mathematics free of a blackboard (I. Gelfand and M. Gromov)
Vaughan Pratt
pratt at cs.stanford.edu
Sat Mar 13 02:05:15 EST 2021
José Manuel Rodríguez Caballero has thrown down the following gauntlet
that I feel surely must be taken up here.
"The main difference between category theory and set theory is the
foundations of mathematics: Category theory is about the mathematical
objects, independently of the representations, whereas set theory is about
the representations of the mathematical object."
If we take the objects of set theory to be the sets of each ordinal rank,
starting with the unique set of rank 0, the unique set of rank 1, the 2
sets of rank 2, the 12 sets of rank 3, the 65,520 sets of rank 4, the
G^197.23... sets of rank 5 where G = 10¹⁰⁰ is a googol, the continuum many
sets of rank ω, and so on through all the ordinals, then since no
nontrivial Lie algebra arose directly in that enumeration, there can be no
doubt that nontrivial Lie algebras could only arise as *representations *by
such sets.
What I have great difficulty with is José's suggestion that, unlike set
theory, category theory somehow transcends representation.
Suppose we accept that the category *Set* is representation-free in that
sense, namely that its objects actually *are *sets, and not mere
representations of them.
What are we to make of the functor category *Set¹* where *1* denotes the
category with one morphism? This category has for its objects the
functors from the category *1* to *Set* and for its morphisms the natural
transformations between those functors.
Since *Set¹* is equivalent to *Set*, yet its objects and morphisms are
clearly not those of *Set* itself, how are we to account for that
difference? Should we say that the objects of *Set* are actual sets while
those of *Set¹* merely representations of them?
Maybe a case could be made for that point of view.
But now consider the category *Grph* of directed graphs permitting multiple
edges between two vertices. Suppose its objects are representation-free in
the same sense as for *Set*.
Let *G* be the category with objects V and E and non-identity morphisms
s,t:V → E. I claim that *Set*^*G* is equivalent to the category *Grph*.
But the objects of *Set*^*G* are functors from *G* to *Set* and its
morphisms are the natural transformations between them.
So unlike *Grph*, the objects of *Set*^*G* are not themselves graphs but
merely representations of them.
Now here's the crucial question.
*In what sense is Grph different from Set^G?*
Unless you have a representation-free conception of *Grph*, how can you
draw the distinction between *Grph* and *Set*^*G* that we imagined we could
draw between *Set* and *Set¹?*
*--------------------------*
Vaughan Pratt
-------------------------------------
>
> The main difference between category theory and set theory is the
> foundations of mathematics: Category theory is about the mathematical
> objects, independently of the representations, whereas set theory is about
> the representations of the mathematical object.
>
> Kind regards,
> Jos? M.
> -
>
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