Re: 'Directed Algebraic Topology'

[Marco Grandis <grandis@dima.unige.it>]

Dear George,

There is indeed such a connection between asymmetric distances
and directed algebraic topology, but is not entirely trivial. The
obvious solution,
by 'left' and 'right topologies' would not work well. The good
solution, in my
opinion, is constructing a 'symmetric topology' and adding
distinguished paths.

All this is in my book, and also in a paper:
M. Grandis,
The fundamental weighted category of a weighted space: From directed
to weighted algebraic topology,
Homology Homotopy Appl. 9 (2007), 221-256.

I am presently working with a colleague. Later, I will be able to
comment more precisely on
these points.

All the best

Marco


On 28 Sep 2009, at 20:43, George Janelidze wrote:

> Dear Colleagues,
>
> In addition to my message of September 22 addressed to Michael Barr
> and all
> of you (concerning 'Directed Algebraic Topology'/quasi-uniform
> spaces) I am
> forwarding a message from Guillaume Brummer:
>
> "...Thank you for copying this interesting material to me, and for
> mentioning quasi-uniform spaces to Michael Barr. Serious early work
> in this
> field was by Leopoldo Nachbin of Rio de Janeiro, published in CRASP
> 226
> (1948) 774-775, -- just 11 years after Andre' Weil's monograph on
> uniform
> spaces. Then came Nachbin's monograph Topologia e ordem (Chicago
> 1950), of
> which an English translation Topology and order was published by Van
> Nostrand (1964). Meantime the book by A'. Csa'sza'r, Fondements de la
> topologie ge'ne'rale, had appeared in Paris (1960)..."
>
> Guillaume Brummer also says:
>
> "...Hans-Peter K"unzi has published several surveys of this field
> since
> 1993, with lots of bibliography..."
>
> However, I still see no connection (which does not mean that it
> will never
> occur!) between "directed/asymmetric algebraic topology" and
> "asymmetric
> general topology". Surely Marco Grandis is the right person to ask
> about
> this. Well, since Marco mentioned preordered topological spaces,
> one could
> think of bitopological spaces and then quasi-uniform spaces might
> occur
> naturally...
>
> George Janelidze
>
>



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