chain complexes

chain complexes

[John Baez]

Dear Categorists -

Who first showed that an internal category in
the category of abelian groups was a 2-term chain
complex of abelian groups?  What's a good reference?

Who first showed that an internal strict omega-category
in the category of abelian groups was a chain complex
of abelian groups?  What's a good reference?

(Of course for "internal X in the category of Y's",
I am willing to accept "internal Y in the category of X's"
as a substitute.)

Best,
jb

Re: chain complexes: in reply to John Baez

[Ronald Brown]

reply to r.brown@bangor.ac.uk

The paper

(R. BROWN and HIGGINS, P.J.), `Cubical abelian groups with
connections are equivalent to chain complexes', Homology, Homotopy
and Applications, 5(1) (2003) 49-52.

has a reference to Grothendieck's `cat\'egorie
cofibr\'ee....'  (1968)  SLNM 79, (the canonical reference for the first
question),
and also to Bourn (JPAA 1990), and gives a proof that 5 different
structures are, in an additive category with kernels, equivalent to chain
complexes.  Among these structures is strict globular omega-categories.
However, this is deduced from some non abelian and more difficult results.

Ronnie Brown







Dear Categorists -
Who first showed that an internal category in
the category of abelian groups was a 2-term chain
complex of abelian groups? What's a good reference?
Who first showed that an internal strict omega-category
in the category of abelian groups was a chain complex
of abelian groups? What's a good reference?
(Of course for "internal X in the category of Y's",
I am willing to accept "internal Y in the category of X's"
as a substitute.)
Best,
jb