Makkai's suggestion

[John Baez <baez@math.ucr.edu>]

Greg writes:


> Of course, after following such a path of least resistance, a journeyman
> categoryist might look at a variety of alternatives and consider their
> trade-offs. In this effort, the very quality you point out is of great
> utility in developing a genuine understanding of the design space. The
> initial presentation, this path of least resistance formulation, however,
> ought to have a precise sense in which it is *initial*, like an initial
> algebra.


Leinster's refinement of Batanin's approach defines weak infinity-categories
as algebras of an "initial globular operad with contractions".

Here "globular" means we're doing infinity-categories in the obvious way,
where given two n-morphisms f,g: x -> y we can talk about (n+1)-morphisms
from f to g.

"Algebra of a globular operad with contractions" means we can compose these
n-morphisms in all the pictorially obvious ways, and every pictorially
plausible law holds *up to a higher morphism*.

"Initial" means we're doing this in exactly the right way: for example,
there aren't any *extra* ways of composing morphisms, and we're not sticking
in *too many* of these higher morphisms.

I am sure people will eventually come up with better ways to do
infinity-category theory.  Eventually most math majors will learn it in
college (unless our current civilization collapses in less than, say, 150
years).    But the approaches we've got right now are already pretty good.
Learn 'em!

Best,
jb


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