Date: Wed, 15 Jan 97 10:19 GMT From: Dr. P.T. Johnstone Not an answer to Bill's question (which I agree is an important one), but a minor correction. Bill wrote: While the K/S definition is right for the construction of the object classifier over an arbitrary base topos (as Gavin showed) and hence for classifiers for various kinds of finitary algebras over an arbitrary base topos, It isn't, and he didn't. Gavin used finite cardinals to construct the object classifier over an arbitrary base topos with NNO (and I subsequently extended the construction to finitary algebraic theories), but it doesn't work over a topos without NNO (and in particular it can't be made to work using K-finiteness). Andreas Blass showed that the existence of an object classifier for toposes over E implies that E has a NNO. Incidentally, I think it is correct to give credit to Kuratowski for the notion of K-finiteness. It's true that Sierpinski's paper was earlier, but his definition was a "global" one (i.e. he defined the class of all finite sets as the sub-semilattice of the universe generated by he singletons), whereas Kuratowski made the crucial observation that the finiteness of a particular set X can be determined locally (i.e. within the power-set of X), without which the notion could never have been imported into topos theory. Peter Johnstone Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Date: Wed, 15 Jan 1997 15:22:00 -0500 (EST) From: F William Lawvere Concerning Peter Johnstone's clarification: Of course I didn't mean that the object classifier could be constructed without an internal parameterizer for the finite objects in the base S .... but what exactly are the finite objects ? While the classifier as a topos is determined by the 2-category of bounded S-toposes , the site for it isn't. I was under the impression that an internal category parameterizing the objects which are both K-finite and separable(=decidable) could be used (while internal presheaves on "all" K=finites would presumably be much bigger..what does IT classify ?) Anyway my point was that at any rate no further extension of the notion of finiteness is needed for classifying in that sense the objects or the group objects in S-toposes, whereas by contrast it seems that to give the mathematically correct notion of "vector space for which there exists a finite basis" does need such an extension. In any topos, a subobject of a nonnon sheaf is always separable ; when is the converse true ? Perhaps there is an internal topos object V which is largest with respect to being fully embedded in the given topos E while at the same time having A as its subtopos of internal nonnon sheaves. Here by A is meant the Boolean internal topos mentioned above which parameterizes the separable K-finites of E (Fred recalled Acunya's work showing among other things that it is Boolean) and to say that V "is" fully embedded in E has sense for any internal category with a terminal object , namely we require that the canonical parametrized (="indexed") functor from V to E is an equivalence E(X,V)--> E/X for each X. The latter functor is defined by merely pulling back the fibration 1/V--> V of pointed objects in V. When the answer to the above question is affirmative, Johnstone's locally separable reflection Vsubqd will consist of subquotients and the K-finites may fit in . It seems that the inclusion of A in V will preserve sums but only certain epis. The idea is that V can't be too large since the inverse to the inclusion will enrich it in A. On Wed, 15 Jan 1997, categories wrote: > Date: Wed, 15 Jan 97 10:19 GMT > From: Dr. P.T. Johnstone > > Not an answer to Bill's question (which I agree is an important one), > but a minor correction. Bill wrote: > > While the K/S definition is right for the construction of > the object classifier over an arbitrary base topos (as Gavin > showed) and hence for classifiers for various kinds of > finitary algebras over an arbitrary base topos, > > It isn't, and he didn't. Gavin used finite cardinals to construct > the object classifier over an arbitrary base topos with NNO (and I > subsequently extended the construction to finitary algebraic > theories), but it doesn't work over a topos without NNO (and in > particular it can't be made to work using K-finiteness). Andreas > Blass showed that the existence of an object classifier for toposes > over E implies that E has a NNO. > > Incidentally, I think it is correct to give credit to Kuratowski for > the notion of K-finiteness. It's true that Sierpinski's paper was > earlier, but his definition was a "global" one (i.e. he defined the > class of all finite sets as the sub-semilattice of the universe > generated by he singletons), whereas Kuratowski made the crucial > observation that the finiteness of a particular set X can be determined > locally (i.e. within the power-set of X), without which the notion > could never have been imported into topos theory. > > Peter Johnstone > Re: Finiteness in Toposes Jan 17 1997 This concerns the possibility , mentioned in my previous message, of two internal toposes of finite objects. The conjecture that there are two natural internal categories of finite objects is partly supported by the fact that there are two natural natural-numbers objects, the usual one N that parameterizes compositional iteration and another semicontinuous one L with the following features: 0) It is a rig, so receives a homomorphism from N and its elementary arithmetic starts out looking very similar. 1) But unlike N it has a least-number-property in the sense that it is inf-complete and better. 2) It can be constructed internally using truth-valued sheaves on N. 3) Hence it also contains a map from (big) omega, which permits (unlike N) the use of the standard method in finite combinatorics where (for example) a binary relation is considered as a matrix which is valued (not only in a rig where 1+1 = 1, but instead) in a rig in which natural numbers are distinct; the resulting generalized characteristic functions are added, multiplied, infed etc. according to the usual methods of arithmetic and analysis and then translated back into the combinatorics of the original finite structures. Of course, in each case one hopes that the answer to a combinatorial problem might turn out decidable, but that shouldnt require us to stay in the bounds of two-valued subsets in the course of a construction. 4) This internally-defined order-complete rig in E has also an external characterization if E is an S-based topos, namely it is the sheaf of germs of S-geometrical morphisms from E to the topos often called S-sets -through-time (I dont think that depends on any presumption that the N in S ,used to parameterize the transitions through time, coincides with its completion in S). In localic or open set terms, there is in S a (T sub zero) space whose points are N, but whose open sets have the usual order on N as their specialization order; continuous functions from any space E to this space are called semi-continuous and there is in E a sheaf of them. 5) The application to the variable linear algebra over algebraic or complex-analytic spaces needs L too, because dimension of a vector space is a semi-continuous function. More precisely, if A is a good module in a ringed topos E, R then for each X and E there should be a map X--> L which is the fiber-wise dimension of X*A. The basic case is perhaps that where E,R is an algebraic affine scheme, and the conceptual problem is to get at what sort of sets contained in A this dimension function is counting (or bounding). One should not expect that equality of dimension will imply isomorphism. This object L has been discussed for 25 years, but I dont know if anyone published the working-out of its properties and role. Bill