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*To*: fom@math.psu.edu*Subject*: Re: FOM: NYC logic conference and panel discussion*From*: Vaughan Pratt <pratt@CS.Stanford.EDU>*Date*: Thu, 09 Dec 1999 13:25:46 -0800*In-reply-to*: Your message of "Thu, 09 Dec 1999 12:15:45 CST." <Pine.SUN.4.10.9912091204230.1104-100000@daisy.uchicago.edu>*Sender*: owner-fom@math.psu.edu

>Matt Frank >I don't think that this is in the book by Lawvere and Schanuel--after all, >Part I of that book is titled "The Category of Sets". On the other hand, >a lot of the book discusses the category of directed graphs instead. In >particular, it discusses truth-value objects in this context, and I found >that useful (the first example of these things that I really liked) >largely because it was so different from the usual set theory. Toposes enter explicitly near the end, where it is pointed out that the book has been concentrating on toposes throughout, in particular the categories of sets, irreflexive graphs, reflexive graphs, and dynamical systems. The authors define a topos as a category C having finite sums and products, function spaces ("map objects") Y^X or X->Y, and a truth value object (\Omega) containing the truth value "true" (the map true:1->\Omega), and also satisfying the interesting condition that for every object X of C, the "slice" category C/X has products. The objects of C/X are the morphisms to X in C, and a morphism of C/X from g:Y->X to g':Z->X is a morphism f:Y->Z such that g'f=g. The subcategory of C/X of primary interest, called P(X) for the "parts" (subobjects) of X, has for its objects monics (in C) to X. P(X) is shown to be equivalent to a poset, namely the poset of predicates on X. Although the book does not take this development further, the experienced reader will immediately infer that P(X) has all infs, i.e. that conjunction is defined on predicates including infinitary and empty conjunction. This makes P(X) a complete semilattice, hence a complete lattice, hence a Heyting algebra. In the category of Sets with \Omega = {0,1} the predicates are Boolean, but in general they are only intuitionistic, or "non-Boolean" as the book puts it. The book is directed to a considerably younger set of readers than usual for introductions to category theory. (The closest the book comes to mentioning Heyting algebras is to point out that not not not P = not P in any topos.) That this is possible without compromising on precision of definition lends support to the thesis that category theory is an elementary subject. >From there one may if so motivated draw the further inference that category theory is a viable candidate for a component of the foundations of mathematics along with arithmetic, set theory, Boolean logic, and other notions sufficiently elementary as to be accessible to at least those high school students who are not overly anxious about mathematics. Vaughan Pratt

**References**:**Matthew Frank**- FOM: NYC logic conference and panel discussion

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