Re: How analogous are categorial and material set theories?

[Patrik Eklund <peklund@cs.umu.se>]

Dear Neil,

Others in this list are much more prominent to answer to your question,
but let me provide one viewpoint.

Set theoryvis really about the theory of sets. Functions and powersets
are there, and complicated structures can evolve. However, a striking
thing about sets is that they are "untyped", which can be given a number
of meanings.

When we move into category theory, the category of sets and functions is
the simplest one. Sets come with no structure, so functions do not
prserve any such structure.

Now, functors over a category become important, the powerset functor
over that most simple and unstructured category of sets being a prime
example.

We need structure, and many real world applications require quite
elaborate structure. Functors over more elaborate categories become
important, where monoidal closed categories as unnderlying categories
bring in fundamental algebraic structures, even for a generalized
powerset functors.

Let me also speak warmly about the term functor, i.e., the functor that
formally constructs terms over a given signature. Such a term functor
over the category of sets and functions produces nothing but
conventional terms, but a term functor over a monoidal category with
more structure can provide terms and expressions with richer structure
and attributes. Stochastic and many-valued aspects are good examples,
and I often refer to nomenclatures in health care, where additions
structure is needed. Expressions e.g. involving diagnoses, functioning
and drugs do not run over the same category, and doing all of it in set
theory is basically ridiculous.

Not sure if these remarks help you at all, so I sincerely hope that more
prominent category theorists subscribing to this mailing list will
provide more enriched comments.

All the best and good luck with your work!

Patrik



On 2017-11-25 00:36, Neil Barton wrote:
> Dear All,
>
> I'm very interested in how categorial and material set theories
> interact, and in particular the advantages of each.
>
> It's well-known that categorial viewpoints are good for isolating
> schematic structural relationships. We can look at sets through this
> lens, by considering a categorial set theory like ETCS (possibly
> augmented, e.g. with replacement). A remark one sometimes finds is
> that once you have defined membership via arrows from terminal
> objects, you could use ETCS for all the purposes to which ZFC is
> normally put.
>
> My question is the following:
>
> (Q) To what extent can you ``do almost the same work'' with a
> categorial set theory like ETCS vs. a material set theory like ZFC?
>
> Just to give a bit more detail concerning what I'm thinking of:
> Something material set theory is reasonably good at is building models
> (say to analyse relative consistency), or comparing cardinality.
> However, there's no denying that for representing abstract
> relationships the language is somewhat clunky, since the same abstract
> schematic type can be multiply instantiated by structures with very
> different set-theoretic properties. So, to what extent can a
> categorial set theory like ETCS supply the good bits of the fineness
> of grain associated with material set theories, whilst modding out the
> `noise'?
>
> For example, the following are easily stated in material set theory:
>
> 1. \aleph_17 is an accessible cardinal.
>
> In material set theory, it's easy to define the aleph function and
> then state that the 17th position in this function can be reached by
> iterating powerset and replacement. But I wouldn't even know how to
> talk about specific sets of different cardinalities categorially. I
> suppose you could say something in terms of isomorphism between
> subobjects, and then exponentials, but it's quite unclear to me how
> the specifcs would go. Is that an easily claim to state (and prove) in
> ETCS?
>
> 2. How would you state that {{}} and {\beth_\omega} are very different
> objects? Set-theoretically, these look very different (just consider
> their transitive closures, for instance). But category-theoretically
> they should look the same---since they are both singletons they are
> isomorphic. So is this a case where their different set-theoretic
> propeties are considered just `noise', or where ETCS just wouldn't see
> a relationship, or where ETCS can in fact see some of these properties
> (and I'm just missing something)?
>
> 3. How would ETCS deal with model theory and cardinality ascriptions?
> (This links to a question asked earlier on this mailing list
> concerning syntactic theories in category theory, and whether from the
> categorial viewpoint we should be taking notice of them at all.) For
> instance, it's an interesting theorem (for characterising structure)
> that a first-order theory categorical in one uncountable power is
> categorical in every uncountable power (Morley's Theorem). But I have
> no idea how one might formalise this in something like ETCS---I know
> of Makkai and Reyes textbook (which I am currently reading) on
> categorial logic (where theories are represented by categories and
> models by functors), but I don't see how you could get
> categoricity-in-power claims out of the set up there. Can this be
> done?
>
> Any help and/or discussion would be greatly appreciated!
>
> Best Wishes,
>
> Neil
>
> --
> Dr. Neil Barton
> Postdoctoral Research Fellow
> Kurt G??del Research Center for Mathematical Logic
> University of Vienna
> Web: https://neilbarton.net/
>
>
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