FOM: small category theory

Stephen G Simpson simpson at
Thu Apr 29 22:21:56 EDT 1999

I said:

 > > 1. A small category is just another kind of algebraic structure, like
 > >    a group or a ring or a vector space.  So it would be possible to
 > >    include an exposition of small category theory in with an
 > >    exposition of the rest of algebra, in a very simple way.  Category
 > >    theory would no longer stick out like a sore thumb.

Carsten Butz replied

 > That's how category theorists think (often) about categories, small or
 > not. There is the fixed technical term of algebraic structures (like
 > groups, rings, distributive lattices, etc, but not local rings, fields,
 > etc.). (Small) categories do not belong into this group, ...

OK, I guess you mean varieties of algebras, in the sense of universal
algebra.  Anticipating this objection, I mentioned a vector space.
Fields do not form a variety, and therefore vector spaces also do not,
because a vector space has two sorts, scalars and vectors, and the
scalars are a field.

 > but there are what are called _essentially algebraic structures_,
 > which roughly means that there might be some operations (here:
 > composition) which are not total, but defined on a definable
 > sub-type. This class of structures behaves very much like algebraic
 > ones (forgetful functors have adjoints, i.e., "free" structures
 > exist, etc). For (small) elementary toposes it was Peter Freyd who
 > first mentioned that they form an essentially algebraic structure.

But fields aren't ``essentially algebraic'' in this sense either, are
they?  But small categories are, right?

Anyway, I apologize for stating my point somewhat carelessly.  But I
still think my point is correct.  When I said that a small category is
an algebraic structure, I meant this in a naive sense, i.e. it is a
set (the set of arrows) together with a partially defined operation
(composition of arrows), and maybe a distinguished subset (the
identity arrows) and is required to satisfy certain simple laws
(associativity etc).  My point is that you could write an algebra book
consisting of chapter 1 on groups, chapter 2 on rings, chapter 3 on
fields, chapter 4 on small categories, etc.  It is all algebra.

Here is another way to make my point.  Logically, the issue of small
(i.e. set size) versus large (i.e. proper class size) is not really
peculiar to category theory.  It applies to any kind of algebraic
structure whatsoever.  If you insist on talking about large
categories, you might as well also insist on talking about large
groups, large rings, large fields, etc.  [ I seem to remember somebody
saying that John H. Conway's surreal numbers form a large field .... ]

This is why I questioned Mossakowski's comment that category theory is
*particularly* concerned with the small/large (i.e. set/class)
distinction.  It seems to me you can make the same distinction in any
branch of algebra, so this distinction does not particularly belong to
category theory.

My feeling is that the *best* place to make the small/large
(i.e. set/class) distinction is set theory, because sets and classes
have the least amount of extra structure, namely none.  Historically I
think set theory is where the distinction *was* first made, and made
very sharply and clearly at that.  It goes back at least to von
Neumann's paper on VNBG, and probably a lot farther back than that,
and over the years axiomatic set theorists have done a lot of
interesting things with it.

The bottom line is that the small/large (i.e. set/class) distinction
belongs to f.o.m.

-- Steve

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