[FOM] R: Preprint: "The unification of Mathematics via Topos Theory"

Sat Jul 10 20:25:09 EDT 2010

Dear Vaughan,

Thank you for your comments. 

You are certainly right in saying that the notion of Morita-equivalence
plays a central role in my approach (indeed, as explained in the paper,
Morita-equivalences and topos-theoretic invariants are the fundamental
ingredients on which the 'machinery' can be put at work), and of course it
would be nice to have more natural (i.e. directly arising from the
mathematical practice and not achieved by topos-theoretic means) examples of
Morita-equivalences. I quote from my paper:

"Morita-equivalence is a general notion of equivalence of mathematical
theories which is ubiquitous in Mathematics (even though there has not been
much interest in the past in identifying Morita-equivalences possibly due to
the lack of a general theory ascribing central importance to this notion and
demonstrating its technical usefulness - one of the purposes of the present
paper is in fact to advocate the extreme importance of investigations in
this area, cfr. sections 6 and 10). As a simple example of theories which
are Morita-equivalent, one can take the theory of Boolean algebras and the
theory of Boolean rings."

On the other hand, as remarked in the paper, Topos Theory itself is a
primary source of Morita-equivalences (in fact, a single mathematical theory
'generates' an infinite number of Morita-equivalences via Topos Theory) so
in many cases one does not need to find 'natural examples' of
Morita-equivalences in order to extract important information about
mathematical theories of interest via the machinery described in the paper. 

In some respects, the example of Boolean algebras and Boolean rings is not
particularly representative of the notion of Morita-equivalence. Indeed, the
classifying topos of the two theories is the same (up to categorical
equivalence) for a 'trivial' reason, in the sense that the equivalence
between the two classifying toposes can already be seen at the level of
sites (indeed, it amounts to a categorical equivalence between the
categories of finitely presentable models of the two theories, while the
Grothendieck topologies on the opposites of these two categories are in both
cases the trivial ones). On the other hand, the most interesting
applications of the methodologies described in the paper arise when we have
two sites of definition of the classifying topos which are 'different
enough' from each other, so that the relationships between the two theories
that one discovers by applying the 'machinery' are *not* naturally visible
(let alone attainable) by working at the level of sites.             

In general, it is worth to note that most of the usual categorical
properties can be lifted (up to Cauchy completion i.e. splitting of
idempotents) to the level of toposes; indeed, the Cauchy completion of a
small category C can be identified (via the Yoneda embedding) with the full
subcategory of the topos [C^op, Set] on the indecomposable projective
objects, and so we can rephrase any property of (the Cauchy completion of) C
as a property of this full subcategory. The point is that, as long as we
restrict our attention to presheaf toposes, we cannot expect topos-theoretic
methodologies to generate insights that could not already be attainable by
the standard means of category theory. On the other hand, when we put
non-trivial Grothendieck topologies on categories, we get an extremely rich
'combinatorics', which can be exploited (as indicated in the paper) to
extract a great variety of non-straightforward insights on
Morita-equivalences.

Olivia   

-----Messaggio originale-----
Da: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] Per conto di
Vaughan Pratt
Inviato: venerdì 9 luglio 2010 8.19
A: Foundations of Mathematics
Oggetto: Re: [FOM] Preprint: "The unification of Mathematics via Topos
Theory"

With the exception of the introductory sections 1 and 2, every section 
of this ten-section paper either refers to or depends in an apparently 
essential way on Morita equivalence.  One would therefore hope for a 
definition of Morita equivalence followed not too long thereafter by an 
example thereof.

The definition would appear to be in section 3, which says "two
geometric theories have equivalent classifying toposes if and only if 
they have equivalent categories of models in every Grothendieck topos E, 
naturally in E. Two such theories are said to be Morita-equivalent." 
Reading between the lines, "two such theories" presumably means two that 
satisfy either side of the if-and-only-if.

In the second-last paragraph of section 4 we read "We can think of each 
site of definition of the classifying topos of a geometric theory as 
representing a particular aspect of the theory, and of the classifying 
topos as embodying those essential features of the theory which are 
invariant with respect to particular (syntactic) presentations of the 
theory which induce Morita-equivalences at the semantical level. We 
shall come back to this point in sections 4 and 6 below."  Presumably 
"section 4" was meant to be "section 5", where the definition is 
repeated in the first paragraph, and an example given in the second 
paragraph.

The example is that of the Morita-equivalence of Boolean algebras and 
Boolean rings.  (Some FOM readers may remember the week-long argument 
Steve Simpson and I had in 1999 as to how best to think of this 
equivalence.)

The question I'd like to raise here is, how representative is this 
example of Morita equivalence?

The examples I'm aware of of Morita-equivalence that are not mere 
equivalences all involve splitting idempotents.  What are the 
idempotents that must be split in order to make this an interesting 
example of Morita-equivalence?

At the end of the week in 1999 Steve and I settled our dispute by 
agreeing that the theories were not isomorphic but were equivalent. 
Steve and I split many hairs during the week, but as I recall no 
idempotents were harmed in order to reach a meeting of the minds.

If these two theories are indeed not only Morita-equivalent but 
equivalent, is there any reason not to suppose, based on the example of 
the concept in the paper, that Morita-equivalence is merely equivalence?

Another core notion of the paper is that of topos-theoretic invariance. 
  This notion is much better served by its examples: Boolean, De Morgan, 
atomic, two-valued, connected, locally connected, compact, local, 
subtopos (of a given topos), classifying (for a given geometric theory), 
etc., which seem more representative of that notion.

It would be helpful to have an equally representative range of examples 
of Morita equivalence, especially ones as accessible as Boolean algebras 
vs. Boolean rings but less degenerate (unless I've overlooked something, 
as I may well have).

Vaughan Pratt

On 7/8/2010 5:05 AM, Olivia Caramello wrote:
> The following paper, recently presented at the International Category
Theory
> Conference 2010, might be of interest to subscribers to this list:
>
> O. Caramello, "The unification of Mathematics via Topos Theory"
>
> Abstract:
> We present a set of principles and methodologies which may serve as
> foundations of a unifying theory of Mathematics. These principles are
based
> on a new view of Grothendieck toposes as unifying spaces being able to act
> as 'bridges' for transferring information, ideas and results between
> distinct mathematical theories.
>
> The paper is available from the Mathematics ArXiv at the address
> http://front.math.ucdavis.edu/1006.3930.
>
> Comments are welcome.
>
> Olivia Caramello
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
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