Re: How analogous are categorial and material set theories?

[Neil Barton <bartonna@gmail.com>]

Dear All,

Thanks so much for your kind and patient responses, and apologies for
the slow reply (I wanted to check out some of Michael's
recommendations before replying).

@Patrik: Thanks for the nice examples and applications. I certainly
don't want to deny that category theory has applications where
material set theory would be inappropriate, but rather to specifically
see if there were any applications to which material set theory was
more suited (and how this can then be incorporated into the structural
setting). However, your examples are useful to see just how different
the two perspectives are (I agree that systematising certain subject
matters in material set theory would be a fruitless project).

@Michael: Thanks very much for the reference. I think I see the proof:
One recovers a model of ZFC not by considering the membership relation
of ETCS (given by various f: 1--> X), but rather by finding the
`membership graph' for the relevant sets in a category satisfying ETCS
(with the replacement stack axiom added).  I do have a question here
though---here we are expected to take the *class* of  all well-founded
extensional accessible pointed graphs, and note that ZFC holds within
this class. Whilst I'm happy that the lemmas showing that the required
graphs exist in the category Set are categorial, it seems to me that
to isolate all these and talk about the *class* of all of them
requires some material-set-theoretic machinery. Is there a *purely
categorial* way of talking about this `collection' of subgraphs in
Set? Or have I missed something?

I suppose this is equivalent to the requirement of asking for a (set)
model of ZFC in material set theory. So could one simply state an
extra axiom to be added to ETCS+: ``Set contains an well-founded
extensional accessible pointed graphs such that...[list the APG ZFC
axioms].''. Is this acceptable in a *pure* categorial framework, or do
you think that presupposes some material set theory? I suppose there
is also the question of how to recover the cumulative hierarchy in
this framework---in the paper you sent me this is done with a
non-categorial theory of ordinals (possibly given by material set
theory).

(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.)

(This in turn relates to the wider question: How can material and
structural set theories inform one another? It *seems* to me (without
any deep arguments for the claim) that material set theory is just
better suited to certain roles (such as the formulation of large
cardinal hypotheses) whilst structural set theory better suited to
systematising the algebraic roles we want sets to perform. This is
despite the fact that we can simulate one perspective within the
other; for example just because you *can* simulate talk of categories
with, say, Grothendieck universes, doesn't mean that it's a
particularly *natural* interpretation.)

@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).

Thanks again!

Best Wishes,

Neil


On 27 November 2017 at 17:49, Steve Vickers <s.j.vickers@cs.bham.ac.uk> wrote:
> Dear Neil,
>
> This is not an answer to your question, so please ignore it if you're not  so
> interested in these broader issues.
>
> My broad question is this. 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?
>
> One thing shared by ETCS and ZFC is the well-pointedness: that the object  is
> determined by its global elements (morphisms from 1).
>
> That can seem obvious if what you're trying to capture is some idea of
> collection, but in fact it breaks down when the collection has topological
> structure. The cohesion between points goes beyond what can be explained in
> terms of the global points themselves, and in point-free topology we see
> non-trivial spaces with no global points at all. This is not necessarily a
> pathology of point-free topology but can be related to topological facts
> such as the existence of principal bundles with no continuous global
> sections. It also feeds back into "sets" as discrete spaces, with
> non-well-pointed toposes of sheaves (= local homeomorphisms = fibrewise
> discrete bundles).
>
...

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