From: street@mpce.mq.edu.au (Ross Street)
To: categories@mta.ca
Subject: Re: query
Date: Thu, 19 Nov 1998 11:15:46 +1000 [thread overview]
Message-ID: <199811190014.LAA28074@macadam.mpce.mq.edu.au> (raw)
>The language of higher category theory in more than analogous
>to homotopy theory, especially in the cellular version. What
>is the appropriate reference for a non-categorical reader?
>Have the A_\infty categories of Smirnov or Fukaya been treated
>in the categorical literature??
Fortunately, since the A_\infty categories (described sketchily in the
preprint I have of Fukaya) use chain complexes rather than topological
spaces (so that I believe a 1-object A_\infty category is an algebra for
the A_\infty non-permutative operad on chain complexes - an A_\infty
DG-algebra), we do not need to realise the whole homotopy types dream (as
described by John Baez) to give a categorical description of A_\infty
categories. Several years ago I remember discussing this by email with Jim
Stasheff. Dominic Verity was here at the time. I cannot remember all the
details but the two basic ingredients were some free strict n-categories
made from the set (whose elements are to be the objects of the A_\infty
category) in much the way the orientals are constructed, and the
construction (below) mentioned in my Oberwolfach Descent Theory notes of
September 1995.
There is a connection between the Gray tensor product and ordinary
chain complexes. Each chain complex R gives rise to a (strict)
omega-category J(R) whose 0-cells are 0-cycles a in R, whose 1-cells b
: a --> a' are elements b in R_1 with d(b) = a'- a, whose 2-cells c
: b --> b' are elements c in R_2 with d(c) = b'- b, and so on. All
compositions are addition. This gives a functor J : DG --> omega-Cat
from the category DG of chain complexes and chain maps. In fact, J :
DG --> omega-Cat is a monoidal functor where DG has the usual tensor
product of chain complexes and omega-Cat has the Gray tensor product. By
applying J on homs, we obtain a (2-) functor J_* : DG-Cat --> V_2-Cat,
where V_2 is omega-Cat with the Gray-like tensor product (extending the
natural tensor product of oriented cubes as described in Sjoerd Crans
thesis). In particular, since DG is closed, it is a DG-category and we
can apply J_* to it. The V_2-category J*(DG) has chain complexes as
0-cells and chain maps as 1-cells; the 2-cells are chain homotopies and the
higher cells are higher analogues of chain homotopies.
Best regards,
Ross
next reply other threads:[~1998-11-19 1:15 UTC|newest]
Thread overview: 20+ messages / expand[flat|nested] mbox.gz Atom feed top
1998-11-19 1:15 Ross Street [this message]
-- strict thread matches above, loose matches on Subject: below --
2009-06-26 15:51 query Tom Leinster
2009-06-26 10:47 query Noson S. Yanofsky
2009-06-24 16:18 query jim stasheff
2008-07-17 8:35 Query Johannes Huebschmann
2003-10-02 12:55 query jim stasheff
2003-05-05 17:46 Query Oswald Wyler
[not found] <199811190226.NAA02248@macadam.mpce.mq.edu.au>
1998-11-20 23:06 ` query Michael Batanin
1998-11-19 9:31 query Marco Grandis
1998-11-18 4:12 query john baez
1998-11-16 22:08 query James Stasheff
1997-10-07 11:30 query categories
1997-10-02 19:52 query categories
1997-10-01 19:50 query categories
1997-07-01 18:14 Query categories
1997-07-01 2:41 Query categories
1997-07-01 2:39 Query categories
1997-06-29 14:39 Query categories
1997-02-10 15:52 query categories
1997-02-10 1:03 query categories
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