John Baez
July 7, 2008
Computation and the Periodic Table
By now there is an extensive network of interlocking analogies
between physics, topology, logic and computer science, which
can be seen most easily by comparing the roles that symmetric
monoidal closed categories play in each subject. However, symmetric
monoidal categories are just the n = 1, k = 3 entry of a hypothesized
"periodic table" of ktuply monoidal ncategories. This
raises the question of how these analogies extend. We present
some thoughts on this question, focusing on how
symmetric monoidal closed 2categories might let us understand
the lambda calculus more deeply.
This is based on work in progress with Mike Stay.
Click on this to see the transparencies of the talk:
For more details try the following papers:

John Baez and James Dolan, Higherdimensional algebra
and topological quantum field theory,
J. Math. Phys. 36 (1995), 60736105.

John Baez and Laurel Langford, Higherdimensional
algebra IV: 2tangles, Adv. Math. 180 (2003), 705764.

John Baez and Mike Stay, Physics, topology,
logic and computation: a Rosetta Stone,
to appear in New Structures in Physics, ed. Bob Coecke.

Albert Burroni,
Higherdimensional word problems with applications
to equational logic, Theor. Comp. Sci. 115 (1993),
4362.

Yves Guiraud, The three
dimensions of proofs, Ann. Pure Appl.
Logic 141 (2006), 266295.

Barnaby P. Hilken, Towards a
proof theory of rewriting: the simplytyped 2λcalculus,
Theor. Comp. Sci. 170 (1996), 407444.

C. Barry Jay and Neil Ghani, The virtues of etaexpansion,
J. Functional Programming 1 (1993), 119.

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. 697701.

R. A. G. Seely, Modeling
computations: a 2categorical framework, in
Proc. Symposium on Logic in Computer Science 1987,
Computer Society of the IEEE, pp. 6571.

Vladimir Voevodsky, A very short note on the
homotopy lambda calculus, Sept. 27, 2006.
Also try my seminar notes from
Fall 2006 and
Winter 2007.
© 2008 John Baez
baez@math.removethis.ucr.andthis.edu