Re: How analogous are categorial and material set theories?

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Dear Neil,

Some comments below.

All the best,

Steve.

On 03/12/2017 16:12, bartonna@gmail.com wrote:
> ...
>
> (This relates to a more general question I have concerning categorial
> foundations: Are there people who claim we should do foundational
> research *solely* in the language of category theory, or does almost
> everyone accept that the `external' (possibly material set-theoretic)
> perspective is also allowed? So, for example, when considering the
> category of sheaves over a topological space, whilst I could take a
> purely categorial outlook, nonetheless sometimes I might want to just
> look at the equivalence class of an open set U relative to a point i
> in the topological space defined in material set theory. The two
> perspectives seem to complement rather than contradict each other, but
> I wonder what the general feeling is concerning the interellation of
> the two foundational systems.)
Yes, I think I would claim that the foundational work should be in the
language of categories. Much of the justification for that is pragmatic
methodology, in that category theory has proved itself effective in
elucidating the underlying mathematics common to different foundations.

I have been hugely influenced in this by my experience of relating
topology to logic, and point-set approaches to point-free. It is
category theory that shows us how point-set and point-free are doing
similar things, using structure shared by the categories of topological
spaces and of point-free spaces such as locales. And we don't have to go
far down this road before the comparison starts to show point-set
topology in an unfavourable light.

A similar example of the categorical analysis of foundations is
Grothendieck's discovery of toposes. (This is my rational
reconstruction, so apologies - to you and to Grothendieck - if it's
bollocks historically.) Grothendieck asked what mathematics was needed
to do sheaf cohomology, known from sheaves over topological spaces, and
gave an answer using categorical structure. Abstracting that gave us
Grothendieck toposes, thus generalized spaces in the sense that you can
still do sheaf cohomology over them.

The methodology is not entirely pragmatic, however. Categorical
structure explicitly describes how objects relate to each other (via
morphisms) as opposed to how they are structured internally. The real
mathematics lies in those mutual relationships. Any foundational
discussion that says the mathematics _is_ the structures - the sets or
whatever - is answering the wrong questions. Those structures are just
particular solutions to the mathematical problem. It is the same as the
difference between the API of a software library (how the calling
program relates to the library routines it calls) and the implementation
of the library routines - which is best left hidden and subject to revision.
>
> ...
>
> @Steve. You ask:  When you look at making set theory more categorical,
> are you just looking for a categorical way to do essentially the same
> thing, or are you trying more deeply to expose possible limitations of
> set theory?
>
> I don't think I'm clearly aiming at either (though I am interested in
> these questions). I'm trying to understand more clearly what purposes
> each foundation is best suited to, and how we can relate the two. I
> suppose it's a mixture of the two---the present paper I'm currently
> writing looks to modify material set theory to get something more
> `structure respecting', but nonetheless facilitating the combinatorial
> power and conceptual simplicity it offers (allowing us to easily work
> with notions such as cardinality). I'm trying to get a better picture
> of the landscape though, and doing this requires understanding the
> other direction (i.e. how one can mimic material set-theoretic claims
> in the structural setting).
Cardinality is in fact what I had in mind when I asked the question. It
is one of those notions that tends to fade away when category theory
takes you from one foundational setting to another, say from classical
maths to constructive, or from point-set topology to point-free.

  From one point of view ("making set theory more categorical") you would
be interested in finding a categorical description that can still deal
with cardinalities. From the other ("exposing possible limitations") you
would use the categories as reason for losing interest in cardinalities.



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