[FOM] R: Preprint: "The unification of Mathematics via Topos Theory"
Vaughan Pratt
pratt at cs.stanford.edu
Sun Jul 11 22:54:50 EDT 2010
Dear Olivia,
> 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.
Well, of course: at least one of the two examples of a Morita-equivalent
pair should not be Cauchy complete. Any category of simplicial sets
with cells of dimension at least one would serve that purpose.
The canonical example is reflexive directed graphs. These are M-sets,
via the monoid M consisting of the three monotone maps of the chain 2,
aka the free bounded distributive lattice on one generator. However
this M is not Cauchy complete. Cauchy completion creates the sort
VERTEX when M had only the sort EDGE (the object of M when M is viewed
as a one-object category).
The undirected case, reflexive undirected graphs, is obtained by
including the twist map on 2, equivalently taking M to be the monoid of
all four functions from the set 2 to itself, aka the free Boolean
algebra on one generator. (Caveat for undirected graphs: the
distinguished self-loop at each vertex, needed for reflexivity, may but
need not be its own opposite, creating the notion of a
semi-distinguished self-loop.) Marco Grandis has developed a rich
theory of higher dimensional such.
> 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.
That's a very good insight abstractly, though I always find it helpful
to consider a representative cross-section of examples, and even to
classify them (analogously to identifying the subdirect irreducibles of
a variety). But merely knowing that there are in principle infinitely
many examples is cold comfort to the reader who has not yet generated
the first one from the axioms.
> In some respects, the example of Boolean algebras and Boolean rings
is not
> particularly representative of the notion of Morita-equivalence.
Yes. I probably should have said so more succinctly, since that was my
main point.
> 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.
Right. If you used say the above graph/M-set example, explained in as
elementary language as possible, you could make these concepts
accessible to a much broader audience than with the present language,
which assumes a great deal of background that I would think can be
dispensed with, at least at an introductory level.
> the indecomposable projective objects
...better known to the FOM community as the free multisorted unary
algebras on one generator per sort.
> and so we can rephrase any property of (the Cauchy completion of) C
> as a property of this full subcategory.
Right. There is neither need nor harm in carving out separate sorts for
the fixpoints of idempotents, since they can always be identified as
those x satisfying fx = x for the relevant idempotent f without needing
their own sort. Cauchy completion is just a natural normalization that
ensures that every kind of such fixpoint has an explicit sort.
> 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.
Indeed, and the same message posted to the categories mailing list would
not have elicited the same response from me. Based on the FOM traffic
it would appear that relatively few FOM subscribers would have the
requisite background to appreciate your topos-theoretic axiomatic
approach. The rest would be better served by starting out in the other
direction.
Because the abstract point of view is strange to most real-world
consumers of Boolean algebra, this is the approach to Boolean algebras
(plural) I've adopted in my Wikipedia article on Boolean algebra in the
section
http://en.wikipedia.org/wiki/Introduction_to_Boolean_algebra#Boolean_algebras
which starts "The term 'algebra' denotes both a subject, namely the
subject of algebra, and an object, namely an algebraic structure.
Whereas the foregoing has addressed the subject of Boolean algebra, this
section deals with mathematical objects called Boolean algebras. We
begin with a special case of the notion definable without reference to
the laws of Boolean algebra, namely concrete Boolean algebras, and then
pass to the general case based on those laws, namely abstract Boolean
algebras."
> 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.
Indeed. Generality is the other benefit of abstraction besides
simplicity. Very nice work, I appreciated the opportunity to see the
paper in advance. Note that I'm not an expert on topos theory, my
interests are too broad to free up the time needed to become one, but
your paper may help me get up to speed.
Incidentally you might enjoy a talk I've given at a few conferences
since April about topoalgebraic categories:
http://boole.stanford.edu/pub/topoalg.pdf
(Slides are organized into columns, making OUTLINE the fourth slide.)
The algebra portion will be very familiar to you, once you've translated
it back into topos language: your work concerns the case P = 0 of no
properties, hence no topology (in this sense). The localic
(topological) portion doesn't so translate, you'll have to generalize
your work from toposes to linearly distributive categories. These are
the appropriate generalization of *-autonomous categories when, as
almost invariably, there is no isomorphism between the sorts S and the
properties P. The case S = P = 1, where S and P are isomorphic (or even
equal) gives rise to the *-autonomous category Set x Set^op (=
Chu(Set,1)) in the same way S = 1, P = 0 gives rise to the topos Set.
Vaughan
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