categories: Re: sum of pointed sets.

["F. William Lawvere" <wlawvere@hotmail.com>]

The meaning of "sum" as coproduct depends on the specification of the
ambient category. If we are dealing with multi-pointed sets as suggested ,
the following would seem reasonable :

An m-pointing of a set X is an arbitrary map from m (as a set) to X, and a
morphism of such is a pair consisting of a map m to m' and a map X to X'
such that the equation expressed by the obvious square holds. In the
category thus defined, it is easily verified that your construction indeed
has the universal mapping property of coproduct.

If we were to restrict m to be finite, but X arbitrary, the above
discussion still applies, except that the category  would not be
closed under the map-space construction although it does have a
truth-value object . On the other hand, if we restrict X to be finite but
leave m unrestricted, we obtain a category satisfying both of these topos
axioms, yet not definable over sets nor indeed over any Boolean base
topos, as Bob Pare showed some years ago.

>From: peter_easthope@gulfnet.sd64.bc.ca 
>To: <CATEGORIES@MTA.CA>
>Subject: categories: sum of pointed sets. 
>Date: Mon, 20 Aug 2001 11:53:33 -0300 (ADT) 

>Hypothesis: the sum of an m-pointed set and an 
>n-pointed set is an m+n-pointed set. 

>Is this true? I've not thought of a counterexample. 

>If this is a theorem, can anyone tell me where to find 
>a proof. Such a proof, presented by a professional 
>mathematician, would be interesting and instructive. 

>Thanks, peter_easthope@gulfnet.sd64.bc.ca