Re: relationship between categorical limts and usual limits
- From: Nath Rao <XrTaHoEnCaAtPhS@xxxxxxxxx>
- Date: Fri, 4 Jan 2008 15:00:34 +0000 (UTC)
harsha wrote:
hi,
i was wondering as to what exactly is the relationship between
categorical limits and the usual notion of limits in topological
spaces.
[Seeing that no one has answered this yet]
It is better to think of mathematical terminology as evolving, same as
other cultural constructs.
more specifically, two naive questions :
can one impose topological structure on a category so that notion of
limits coincide?
I doubt this, but I cannot give a counter-example offhand. Since
fixed-point sets of monoid actions can be described as limits, it should
be possible to do so.
can one derive a category from the category of open sets in a
topological space in which somehow points correspond to limits?
There is a concept of locales which are very close to (sober) spaces.
There must be a canonical way of getting points from spatial locales.
The omnibus reference is Johnstone's "Sketches of an elephant".
---
To go back to my first comment: the terms "limit" and "colimit" replaced
(and generalized the precategorical versions of) "inverse limit" and
"direct limit". Inverse limits arose as generalization of intersection
of (compact) subsets of a some space. For the last, you might be able to
cook up something using the Hausdorff topology. There are recent books
on history of topology that may actually indicate how. If it fails, you
can try plowing through Lefshetz's Colloquium lectures on algebraic
topology (available free, in pdf format, thru AMS web site)
Regards
Nath Rao
.
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