FOM: Category or collection; chicken or egg?

Vaughan R. Pratt pratt at cs.Stanford.EDU
Thu Nov 20 13:16:03 EST 1997

I'd like to address the question, what are our psychological
priorities, as a function of "our".

>From: Solomon Feferman <sf at Csli.Stanford.EDU>
>Date: Wed, 19 Nov 1997 23:46:02 PST.
>
>The *logical* and *psychological priority* if not primacy of
>the notions of operation and collection is thus evident."

I feel strongly myself that psychological priority is crucial here, and
am more than happy to focus on it.  This means in particular that I
have no problem with Sol's

>I do not mean 'collection' in the
>sense of 'set' or 'class' in an axiomatic theory like ZF.  I mean it in an
>informal pre-axiomatic sense, which could be clarified in various
>ways by axiomatic foundations, but not necessarily founded in terms of
>something more basic.

together, but am very comfortable separating them and focusing on the
former.

I would have no problem with a claim that "collection" and "operation"
have psychological priority for those for whom mathematics is developed
from a set theoretic perspective.

But as Torkel Franzen points out,

>Date: Thu, 20 Nov 1997 09:11:57 +0100
>Here the problem is that many, perhaps most people, have grown up with
>a set-theoretic view of things, and so are disinclined to suppose that
>categories could be just as basic.

What I would have more of a problem with would be any claim that
"collection" and "operation" have psychological priority for category
theorists.

What takes its place?  I'm not really sure, a category theorist would

My own psychological priority puts two primitive notions ahead of the
rest, collections and geometry.  The former allows me to collect my
thoughts in terms of sets, functions (set transformers), and predicates
(set carvers).  The latter lets me see things in terms of objects
(points), morphisms (edges between points), and 2-cells (surfaces
between edges).  These are simultaneously pre-axiomatic and axiomatic
for me.

Neither is independent of the other psychologically.  I need geometry,
specifically area, to understand collections psychologically.  I
visualize a set as made up of separated points, but to visualize the
separation I imagine the points as being spread out.  For this I use
area, a geometric notion.  For posets the area becomes even more
critical because I need somewhere to draw the edges.

Conversely, I depend on operations to understand discrete geometry.  I
understand edges logically as composing via a binary operation, and as
having a source and target given by two unary operations.  Surfaces
compose via two binary operations, horizontal and vertical composition,
the former of which is the same composition as for the edges (which can
be understood as zero-area nonzero-length surfaces).  These
compositions are governed by

(i) identity laws for each composition (that zero-area surfaces act as
identity for vertical composition while zero-length edges act as
identity for horizontal composition),

(ii) an associative law for each composition; and

(iii) an interchange law for the interaction of the two compositions, a
variant of associativity whose essential content is

+---+ +---+     +---+---+     +---+---+
| A | | B |     | A | B |     | A | B |
+---+ +---+  =  +---+---+  =  +---+---+
| C | | D |     | C | D |     +---+---+
+---+ +---+     +---+---+     | C | D |
+---+---+

This is all there is to 2-category theory: it is simply-connected
discrete two-dimensional geometry of an oriented kind in which motions
compose in the manner of a monoid rather than a group.

Starting from these geometric intuitions, which are very strong for me,
I find it *much* easier to pass to the notion of adjunction than from a
set-theoretic perspective.  Adjunction is to isomorphism as isomorphism
is to identity.  Just as an isomorphism is a pair f,g of edges with f.g
and g.f (horizontal compositions) being the respective identities at
each end, so is an adjunction a pair \eta,\epsilon of surfaces with the
vertical compositions \eta o\epsilon and \epsilon o\eta being the
respective identities at each end.

I would find it very confining to have to suppress either collections
or geometry as the psychological underpinnings of my grasp of
mathematics.  I use a lot of both.