FOM: small category theory

[Stephen G Simpson]

I now have Adamek-Herrlich-Strecker ``Abstract and Concrete
Categories'' (1990) and Herrlich-Strecker ``Category Theory'' (1973)
out of the library.  I thank Mossokowski for recommending these books.
I like these books a lot.  There is a lot of mathematical meat in
them.  They also exhibit an appropriate attitude of reverence and
respect toward the set-theoretic foundations of category theory.

The part of A-H-S entitled ``Foundations'' (pages 5-9) has three
sections: 1 on sets, 2 on classes, 3 on conglomerates.  The approach
is informal but it is pretty clear from sections 1 and 2 that the sets
and classes are supposed to satisfy the usual VNBG axioms.  At the end
of section 2, the authors say that sets and classes alone would
suffice for most of the book:

  The framework of sets and classes described so far suffices for
  defining and investigating such entities as the catogory of sets,
  ..., functors between these categories, and natural transformations
  between such functors.  Thus for most of this book we need not go
  beyond this stage.  Therefore we advise the beginner to skip from
  here, go directly to [sub]section 3, and return to this [sub]section
  only when the need arises.

  The limitations of the framework described above become apparent
  when we try to perform certain constructions with categories, e.g.,
  when forming ``extensions'' of categories or when forming categories
  that have categories or functors as objects. ...

Unfortunately ``extensions'' are not in the index, so I don't know for
sure what the authors are talking about.  My best guess is that they
are referring to Kan extensions, but Kan extensions do not seem to be
in the book, so I am not sure exactly why the authors insist on going
beyond sets and classes.

In any case, it appears that a formalization of the A-H-S informal
theory of sets, classes and conglomerates would be interpretable in
ZC+I, Zermelo set theory plus the axiom of choice plus one
inaccessible cardinal, kappa.  The A-H-S ``sets'' would be interpreted
as elements of V_kappa, the A-H-S ``classes'' as subsets of V_kappa,
and the A-H-S ``conglomerates'' as arbitrary sets.

Later, on page 31, the authors define a quasicategory to be something
like a category, except that categories are sets or classes while
quasicategories are allowed to be conglomerates.  They then say:

  Virtually every categorical concept has a natural analogue or
  interpretation for quasicategories. .... Because the main object of
  our study is categories, most notions will only be specifically
  formulated for categories.  However, we will freely make use of the
  implied quasicategorical analogues ....

So obviously the treatment will have to contain logical gaps.  To fill
in these logical gaps, you would apparently need to carry along the
quasicategorical baggage through the entire book.

It seems clear that the ``small category'' foundation that I have
advocated would be logically cleaner.  Simply work in the usual ZC
framework, define a category to be a set-size algebraic structure of
an appropriate kind, and postulate strong limit cardinals or
inaccessible cardinals if and when needed.

Obviously this ``small category'' foundation would work, except that
it would not let you talk about the categories of *all* sets, *all*
groups, etc.

I wonder why category theorists think they cannot live with this
straightforward foundational approach?  Perhaps category theorists
have a psychological need to be told that categories are inherently
bigger or more inclusive than other mathematical objects.

:-)

Regarding the alleged need for inacessibles, Mossokowski 05 May 1999
00:04:05 and 11 May 1999 23:18:47 cited a theorems on page 200 and 385
of A-H-S.  I still don't see why these theorems could not be
formulated in a meaningful way for categories of size kappa where
kappa is not inaccessible.

I will post more later about the Herrlich-Strecker appendix on
foundations.  It is pretty interesting.

-- Steve