Subclassifier object question
Hello, First of all sorry for the beginner's question. I have been reading various sources about the notion of a subobject classifier, e.g. McClarty's book, Lawvevre's book. Let assume the following situation: 1) Has a subobject classifier, true : 1 ---------> Omega. 2) Have an object X and object A and a monomorphism subX: A------->X. 3) With 2), we will have the pullback diagram with the corners of the pullback being the subobject classifier and the unqiue classifier/classifier X------>Omega 4)We have either a element, el: 1------>X or a generalized el: element H------->X. Question: Let's assume for simplicity that our category is Set. The problem is I don't understand how the notion of a subobject classifier determines whether "el" does or does not belong to the aforementioned subobject, "subX"! (I know that the outer commuting square figures in this, but it seems like every element will be "classified" as belonging to "subX"). Thanks and regards, Bill Halchin 20-Aug-2001 18:17:23 -0300,1833;000000000000-00000011
Bill Halchin asks
I don't understand how the notion of a subobject classifier determines whether "el" does or does not belong to the subobject
U ----------> 1 V _| | | | true | | V chi V X --------> Omega If any element "belongs to U" in the arrows sense that it factors through U, ie there is a map from 1 (or somewhere) to U, then the composite with chi:X->Omega is equal to (the composite with) true. Conversely, if an element of X "belongs to U" in the logical sense that its composite with chi is equal to (the composite with)true, then we have a commutative square that can be compared to the pullback. For more about the logical consequences of this property of Omega, and in particular the mysterious formula a & F(a) = a & F(true) see "Geometric and Higher Order Logic ..." in TAC, v7 (2000) pp 284--338. For a presentation of set theory (or, as I call it, Zermelo type theory) in a form that is both the way that ordinary mathematicians use it, and directly related to the topos-theoretic way of saying things, see Section 2.2 of my book "Practical Foundations of Mathematics". Both accessible via my home page at http://www.dcs.qmw.ac.uk/~pt Paul 20-Aug-2001 18:17:29 -0300,5651;000000000000-00000012
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Galchin Vasili -
Paul Taylor