------------- Does anyone have any advice about this following proposed terminology? I think I want to use the term "weak topos" to mean a cartesian closed category with all finite limits and colimits, stable surjections, and a weak power object. By a "weak power object" I mean an object W with a monic M>--->W such that every monic in the category is a pullback of M in at least one way. The reasons for this are that categories of assemblies (including assemblies for extensional or modified realizability) have these properties and: 1) Except for the part about colimits this is just what you need to show the effective reflection is a topos. 2) Using the presence of coequalizers this gives a usable generalization of geometric morphisms. If a functor between weak toposes in this sense has a left exact left adjoint then that adjoint is regular (pres. finite limits and surjections) and the adjunction lifts to a geometric morphism between the toposes. And in fact these arise among categories of assemblies (as is in effect remarked by Pitts and more recently Oosten). Thanks, Colin ==============================================================================
In the early 60's Eilenberg, Mac Lane and I had lunch in a Denver cafeteria to see if we could agree on various items of terminology. There was one thing we all agreed on: the prefex "weak-" was an operator on definitions that removed uniqueness conditions. Colin's candidate for "weak topos" is only missing one of the uniqueness conditions, so I would hesitate to use the term. How about "near topos"? Best thoughts, peter ==============================================================================
Peter Freyd says,
Colin's candidate for "weak topos" is only missing one of the uniqueness conditions, so I would hesitate to use the term. How about "near topos"?
Two counterarguments to this (without intending to express a definitive view): (1) There's always a danger of proliferating prefixes for "inferior" forms of things - pseudo, quasi, pre, weak, near, etc. Are you, Peter, willing to propose a definitive meaning for "near" analogous to your (& Sammy & Sauders') meaning for "weak"? Do you think we should use up one of these words for this purpose? (2) You say that a weak topos should be something satisfying the definition of a topos with uniqueness deleted *everywhere*. The first clause of this definition is that it's a category; do you intend it to be a weak category? What is that? Something (like homotopies, perhaps) which has non-unique composition? I suspect this leads us to a structure bearing little relationship to simple type theory or higher order predicate calculus. We often think of categorical definitions such as toposes in a hierarchical way, eg "A TOPOS is a cartesian category in which each object has a powerobject" (Categories, Allegories, 1.9). Here "cartesian category" is a background definition (genus in philosphical jargon) and "... powerobject" is the distinguishing property we have in mind (species) (cf bounded comprehension) So, if we want to weaken the definition, we do so by weakening the distinguishing property. n'est ce pas? Maybe this is a question that should be answered by examining what interesting phenomena arise in the literature. Raymond Hoofman's thesis ("Non-stable Models of Linear Logic", R.U. Utrecht, 1992) might be a good place to start. He considers weak cartesian closed categories - useful for modelling beta without eta in the lambda calculus. Exactly what to weaken is not, a priori, a clear cut issue. Paul ==============================================================================
Paul Taylor <pt@doc.ic.ac.uk> writes,
Peter Freyd says,
Colin's candidate for "weak topos" is only missing one of the uniqueness conditions, so I would hesitate to use the term. How about "near topos"?
...
Are you, Peter, willing to propose a definitive meaning for "near" analogous to your (& Sammy & Sauders') meaning for "weak"?
...
Exactly what to weaken is not, a priori, a clear cut issue.
The prefix word "weak" is defined to mean "missing uniqueness", so it seems appropriate for Colin's work (or am I mistaken here?). The problem seems to be that there are degrees of "weak" imaginable. Why not provide an outside tag to identify the terminology? Colin should use "weak topos" throughout his own work; works by third persons should refer to "weak topos (as per McLarty)". ==============================================================================
Paul Taylor has given good evidence on why one should avoid using the prefix "weak" when it leads to ambiguities. He says: "A TOPOS is a cartesian category in which each object has a powerobject" (Categories, Allegories, 1.9). Here "cartesian category" is a background definition (genus in philosphical jargon) and "... powerobject" is the distinguishing property we have in mind (species) (cf bounded comprehension) So, if we want to weaken the definition, we do so by weakening the distinguishing property. This gives a different notion of weak topos than McLarty's. In particular, it would have only weak exponentials whereas McLarty's has plain exponentials and a weak subobject-classifier (hence weak powerobjects). La Monte Yarroll says: Why not provide an outside tag to identify the terminology? Colin should use "weak topos" throughout his own work; works by third persons should refer to "weak topos (as per McLarty)". The trouble is, of course, the last phrase will inevitably be changed to "McLarty-weak topos". ==============================================================================
participants (4)
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cxm7＠pop.cwru.edu -
Paul Taylor -
piggy＠hilbert.maths.utas.edu.au -
pjf＠saul.cis.upenn.edu