Re: Non-Cartesian Homological Algebra

["Stephen Urban Chase" <suc1@cornell.edu>]

Here are some further remarks on the interesting concepts discussed in
George's post and the replies to it.  Incidentally, I prefer the term
"non-commutative category theory" to "non-Cartesian ...", but that's a
personal matter.

If A is an algebra over a field  k, then, in the context of the internal
category theory introduced in Aguiar's 1997 thesis, an  A-coring is simply
an internal category in the dual of the monoidal category  Vec(k)  of 
k-spaces.  On the other hand, the notion of internal category in Vec(k)
itself includes (but is more general than) the small k-linear categories
in Mitchell's 1972 paper, "Rings with Several Objects".  I think it is
useful to view these 2 concepts as special cases of the same general
theory.  Some of the constructions for corings in the book of Brzezinski
and Wisbauer are special cases of categorical notions developed in
Aguiar's thesis.

Corings have been around for awhile, as Sweedler's original paper
introducing them was published in 1975, and they appeared in at least a
few other papers during the ensuing 20 years (e.g., Takeuchi's
monograph on a Morita theory for monoidal categories of bimodules, which I
think was published in the Journal of the Math. Society of Japan). 
Sweedler's version of Jacobson's theorem in his paper seems to be a sort
of non-commutative analogue of the connection between equivalence
relations and quotient spaces.

Aguiar's framework is probably not general enough to cover some cases of
interest.  It may be that one should begin simply with a monad in an
arbitrary bicategory, since at least a few of the basic concepts for
internal categories can be developed in that setting.  Taking the
bicategory to be Vec(k) with a single object, the important notion of
entwined structure, discussed in work of Caenepeel, Brzezinski, Pareigis,
and others, then appears as a sort of distributive law, but relating a
monad and a comonad rather than 2 monads.  The theorem that an entwined
structure is equivalent to a certain type of coring is then apparently an
analogue of Beck's theorem.

One would like the theory to include Takeuchi's notion of X-bialgebra
(which generalizes concepts developed earlier by Sweedler and,
independently, David Winter).  However, to that end it appears useful to
enrich the bicategory so that the
1-endomorphisms of an object (and 2-morphisms between them) constitute a
2-monoidal category in the sense of [M. Aguiar and S. Mahajan, Hopf
Monoids in Species and Associated Hopf Algebras] (see Chapter 5, although
most of that monograph, not yet completed, is on a quite different
subject).  There is a link to the monograph from Aguiar's website.  He and
I have had some discussions on these matters during the summer, but there
is still much about this situation that we (or, at least, I) don't
understand.

Steve Chase