[FOM] Categories in NBG
martdowd at aol.com
martdowd at aol.com
Fri May 17 15:30:34 EDT 2013
Colin McLarty's recent result in
A FINITE ORDER ARITHMETIC FOUNDATION FOR COHOMOLOGY
http://arxiv.org/pdf/1102.1773v3.pdf,
that the cohomology needed for FLT can be developed in a fragment of ZFC,
is clearly of interest.
The discussion of whether it could be developed in ZFC was, however, a
tempest in a teapot. It can straightforwardly be developed in NBG, and
universes are unnecessary. This doubtless is true for category-theoretic
methods in mathematics in general. This observation is at least partially
made in the literature already.
In his 2010 paper
WHAT DOES IT TAKE TO PROVE FERMAT'S LAST THEOREM?
GROTHENDIECK AND THE LOGIC OF NUMBER THEORY
Bull. Symbolic Logic Volume 16, Issue 3 (2010), 359-377
McClarty states "Probably everything in [Hartshorne, 1977] can be formalized
in NGB". The same claim applies to Grothendieck duality as presented in
Altman and Kleiman.
Important examples of the methods required can be found in
SET THEORY FOR CATEGORY THEORY
http://arxiv.org/pdf/0810.1279v2.pdf
by Michael Shulman. First, if $I$ is an index class then a collection
$\{C_i: i\in I\}$ of classes can be given as a single class
$\{\langle i,x\rangle: i\in C_i\}$. On page 13 it is stated that
"We can choose an element from each of any collection of nonempty classes."
Shulman goes on to show that if limits exists then a limit functor can be
chosen.
Expanding on McLarty's remark, the category theory developed in Hartshorne,
and in Altman and Kleiman, can undoubtedly be carried out in NBG. As noted
in McLarty 2010, it is necessary to check that transfinite inductions are
performed only on $\Delta^1_0$ predicates; but I doubt if there is any
]transfinite induction in either of these two references.
Derived categories are not used in the above two references, but thet can be
treated in NBG, indeed this is done in Shulman. Define a pre-category to be
a sextuple of classes < Obj, Arr, Dom, Codomain, Comp, Id >, satisfying the
usual axioms. A category is a pre-category with small Hom sets.
This requirement is enforced by many authors, for example Mitchell in his
"Theory of Categories". It allows the definition of the Hom functor for
example. Many definitions, for example functors and natural transformations,
can be given for pre-categories, though,
Using choice, the localization of a category may be defind and is a
pre-category. There are general theorems giving sufficient conditions for
it to be a category, for example in
http://www.uni-math.gwdg.de/theo/intro-derived.pdf
http://hopf.math.purdue.edu/Christensen/derived.pdf
- Martin Dowd
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