Subject: categories: Geocal 06

From: "John Baez" <[email-masked]>
Date: Sun, 26 Mar 2006 13:50:19 -0800 (PST)

Here's some stuff about Geocal 06 and n-categories in proof theory.

Best,
jb

Also available at http://math.ucr.edu/home/baez/week227.html

March 12, 2006
This Week's Finds in Mathematical Physics (Week 227)
John Baez

[stuff about physics deleted]

I had a great time in Marseille.  The area around there is great for
mathematicians.  Algebraists can visit the beautiful nearby city of
Aix - pronounced "x".  Logicians will enjoy the dry, dusty island of
If - pronounced "eef", just like a French logician would say it.
And everyone will enjoy the medieval hill town of Les Baux, which looks
like something out of Escher.

Actually the Chateau D'If, on the island of the same name, is where
Edmond Dantes was imprisoned in Alexander Dumas' novel "The Count of
Monte Cristo".  It's in this formidable fortress that the wise old
Abbe Faria tells Dantes the location of the treasure that later made
him rich.

I guess everyone except me read this story as a kid - I'm just reading
it now.  But how many of you remember that Faria spent his time in
prison studying the works of Aristotle?  There's a great scene where
Dantes asks Faria where he learned so much about logic, and Faria replies:
"If - and only If!"

That Dumas guy sure was a joker.

Luckily I didn't need to be locked up on a deserted island to learn
some logic in Marseille.  There were lots of great talks on this topic
at the conference I attended:

7) Geometry of Computation 2006 (Geocal06), http://iml.univ-mrs.fr/geocal06/

For example, Yves Lafont gave a category-theoretic approach to Boolean
logic gates which explains their relation to Feynman diagrams:

8) Yves Lafont, Towards an algebraic theory of Boolean circuits,
Journal of Pure and Applied Algebra 184 (2003), 257-310.  Also
available at http://iml.univ-mrs.fr/~lafont/publications.html

and together with Yves Guiraud, Francois Metayer and Albert Burroni,
he gave a detailed introduction to the homology of n-categories and
its application to rewrite rules.  The idea is to study any sort of
algebraic gadget (like a group) by creating an n-category where the
objects are "expressions" for elements in the gadget, the morphisms
are "ways of rewriting expressions" by applying the rules at hand,
the 2-morphisms are "ways of passing from one way of rewriting
expressions to another" by applying certain "meta-rules", and so on.
Then one can use ideas from algebraic topology to study this
n-category and prove stuff about the original gadget!

To understand how this actually works, it's best to start with Craig
Squier's work on the word problem for monoids.  I explained this
pretty carefully back in "week70" when I first heard Lafont lecture
on this topic - it made a big impression on me!  You can read more here:

9) Yves Lafont and A. Proute, Church-Rosser property and homology of
monoids, Mathematical Structures in Computer Science, Cambridge U.
Press, 1991, pp. 297-326.  Also available at
http://iml.univ-mrs.fr/~lafont/publications.html

10) Yves Lafont, A new finiteness condition for monoids presented by
complete rewriting systems (after Craig C. Squier), Journal of Pure and
Applied Algebra 98 (1995), 229-244.  Also available at
http://iml.univ-mrs.fr/~lafont/publications.html

Then you can go on to the higher-dimensional stuff:

11) Albert Burroni, Higher dimensional word problem with application
to equational logic, Theor. Comput. Sci. 115 (1993), 43-62.
Also available at http://www.math.jussieu.fr/~burroni/

12) Yves Guiraud, The three dimensions of proofs, Annals of Pure
and Applied Logic (in press).  Also available at
http://iml.univ-mrs.fr/%7Eguiraud/recherche/cos1.pdf

13) Francois Metayer, Resolutions by polygraphs, Theory and
Applications of Categories 11 (2003), 148-184.  Available online
at http://www.tac.mta.ca/tac/volumes/11/7/11-07abs.html

I was also lucky to get some personal tutoring from folks including
Laurent Regnier, Peter Selinger and especially Phil Scott.  Ever since
"week40", I've been trying to understand something called "linear logic",
which was invented by Jean-Yves Girard, who teaches in Marseille.
Thanks to all this tutoring, I think I finally get it!

To get a taste of what Phil Scott told me, you should read this:

14) Philip J. Scott, Some aspects of categories in computer science,
Handbook of Algebra, Vol. 2, ed. M. Hazewinkel, Elsevier, New York, 2000.
Available as http://www.site.uottawa.ca/~phil/papers/

Right now, I'm only up to explaining a microscopic portion of this
stuff.  But since the typical reader of This Week's Finds may know more
about physics than logic, maybe that's good.  In fact, I'll use this
as an excuse to simplify everything tremendously, leaving out all sorts
of details that a real logician would want.

Logic can be divided into two parts: SYNTAX and SEMANTICS.  Roughly
speaking, syntax concerns the symbols you scribble on the page,
while semantics concerns what these symbols mean.

A bit more precisely, imagine some kind of logical system where you
write down some theory - like the axioms for a group, say - and use
it to prove theorems.

In the realm of syntax, we focus on the form our theory is allowed
to have, and how we can deduce new sentences from old ones.  So, one
of the key concepts is that of a PROOF.  The details will vary depending
on the kind of logical system we're studying.

In the realm of semantics, we are interested in gadgets that actually
satisfy the axioms in our theory - for example, actual groups, if we're
thinking about the theory of groups.  Such a gadget is called a MODEL
of the theory.  Again, the details vary immensely.

In the realm of syntax, we say a list of axioms X "implies" a sentence
P if we can prove P from X using some deduction rules, and we write this
as

X |- P

In the realm of semantics, we say a list of axioms X "entails" a sentence
P if every model of X is also a model of P, and we write this as

X |= P

Syntax and semantics are "dual" in a certain sense - a sense that can
be made very precise if one fixes a specific class of logical systems.
This duality is akin to the usual relation between vector spaces and
their duals, or more generally groups and their categories of
representations.  The idea is that given a theory T you can figure
out its models, which form a category Mod(T) - and conversely, given the
category of models Mod(T), perhaps with a little extra information,
you can reconstruct T.

A little extra information?  Well, in some cases a model of T will be
a *set* with some extra structure - for example, if T is the theory of
groups, a model of T will be a group, which is a set equipped with some
operations.  So, in these cases there's a functor

U: Mod(T) -> Set

assigning each model its underlying set.  And, you can easily reconstruct T
from Mod(T) *together* with this functor.

This idea was worked out by Lawvere for a class of logical systems
called algebraic theories, which I discussed in "week200".   But, the
same idea goes by the name of "Tannaka-Krein duality" in a different
context: a Hopf algebra H has a category of comodules Rep(H), which
comes equipped with a functor

U: Rep(H) -> Vect

assigning each comodule its underlying vector space.  And, you can
reconstruct H from Rep(H) together with this functor.  The proof is
even very similar to Lawvere's proof for algebraic theories!

I gave a bunch of talks in Marseille about algebraic theories, some
related logical systems called PROPs and PROs, and their relation to
quantum theory, especially Feynman diagrams:

14) John Baez, Universal algebra and diagrammatic reasoning, available
as http://math.ucr.edu/home/baez/universal/

I came mighty close to explaining how to compute the cohomology of
an algebraic theory... and you can read more about that here:

15) Mauka Jibladze and Teimuraz Pirashvili, Cohomology of algebraic
theories, J. Algebra 137 (1991) 253-296.

Mauka Jibladze and Teimuraz Pirashvili, Quillen cohomology and
Baues-Wirsching cohomology of algebraic theories, Max-Planck-Institut
fuer Mathematik, preprint series 86 (2005).

But alas, I didn't get around to talking about the duality between
syntax and semantics.  For that Lawvere's original thesis is a good
place to go:

16) F. William Lawvere, Functorial Semantics of Algebraic Theories,
Ph.D. thesis, Columbia University, 1963.  Also available at
http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html

Anyway, the stuff Phil Scott told me about was mainly over on the
syntax side.  Here categories show up in another way.  Oversimplifying
as usual, the idea is to create a category where an object P is a
*sentence* - or maybe a list of sentences - and a morphism

f: P -> Q

is a *proof* of Q from P - or maybe an equivalence class of proofs.

We can compose proofs in a more or less obvious way, so with any
luck this gives a category!  And, different kinds of logical system
give us different kinds of categories.

Quite famously, intuitionistic logic gives cartesian closed categories.
The "multiplicative fragment" of linear logic gives *-autonomous categories.
And, full-fledged linear logic gives us certain fancier kinds of categories.
If you want to learn about these examples, read the handbook article by
Phil Scott mentioned above.

One thing that intrigues me is the equivalence relation we need to get
a category whose morphisms are equivalence classes of proofs.  In
Gentzen's "natural deduction" approach to logic, there are various
deduction rules.  Here's one:

P |- Q    P |- Q'
------------------
  P |- Q & Q'

This says that if P implies Q and it also implies Q', then it implies
Q & Q'.

Here's another:

P |- Q => R
------------
P and Q |- R

And here's a very important one, called the "cut rule":

P |- Q    Q |- R
-----------------
     P |- R

If P implies Q and Q implies R, then P implies R!

There are a bunch more... and to get the game rolling we need
to start with this:

P |- P

In this setup, a proof f: P -> Q looks vaguely like this:

    f-crud
    f-crud
    f-crud
    f-crud
    f-crud
    f-crud
 -------------
    P |- Q

The stuff I'm calling "f-crud" is a bunch of steps which use the
deduction rules to get to P |- Q.

Suppose we also we also have a proof

g: Q -> R

There's a way to stick f and g together to get a proof

fg: P -> R

This proof consists of setting the proofs f and g side by side and then
using the cut rule to finish the job.  So, fg looks like this:

     f-crud     g-crud
     f-crud     g-crud
     f-crud     g-crud
     f-crud     g-crud
    --------   --------
     P |- Q     Q |- R
   ---------------------
           P |- R

Now let's see if composition is associative.  Suppose we also have a
proof

h: R -> S

We can form proofs

(fg)h: P -> S

and

f(gh): P -> S

Are they equal?  No!  The first one looks like this:

     f-crud     g-crud
     f-crud     g-crud
     f-crud     g-crud       h-crud
     f-crud     g-crud       h-crud
    --------   --------      h-crud
     P |- Q     Q |- R       h-crud
   ---------------------   -----------
           P |- R            R |- S
         ----------------------------
                    P |- S

while the second one looks like this:

                g-crud       h-crud
                g-crud       h-crud
    f-crud      g-crud       h-crud
    f-crud      g-crud       h-crud
    f-crud     --------     --------
    f-crud      Q |- R       R |- S
   ---------   ---------------------
    P |- Q            Q |- S
   ----------------------------
            P |- S

So, they're not quite equal!  This is one reason we need an
equivalence relation on proofs to get a category.  Both proofs
resemble trees, but the first looks more like this:

\  /  /
 \/  /
  \ /
   |

while the second looks more like this:

\  \  /
 \  \/
  \ /
   |

So, we need an equivalence relation that identifies these proofs
if we want composition to be associative!

This sort of idea, including this "tree" business, is very familiar
from homotopy theory, where we need a similar equivalence relation if we
want composition of paths to be associative.  But in homotopy theory,
people have learned that it's often better NOT to impose an equivalence
relation on paths!  Instead, it's better to form a *weak 2-category* of paths,
where there's a 2-morphism going from this sort of composite:

\  /  /
 \/  /
  \ /
   |

to this one:

\  \  /
 \  \/
  \ /
   |

This is called the "associator".  In our logic context, we can think of
the associator as a way to transform one proof into another.

The associator should satisfy an equation called the "pentagon identity",
which I explained back in "week144".  However, it will only do this if
we let 2-morphisms be *equivalence classes* of proof transformations.

So, there's a kind of infinite regress here.  To deal with this, it
would be best to work with a "weak omega-category" with

sentences (or sequences of sentences) as objects,
proofs as morphisms,
proof transformations as 2-morphisms,
transformations of proof transformations as 3-morphisms,...

and so on.   With this, we would never need any equivalence relations:
we keep track of all transformations explicitly.  This is almost beyond
what mathematicians are capable of at present, but it's clearly a good
thing to strive toward.

So far, it seems Seely has gone the furthest in this direction.
In his thesis, way back in 1977, he studied what one might call "weak
cartesian closed 2-categories" arising from proof theory.  You can read
an account of this work here:

17) R.A.G. Seely, Weak adjointness in proof theory, in Proc. Durham Conf.
on Applications of Sheaves, Springer Lecture Notes in Mathematics 753,
Springer, Berlin, 1979, pp. 697-701.  Also available at
http://www.math.mcgill.ca/rags/WkAdj/adj.pdf

R.A.G. Seely, Modeling computations: a 2-categorical framework, in
Proc. Symposium on Logic in Computer Science 1987, Computer Society
of the IEEE, pp. 65-71.  Also available at
http://www.math.mcgill.ca/rags/WkAdj/LICS.pdf

Can we go all the way and cook up some sort of omega-category of proofs?
Interestingly, while the logicians at Geocal06 were talking about
n-categories and the geometry of proofs, the mathematician Vladimir
Voevodsky was giving some talks at Stanford about something that sounds
pretty similar:

18) Vladimir Voevodsky, lectures on homotopy lambda calculus,
notice at http://math.stanford.edu/distinguished_voevodsky.htm

Voevodsky has thought hard about n-categories, and he won the
Fields medal for his applications of homotopy theory to algebraic
geometry.

The typed lambda calculus is another way of thinking about intuitionistic
logic - or in other words, cartesian closed categories of proofs.  The
"homotopy lambda calculus" should thus be something similar, but where
we keep track of transformations between proofs, transformations
between transformations between proofs... and so on ad infinitum.

But that's just my guess!  Is this what Voevodsky is talking about???
I haven't managed to get anyone to tell me.  Maybe I'll email him and ask.

There were a lot of other cool talks at Geocal06, like Girard's talk
on applications of von Neumann algebras (especially the hyperfinite
type II_1 factor!) in logic, and Peter Selinger's talk on the category of
completely positive maps, diagrammatic methods for dealing with these
maps, and their applications to quantum logic:

19) Peter Selinger, Dagger compact closed categories and completely
positive maps, available at http://www.mscs.dal.ca/~selinger/papers.html

But, I want to finish writing this and go out and have some waffles
for my Sunday brunch.  So, I'll stop here!

-----------------------------------------------------------------------

Addendum: An anonymous correspondent had this to say about the
"homotopy lambda calculus":

  Several years ago, Kontsevich explained to me an idea he had about
 "homotopy proof theory" (or model theory, or logic, ...).  As soon as I
  saw Voevodsky's abstract it reminded me of what Kontsevich said; perhaps
  it's a well-known idea in the Russian-Fields-medallist club.  Somewhere
  I have notes from what Kontsevich said, but as far as I remember it went
  roughly like this.

  In certain set-ups (such as Martin-Lof type theory) every statement
  carries a proof of itself.  Of course, a statement may have many proofs.
  If we imagine that all the statements are of the form "A = B", then what
  we're saying is that every equals sign carries with it a *reason* for
  equality, or proof of equality.  If I remember rightly, Kontsevich's
  idea was to do a topological analogue, so that every term (like A and B)
  is assigned a point in some fixed space, and equalities of terms induce
  paths between points.  There was more, pushing the idea further, but I
  forget what.

-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at

http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try

http://math.ucr.edu/home/baez/twfcontents.html

A simple jumping-off point to the old issues is available at

http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

http://math.ucr.edu/home/baez/this.week.html