* chain complexes
@ 2003-07-14 7:29 John Baez
2003-07-20 16:43 ` chain complexes: in reply to John Baez Ronald Brown
0 siblings, 1 reply; 2+ messages in thread
From: John Baez @ 2003-07-14 7:29 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 2+ messages in thread
* Re: chain complexes: in reply to John Baez
2003-07-14 7:29 chain complexes John Baez
@ 2003-07-20 16:43 ` Ronald Brown
0 siblings, 0 replies; 2+ messages in thread
From: Ronald Brown @ 2003-07-20 16:43 UTC (permalink / raw)
To: categories
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
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