Generalizing bicategories

Generalizing bicategories

[George Janelidze]

Dear All,

In addition to all comments and references given to John Baez, I would like
to mention MSc Thesis of my student Nelson Martins-Ferreira, and the paper

N. Martins-Ferreira: "Weak categories in additive categories with kernels",
Fields Institute Communications, Vol. 43, 2004, 387-410

(and its references). Nelson uses a notion of an internal weak category in a
2-category there (Page 391, Definition 6) and several related/more special
notions to develop a convenient setting for 2-dimensional "abelianization".
And I would like to use this opportunity to repeat few remarks on the story
of abelianization of categorical structures, which I briefly made in my talk
on CT1995 in Halifax:

(i) It is well known (precise references to be found in various surveys of
Ronnie Brown) for a long time that internal 2-categories in an additive
category A with kernels can be identified with composable pairs (f,g) of
morphisms in A with fg = 0, and that the same is true for n-categories, with
n-sequences instead of pairs. Hence, whenever somebody comes up with, say, a
new notion of a weak n-category, I would ask: what are the internal weak
n-categories in YOUR sense in the category Ab of abelian groups? (Of course
this question is "Yoneda invariant", and so there is no difference between
considering the category Ab and considering an abstract additive category
with kernels here)

(ii) Why do we expect a simple answer to the question above? The point is,
that most of higher categorical structures involve composition/coherence
maps between pullbacks of split epimorphisms - and in the additive case,
using kernels of those morphisms, one presents such pullbacks as direct sums
and the composition/coherence maps as matrices. Therefore I expect an
internal weak n-category in YOUR sense in the category of abelian groups to
be nothing but an additive functor from a fixed finitely generated category
X to Ab. The only question is: what is X?

(iii) There are many examples showing that making comparisons between higher
categorical structures might be highly nontrivial; so why not examining
first what will happen to them in the simple additive/abelian world?
Furthermore, and more generally, if T is a finite limit theory, then the
free-forgetful adjunction between Sets and Ab "induces" an adjunction
between Models(T) = Models(T,Sets) and Models(T,Ab), and the abelianization
(=the left adjoint in that adjunction) Models(T) ---> Models(T,Ab) should
help to study the category Models(T). Note that when T is the theory of
groups, the abelianization functor becomes the usual one, and so it
coincides with the first homology group functor (with coefficients in the
additive group of integers).

Answering the question from (i), one would certainly begin with
bicategories, and, as far as I know, Nelson was the first who has described
internal bicategories in additive categories with kernels, and I gave a talk
on Nelson's work on Australian Category Seminar in March 2002 - before
Nelson himself presented his more general results on the Meeting on Galois
Theory, Hopf Algebras, and Semiabelian Categories at Field Institute
(Toronto, September 2002).

Now results:

As shown by Nelson in the abovementioned paper (Section 5.2), an internal
weak category in Mor(Ab) (=Cat(Ab)) (which is a special case of Nelson's
more general result!) can be identified with a diagram in Ab consisting of
morphisms

d : A_1 ---> A_0, d' : B_1 ---> B_0, k : A_1 ---> B_1, k' : A_0 ---> B_0,

l, r : A_0 ---> A_1, and h : B_0 ---> A_1

with k'd = d'k, kl = kr = 0 and kh = 0. It becomes

(a) a bicategory, if B_1 = 0;
(b) a double category if l = r = 0 and h = 0;
(c) hence, a 2-category if B_1 = 0, l = r = 0 and h = 0.

Nelson also examines what would happen without the coherence conditions, and
then gets much more complicated formulas (See Proposition 6 in his paper).

What Nelson has not done is what Tom Leinster calls "weakening in both
directions". Another temptation is the (non-abelian) group case, hence
generalizing crossed complexes. Let me also mention

S. E. Crans: "Teisi in Ab", Homology, Homotopy and Applications 3, 2001,
87-100,

although I do not know if anyone has considered a "double-" version of
Crans' teisi.

George Janelidze