Re: Evil in bicategories

[Michael Shulman <shulman@math.uchicago.edu>]

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


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