On Mon, Sep 13, 2010 at 11:08 PM, Michael Shulman <shulman@math.uchicago.edu> wrote:
This is also true of Batanin's definition, which takes as basic underlying data a globular set....potentially including (I believe) pretty much all definitions of higher category.
I wrote this without thinking hard enough; sorry. Of course, one also has to consider the extra structure placed on the underlying data. Extra structure of the "horn-filling" variety, as in Joyal's and Street's definitions, consists of conditional assertions that certain dependent types are inhabited, which is certainly "non-evil." I would expect that all the "non-algebraic" definitions could be dealt with similarly; for instance, the Simpson-Tamsamani definition involves also the assertion that certain maps of (n-1)-categories are equivalences, which should itself be a "non-evil" assertion based again on inhabitation of certain dependent types. But it would be tricky to write all of that out carefully. For Batanin-type definitions, it is going to depend on what operad you pick; for instance strict omega-categories are definitely "evil." But I would guess that if you use a "CW operad" which is built up freely, as an operad, by attaching operations of successively higher dimension whose boundaries are composite operations of lower dimension (which is how I usually think of an operad for weak higher categories), then its algebras should also be definable in a "non-evil" way. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]