Date: Thu, 18 Jun 1998 13:44:30 -0400 (EDT) From: Michael Barr Subject: categories: Chu(Ab,circle) is abelian The category of topological abelian groups is not abelian. The reason is not hard to explain. In topological spaces, points are primordial and a monomorphism of points need not be a monomorphism of the attached frames (that is a surjection of the open set lattices). When it is, then ignoring the usual separation axiom, it is a subspace and thereby a kernel. In frames, the reverse is true. Now the open sets are all and you ignore the points. In Chu(Ab,circle), the points and open sets are treated with equal respect. Now a monomorphism must be injective on the points and surjective on the opens and is obviously a kernel. The dual is also true. At another conceptual level, abelianess is defined by certain exactness condition, which concerns canonical arrows, usually between limits and colimits. A category is pointed if the canonical map 0 --> 1 is an isomorphism and there is then a canonical arrow A + B --> A x B and when that is an isomorphism, it is additive. There is a map from the domain of any monomoprhism to the kernel of its cokernel.... The conditions are self dual and a limit is computed in a Chu category as the limit of its first component and colimit of its second and all the required isomoprhisms remain isomorphisms. Of course, this is just as true of Chu(A,_|_) whenever A is abelian (and, of course, closed monoidal). The contrast with the case of topological and that of localic abelian groups is striking. I realized all this as a result of listening to Peter Freyd's lecture in Saint John, NB last week. He was trying to discover the initial abelian category with one object. It is self dual, but not this one since this contains no non-zero bijective (that is objects that are simultaneously injective and projective) while Freyd's category has enough of them. Still it might be interesting. And to anticipate Vaughan's question, no chu(Ab,circle) is not abelian, roughly for the same reasons as topological and localic abelian groups. Subject: categories: Re: Chu(Ab,circle) is abelian Date: Fri, 19 Jun 1998 23:33:55 -0700 From: Vaughan Pratt >And to anticipate Vaughan's question, no >chu(Ab,circle) is not abelian, roughly for the same reasons as topological ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ >and localic abelian groups. Actually I was going to ask about Ab. I thought it was abelian groups but (maybe I'm just the last to be told) the context suggests Ab = topological abelian groups. Is there life in discrete circles? (To reconcile the subject line with the underlined line you need to know Mike's usage: the large print giveth and the small print taketh away. Chu is to chu as preordered sets are to posets, or topological spaces to T_0 spaces.) Vaughan Date: Sat, 20 Jun 1998 20:53:18 -0400 (EDT) From: Michael Barr Subject: categories: Re: Chu(Ab,circle) is abelian Ab is discrete abelian groups. Thus the circle has, for these purposes, the discrete topology. Since you are mapping discrete groups to it, the topology is irrelevant. But the small chu category is much more like hausdorff topological abelian groups (in fact, is equivalent to two full subcategories of them) and is not abelian for similar reasons. One take on this is that separation conditions are more or less incompatible with effective equivalence relations (= monics are kernels in the additive case). And extensionality is incompatible with the dual. On Fri, 19 Jun 1998, Vaughan Pratt wrote: > > >And to anticipate Vaughan's question, no > >chu(Ab,circle) is not abelian, roughly for the same reasons as topological > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ > >and localic abelian groups. > > Actually I was going to ask about Ab. I thought it was abelian groups > but (maybe I'm just the last to be told) the context suggests Ab = > topological abelian groups. Is there life in discrete circles? > > (To reconcile the subject line with the underlined line you need to know > Mike's usage: the large print giveth and the small print taketh away. > Chu is to chu as preordered sets are to posets, or topological spaces > to T_0 spaces.) > > Vaughan >