Does any body know the answer to the following question ? Is there a locally cartesian closed category C such that there is a full sub-left-exact category D of C which is non-reflective and forms an exponen- tial ideal, i.e. whenever A is an object in D and X is an object in C then [X->A] is (isomorphic to) an object in D ? The background to this question is the following. If something as described above would exist then one can perform the following construction. A morphism f : Y -> X in C is a DISPLAY MAP iff for any g : Z -> X : Pi (!Z) (g*f) : P -> 1 with P in D . For the collection DISP of display maps then one can prove : i) display maps are stable under arbitrary pullbacks ii) for any I in C : DISP / I ---> C / I reflects finite limits (iii) if f : X -> I is a display map and g : I -> J is an arbitary morphism in C then Pi (g) (f) is a display map as well . Now if D forms an exponential ideal then DISP / 1 is equivalent to D and we would get that the full subcategory of C (in the fibrational sense) is not reflective w.r.t. C / C . This would give an answer to a question I raised on this forum about one year ago. Thomas Streicher ======================================
The method I described to get examples for Streicher's request can, in fact, be easily modified to achieve full completeness. Let me give a specific example. Let D be the opposite of the ordered set of all ordinals. Except for the existence of a bottom element it is a Heyting algebra, therefore a local CCC. Let C be the result of adjoining a bottom element. Then D is a full left-closed exponential ideal in C. The bottom of C does not have a reflection in D. QEF Best thoughts peter freyd ======================================
Thanks for the nice counterexample ! I think what you meant is that the constru- ction works for all Heyting algebras (I mean the finitary version : inf-semi-lat tice with exponentiation, no infinite sups or infs are claimed) without a least element. So the archetypical example is the natural numbers with an element for infinity together with the inverse ordering. By the way the display maps in a ll these examples are definable in the sense that whenever f : X -> I is an arbitrary map then there is a greatest subob- ject m : J >--> I such that m*f is a display map. Thus we have definability in the sense of Benabou. In all these cases either m is identity or J = 0 . May it be correct that there are only posetal counterexamples because in this case equalizers do not cause any problems ?? Thomas Streicher ======================================
[ the first of Peter's messages did not arrive the first time, and the moderator missed that reference in the second. Sorry about that. Bob Rosebrugh ] I sent two messages to the categories net only the second of which seems to have been distributed. The second begins by referring to the first. The following is a slightly edited version of the first message: ******************************************************************** I wonder if Thomas Streicher's question is really as he states it. Take any Heyting Algebra and "ideal", that is, a subset closed under finite meet and hereditary upwards. Then the ideal is a full "sub-left-exact" subcategory and it is an exponential ideal. The bottom element of the Heyting Algebra has a reflection in the ideal iff the ideal is principal. If one wants the subcategory to be closed under the formation of arbitrary limits (not just finite limits) then there are counterexamples but none that I know that can be so easily described. Best thoughts, peter freyd ======================================
With regard to the Thomas Streicher's question: May it be correct that there are only posetal counterexamples because in this case equalizers do not cause any problems ?? There may well be a sense in which the answer is yes. But consider the following: let C with subcategory D be a counterexample. Let A be any local CCC. Then A x C with subcategory A x D is also a counterexample. ======================================
I must be suffering from schizophrenia, because I can't remember this counterexample. Perhaps you'd remind me? PS What was the simplifying lemma about display maps which you presented at a PSSL in Cambridge in 1987? ======================================
Thanks again. Due to this observation by choosing for A the categories Set , omega-Set or PER-omega we get counterexamples which are very rich. IsnBt it the case that by taking the product of some lccc A with e.g. the nat ural numbers with inverse order one gets some aspect of "time" into realizabili ty models which destroy the existence of existential quantifiers. Maybe it is worthwhile to investigate connections with Martin-LoefBs work in mathematics of infinity where he considers sequences of models of type theory in order to explain infinite objects. Thomas Streicher P.S. I think a further interesting counterexample is obtained by taking for C the product of 0mega-Set with (N u {infinity})* and for D the full subcatego ry which is the cartesian product of PER-omega with (N u {infinity})*. ======================================
participants (4)
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Paul Taylor -
pjf＠linc.cis.upenn.edu -
pjf＠saul.cis.upenn.edu -
Thomas Streicher