re: the definition of "evil"

[John Baez <john.c.baez@gmail.com>]

Dear Categorists -

I'm glad Peter is trying to formulate a definition of structures that can be
transported along equivalences, and I like the spirit of his definition,
namely in terms of a "lifting property" where one has a 2-functor

U: XCat -> Cat

and one is trying to lift equivalences from Cat to XCat.

But it makes me nervous when he says "isomorphic [not equivalent!]".  Just
as evil in category theory typically arises from definitions that impose
equations between objects instead of specifying isomorphisms, evil in
2-category theory typically arises when we specify isomorphisms between
objects instead of specifying equivalences.

It would be sad, or at least intriguing, if the definition of "evil" was
itself evil.

Best,
jb

  DEFINITION. Let X be some structure on categories. By this, I mean
>  that there is a given 2-category called X-Cat, whose objects are
>  called X-categories, whose morphisms are called X-functors, and whose
>  2-cells are called X-transformations, and for which there is a given
>  2-functor U to Cat, called the forgetful functor.
>
>  We say that X is "transported along equivalences of categories" if the
>  following holds. Given an X-category D', with underlying category D =
>  U(D'), and a category C, and an equivalence (F,G,e,h) of categories D
>  and C, where F: D -> C, G: C -> D, and e: FG -> id_C, h: GF -> id_D,
>  it is then possible to find:
>
>  (1) an X-category C' whose underlying category U(C') is isomorphic
>     [not equivalent!] to C. Let c : U(C') -> C be the isomorphism
>     (i.e., an invertible functor in Cat) with inverse c': C -> U(C');
>
>  (2) an X-equivalence of X-categories (F',G',e',h'), where
>     F': D' -> C', G': C' -> D', e': F'G' -> id_C', and h': G'F' -> id_D'
>     [the concept of equivalence makes sense in any 2-category];
>
>  such that
>
>  (3) (U(F'), U(G'), U(e'), U(h')) = (cF, Gc', cec', h).
>
>  Here, cF and Gc' denotes composition of functors, and cec' denotes
>  whiskering.
>
>  The structure X is called "evil" iff it is not transported along
>  equivalences of categories.
>
>  This finishes the definition.
>
>


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