From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: Lie Groups
Date: 02 Jul 2000 16:39:16 GMT
Newsgroups: sci.math
Summary: [missing]

A category is a collection of objects and arrows, where each arrow has
a source and a destination, and a way of combining two arrows if the
destination of the first one is the same as the source of the second
one, where each object X has an identity arrow from X to X which is
an identity for composition, and where the composition is associative.

Manifolds form a ("large") category, where the objects are the
manifolds (and there are too many for there to be a set of all
manifolds, which is why the category is "big") and the arrows are the
smooth maps between manifolds, with composition being the same as
ordinary function composition.

This is all defined anywhere you'd look it up. But to appreciate the
definition "it's a group object in the category of manifolds", you'd
also need to know the definition of "group object in a category".

If you care, a group object in a category is defined to be analogously
to the way a group (in the category of sets) is defined. It's an
object X, with a multiplication m:XxX->X and inverse map i:X->X
satisfying identity, associativity, and inverse properties. The wrinkle
is that XxX and XxXxX (needed to consider associativity) cannot be
defined as sets of ordered pairs, because X is just an object in a
category and might not be a set of anything.

Instead, we define an abstract product object XxY of two objects in a
category to be an object with two projection mappings p1:XxY->X and
p2:XxY->Y with the property that for any two mappings f:Z->X and
g:Z->Y there exists a unique map fxg such that the compositions of
fxg with p1 and p2 are f and g respectively. We can show that any
two products of X and Y are isomorphic, and (XxY)xZ and Xx(YxZ) are
also isomorphic.

In the case of manifolds, though, this is overkill. The product of
two manifolds X and Y consists of pairs of points (x,y) with
coordinate charts built up by sticking together coordinates for x and
coordinates for y. A group object is simply a group, and that it's in
this category just means that the multiplication and inverse maps are
smooth maps.

Keith Ramsay