🚀 go-pugleaf

In article <1992Nov5.022710.10234@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <abian.720910802@pv343f.vincent.iastate.edu> abian@iastate.edu (Alexander Abian) writes:
>>
>>   So, MY DEFINITION is:
>>
>>      An n-dimensional differentiable manifold  M  is a connected
>> subdomain of  an  n+k  dimensional  Euclidean space such that
>> at every point,   M  has an n-dimensional tangent hyperplane.
>
>If you require manifolds to be connected, what does coproduct in the
>category of manifolds become?  (I.e. how do you add two manifolds
>together?)  Does coproduct even exist then?  Seems unlikely.
>
>I'm guessing connectedness doesn't damage coequalizers.  Anyone know
>for sure?  John?

Geez, some people wouldn't understand how to tie their shoes until it
was explained to them in the language of categories!!  :-)  Let us pause
to recall the words of Goethe: "Computer scientists are like Frenchmen:
whatever you say to them they translate into their own language and
forthwith it is something entirely different."  (Okay, Goethe said
"mathematicians," not computer scientists.)  

As far as I can tell, manifolds don't have equalizers or coequalizers.
I have never messed with coequalizers but I guess you just turn all the
arrows around in the definition of equalizers... so given two smooth
maps f,g from X to Y, the coequalizer would be a manifold Z with a smooth
map from Y to Z such that any map h from Y such that hf = hg factors uniquely
through Z.  So it seems like we are taking a quotient of Y, identifying
the point f(x) with the point g(x) for all x in X.  This quotient won't
usually be a manifold.  

Of course, there are spaces that are a little nastier than manifolds but
still interesting.