the sets
we find therein are all formalisms of the theory.
What does it mean, then, to say that a category is just a set of
objects which have morphisms between them which are just functions? If
I have suggested anything at this point, I have suggested this: things
like GROUPS, are not sets at all. In fact, they are entities for which
I have a theory and, consequently, a model. The MODEL is certainly a
description of that entity within the theory of sets, however the
entity itself, the one that I am using in my head, is not a set.
I know I am missing something about category theory, or Lawvere's
model theory, because, as Baez points out,
>taking C to be the theory of groups and X to be the category of sets, we get
>the usual category of groups:
>Mod(Th(Grp),Set) = Grp
He then goes on to say,
>That's reassuring, and that's how it always works.
Sadly, I am not reassured. Shouldn't the point of category theory be
to free us entirely from the confines of set. Certainly, any new
fangled construction for handling theories as if they were entities
should properly include a theory of sets. There is certainly more to
algebraic structures than sets with added structure. In my mind, if we
are going to use some new abstraction (in this case category theory, or
model theory) to discuss mathematical entities, it should transcend set
theory altogether and properly include set theory.
All of this is highliting a very important relationship between
discourse and mathematics. It is the case that any symbolic system,
for which participants jointly (or even singularly) agree upon the
semantics of the symbols, should be seen as belonging properly to a set
theory. The implications of category theory, as an alternate
foundation for mathematics, should toy with the possibility that there
is a mathematics that is independant of our talk (jibber jabber) and
sybolic notation (#,M,<,>,%,T). Personally, I got the feeling for this
when considering homotopy equivalence classes (fundamental group for 2
dimensional surfaces).
Unfortunately, even the arrow and dot drawings (basically directed
graphs), which innocently convey category theory, are subject to these
very same problems. Thus, it may be that everything that has thus far
been said about the theory of categories, comprises only one or more
models of the theory of categories.
Ours is a ripe time for the philosophy of mathematics, we are
enduring a re-animation of a science of mathematics, ( best exemplified
by a cognitive science of mathematics). I implore each of you to start
digging.
Flame on!
Gonçalo Rodrigues
fish to the penguins:
If I understood you, no you're not missing anything. An algebraic
theory TH (in Lawvere's sense) is a special type of category - I will
also call it TH. A model of TH in a category A is a functor
TH -> A
satisfying some properties. The case Baez discussed is where A is Set
but there is nothing to prevent us from discussing models of TH in
other categories - and it *is* useful and interesting to do so. Of
course, for the theory to develop smoothly your A has to have some
strong (co)completeness properties.
>>taking C to be the theory of groups and X to be the category of sets, we get
>>the usual category of groups:
>
>>Mod(Th(Grp),Set) = Grp
>
>He then goes on to say,
>
>>That's reassuring, and that's how it always works.
>
This is telling us that when A = Set, the models of TH(Grp) *in the
category of sets* are just groups as we usually understand them: a set
plus a binary operation satisfying some properties. Or, that the two
different ways to formalize the idea of a group lead to the same
results.
Hope it helps, best regards,
G. Rodrigues