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

Vaughan Pratt pratt at cs.stanford.edu
Fri Jul 9 03:18:44 EDT 2010

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 

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 

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
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