[16:54] i don't have a nifty text document to cut and paste from this time [16:54] so you'll just have to put up with my lousy typing [16:55] for anyone who doesn't know it [16:55] the point of all this generalized abstract nonsense [16:56] is to show that there are models of set theory with distinctly geometric features [16:56] last time, i introduced 14 examples of topoi [16:57] I showed how in each one pullbacks and terminal objects always exist [16:57] so by an argument in elementary category theory, finite limits always exist [16:57] this gives us a lot of set-theoretic features [16:58] like products, intersections, and singletons [16:59] but to really take off, any model of set theory has to have a notion of subsets [16:59] today, we construct that notion in the category, Sets [17:00] we can use that as a guide to define a subobject classifier for a category [17:00] we give some conditions for the existence of subobject classifiers [17:01] then we show how subobjects exist in several topoi [17:01] there is a set of notes I wrote [17:02] at http://home.earthlink.net/~sabean/topos.[tex|dvi|ps|pdf] [17:02] <^LoNeR^> ! [17:02] you won't be able to compile the tex unless you have Paul Taylor's diagram package [17:02] yeh ^LoNeR^ [17:02] http://home.earthlink.net/~sabean/topos2.[tex|dvi|ps|pdf] [17:03] sorry [17:03] <^LoNeR^> Are those the same notes as the previous one, or expansion of them [17:03] the first URL I gave was for the diagram from last time [17:03] n.b. the 2 [17:04] ok I'll wait a sec for anyone who wants to download [17:04] let us begin [17:05] we have two ways of defing subsets in Sets [17:05] as a monic (inclusion) [17:05] or as a characteristic function [17:05] <^LoNeR^> hmm claims 404 [17:05] dang [17:06] another typo [17:06] sad [17:06] http://home.earthlink.net/~ssabean/topos2.[tex|dvi|ps|pdf] [17:06] that's me ssabean [17:06] I forgot the s [17:07] I should have just pasted from my browser [17:07] oh well [17:07] i'm scared to proceed [17:07] did that work out for you? [17:08] <^LoNeR^> yes [17:08] ok [17:08] anyhow, notice that 0 is the value of true for our characteristic function [17:08] ok so then true is a monic [17:09] {0} --> {0,1} defined by 0 goes to 0 [17:10] and every subset of S of a set X is the pullback of true along the characteristic function [17:10] pullbacks and singletons always exist, so no problems [17:11] so the truth object in Sets is 2 [17:11] but this is not true in topoi in general [17:12] one example of prime imprtance to set theorists is when the truth object is a complete boolean algebra [17:12] other than the minimal algebra, obviously [17:13] so in definition 1, we basically say what a subobject classifier is [17:13] just a monic from a terminal to the truth object [17:14] which if we have some other monic S-->, then there is unique "characteristic function" that makes the square a pullback [17:15] i'll get into some concrete examples of subobject classifiers in a bit [17:16] we note that in general we have more than one monic corresponding to the same subobject [17:16] so we define an equivalence relation to deal with that issue [17:17] so then a subobject of X is an equivalence class of monics to X [17:18] by abuse, we refer to the subobject S or m [17:19] next we need to provide some sort of existence conditions for subobject classifiers [17:20] to do this, we note that the the subobject functor is representable [17:20] i.e., isomorphic to a Hom-functor [17:21] we let Sub_C X be the set of all subobjects of X [17:21] i.e., the set of all equivalence classes of monics to X [17:22] we say that C is well-powered if Sub_C X is a set for all X [17:23] All the categories in our example list are well-powered [17:24] so we can show that Sub_C: C^op ---> Sets is a functor [17:24] by pullback [17:24] this idea gets wrapped up into theorem 1 [17:25] If we have a category C, with all finite limits and small Hom-sets [17:25] the C has a subobject classifier iff there is some Omega in C and some natural isomorphism [17:26] in eq 4 [17:26] and when that's true [17:26] C is automatically well-powered [17:27] i'm not going through the proof, but it is in my notes [17:28] I point out that when the subobject classifier exists, it is unique up to isomorphism [17:28] then about half way down on page 3 I make some remarks about classifying bundles of Lie groups [17:30] i won't go into that much now, but the application to principal O(k)-bundles over n-complexes is kind of cool [17:30] so you may want to have a look [17:31] the point there is that the subobject classifier idea is sort of built on the idea of classifying bundles in topology [17:31] the topic of classifying topoi is an industry [17:32] so then I turn to some examples [17:32] Sets is already taken care of [17:32] and that means that FinSets is done too [17:32] the subobject classifier is the same [17:33] in Sets x Sets [17:33] the truth object is just the pair (2,2) [17:33] so there are 4 truth values [17:34] and this generalizes [17:34] in Sets^n, n fixed [17:34] there are 2^n truth values [17:35] as i will get into another time, it is just the power set of n [17:35] think complete boolean algebra [17:35] then there's the category of representations of a group G [17:36] on a set X [17:36] subobjects here are merely subsets of X that are closed under the group action [17:37] if S is a subobject, then the relative complement of S is also closed under the action of G [17:38] this means we can use the same old characteristic function [17:38] the truth object is again 2 [17:38] the monic true is the same too with G acting trivially on 1 and 2 [17:39] but [17:39] in the category of representations of a fixed monoid M [17:39] if S is a subobject, then the complement of S need not be closed [17:40] but we can fix this up by looking at "right ideals" of M [17:41] our truth object then is just the set of right ideals of M [17:41] and the subobject classifier is a monic from 1 into it [17:42] defined by the one point goes to M (as a "maximal" ideal) [17:43] we note that the definition of subobjects in BM agrees with the definition of subobjects of BG [17:44] so all is well for consistency's sake [17:44] so next I look at Sets^2 [17:45] the truth object turns out to be a map, naturally [17:45] 3 --> 2 [17:46] and the subobject classifier is the map {0} goes to {0} [17:47] the interesting thing here is that the characteristic function [17:48] is a map which appears to tell us whether, given a subobject S, x \in S is always true, true at 1, or never true [17:48] this makes connection to Lawvere's "sets through time" analogy I mentioned last time [17:49] and it extends to Sets^n [17:50] the truth object here is sequence of the map tau [17:50] on omega + 1 [17:51] then 1 is just a subobject of Omega [17:51] and our subobject classifier, under this analogy, becomes "time till truth" [17:52] alas, I did not have time to treat subobjects in the category of presheaves on a category C [17:52] but I will get that next time around [17:53] subobjects in a category of sheaves over a topological space will come even later [17:53] but with that, we might develop subobjects in the category of sheaves over a site [17:53] using, as a guide, the ever popular Zariski site [17:54] turns out to be useful [17:54] that's all i have [17:54] thanks for letting me rugurgitate yet more category theory [17:55] <^LoNeR^> Thank you Ring, are there any questions from the audience? [17:55] audience? [17:56] hehe [17:56] you have high hopes [17:56] <^LoNeR^> where can taylor's package be found? [17:57] hang on [17:58] http://www.dcs.qmul.ac.uk/~pt/diagrams/ [18:11] here are some exercises on subobjects for the extra motivated [18:11] prove that FinSets^N has no subobject classifier [18:12] R a ring prove that the category of left R-modules has no subobject classifier [18:13] If A and B are equivalent categories prove that subobject classifier for the one gives a subobject classifer for the other [18:13] that's a lot [18:14] trying to chase down the s.c. in FinSets^N is instructive