Dear John, Ok! i'm interested. i know you've probably given this (on this list even) dozens of times before, but is there a self-contained account of Leinster's refinement of Batanin's approach? For example, arXiv has a link to the promisingly titled Higher Operads, Higher Categories. Would this be a place to begin? i note the Eugenia Cheng has a survey paper on weak n-categories. Is this a place to begin? Best wishes, --greg On Thu, Sep 2, 2010 at 9:04 PM, John Baez <baez@math.ucr.edu> wrote:
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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