[Note from moderator: this is two postings from Peter...an error on my part resulted in the first not being circulated until now.] Date: Tue, 28 Oct 1997 08:50:20 -0500 (EST) From: Peter Freyd <pjf@saul.cis.upenn.edu> Vaughan points out the many remarkable similarities between topoi and abelian categories. I've always thought that the most remarkable is the fact that if one forms the pushout of a pair of maps one of which is monic then the result is a pullback. The known proofs for the two cases are remarkably different. When both maps are monic this is known as the amalgamation property (at least it's so known in the category of belian groups, that is, groups abelian or not). It's also right here that a big difference shows up. For abelian cats the lemma remains true when one relaxes the hypothesis from "a pair of maps one of which is monic" to "a pair of maps that are jointly monic." Such a lemma is very wrong for topoi. (For a counterexample in sets pushout any jointly monic pair of maps from a 3-element set. The result is a pullback iff one of the given maps is already monic. Such a counterexample sits, in fact, in any non-degenerate topos.) Vaughan asked: What can be said of the class of those categories having all the elementary properties common to toposes and abelian categories? In particular does it contain anything other than toposes and abelian categories? And if so, does this outcome change when the language is extended to say second order logic? The answer to the second question is no (hence so is the answer to the third sentence). There is a complete answer to the first question. You won't like it. For a 1-sentence axiom of pratt categories first take a 1-sentence elementary axiom, T, for elementary topoi and a 1-sentence elementary axiom, A, for abelian categories Then take the sentence: T or A. Sorry. Vaughan goes on to say: To the extent that both toposes and abelian categories share much pleasant structure, the models of the intersection of their theories, for a suitable choice of language, would seem to be a nice class in its own right. This strikes me as an interesting topic (and perhaps we should use the phrase "pratt categories" for what emerges as the right choice of this class). My first choice for the suitable choice of language would be the set of universal Horn sentences in the predicates of finite limits and colimits (where it is understood that we already have the axioms for finite bicompleteness). Can one prove the expected representation theorem: does every pratt category have a faithful limit- and colimit- preserving functor into a product of an abelian category and a topos? Date: Fri, 31 Oct 1997 05:35:00 -0500 (EST) From: Peter Freyd <pjf@saul.cis.upenn.edu> To: cat-dist@mta.ca Somehow my first answer to Vaughan's question went astray. He had asked: What can be said of the class of those categories having all the elementary properties common to toposes and abelian categories? In particular does it contain anything other than toposes and abelian categories? And if so, does this outcome change when the language is extended to say second order logic? The answer to the second question is no, hence so is the answer to the third. And you won't like the answer to the first. Given any two elementary theories, *A* and *T*, let *AoT* be the set of all sentences of the form "A or T", where "A" is a sentence in *A* and "T" is a sentence in *T*. Clearly every model of *A* is a model of *AoT* and so is every model of *T*. Almost as clearly: every model of *AoT* is either a model of *A* or of *T*. Since the elementary theories of abelian cats and topoi can each be finitely axiomatized, there's a single elementary sentence common to topoi and abelian cats whose only models are either topoi or abelian cats. * * * A few misstatements from my previous post. My attempt to abbreviate V4 didn't work. I want to say that the diagram is a pushout (of course it's a pullback). So it should be: V4) If A -> C is monic and A / \ B C \ / D is a pushout then and so is 0xA / \ 0xB 0xC \ / 0xD. I wrote: And note that the type-A objects can not be reflective: if 1 has a map to any type-A object the entire category collapses. I should have written: And note that the type-A objects can not be reflective unless all objects are type-A: if 1 has a map to any type-A object then 0 = 1. Finally (I must be kidding), as it stands the P-E-l-r-/\ structure is not properly fixed for abelian categories. Adjust as follows: /\ is defined for pairs f:R -> Y, g:R -> X only when both R and Y are of type-T. Note that the equation in V9 is a directed equality: if /\(f,g) is defined then the equality holds. V10 must be modified by strengthening the hypothesis to include the condition that Y is type-T (at which point the condition on R becomes redundant).