Talk:Free Boolean algebra
Disputed
I disputed myself here because I'm not quite sure of this fact--is this really sufficient to characterize free BAs? It's not obvious that this condition implies that any permutation of the generators extends to an automorphism, which is surely a property we want. Can't find any clear references at the moment. --Trovatore 00:13, 2 November 2005 (UTC)
- It sounds right, but (i) it's a new criteria to me and (ii) I haven't yet had my morning tea, so I will remove the tag once I have double-checked it. --- Charles Stewart 15:12, 2 November 2005 (UTC)
- Thanks, I appreciate it. Looking at your changes so far, I'm a bit concerned by the change to "initial object". From my dimly remembered category theory I thought an initial object was just one from which there was an arrow to any other object in the category. If the arrows are just BA homomorphisms, that's not nearly strong enough. --Trovatore 16:13, 2 November 2005 (UTC)
- No, you've forgotten the condition on cones, and isomorphism does follows from initiality. --- Charles Stewart 16:20, 2 November 2005 (UTC)
- Thanks, I appreciate it. Looking at your changes so far, I'm a bit concerned by the change to "initial object". From my dimly remembered category theory I thought an initial object was just one from which there was an arrow to any other object in the category. If the arrows are just BA homomorphisms, that's not nearly strong enough. --Trovatore 16:13, 2 November 2005 (UTC)
- The "condition on cones" doesn't ring a bell. I'd ask you to explain it to me, but maybe it would be almost as easy, and much more useful, to add some text to the article about it? At the moment I don't know how to reconcile what's here with what's in the initial object article. --Trovatore 17:19, 2 November 2005 (UTC)
- The discussion at initial object is fine. The point I was alluding to is that the initial object construction is a special case (indeed the simplest case) of a colimit, which tells us that not only is there always such an arrow, but that for every such arrow the cone diagram (actully cocone diagram, since we are doing colimits) commutes. In this case that is the same as saying that there is a unique arrow from the IO to every object, which is the same as saying that any two arrows from
theany IO to another given object is equal. This is just what you need to prove 'equlity up to isomorphism', noting that the arrow fromtheany IO to itself must be the identity arrow. --- Charles Stewart 19:03, 2 November 2005 (UTC)
- The discussion at initial object is fine. The point I was alluding to is that the initial object construction is a special case (indeed the simplest case) of a colimit, which tells us that not only is there always such an arrow, but that for every such arrow the cone diagram (actully cocone diagram, since we are doing colimits) commutes. In this case that is the same as saying that there is a unique arrow from the IO to every object, which is the same as saying that any two arrows from