[16:00] <^LoNeR^> Today we will have R[[x]] tallking on: Sets and topoi, subobjects in presheaf categories [16:00] <^LoNeR^> :et's give him a big hand [16:01] * KimJ claps [16:01] <^LoNeR^> clap clap clap! [16:01] * [-K-] claps [16:01] * Evander applauds [16:01] * R[[x]] applauds wildly [16:01] oh [16:01] wait [16:01] <^LoNeR^> waiting ..... [16:01] ok so today i was gonna talk about subobjects again [16:02] dpecifically subojects in presheaf categories [16:02] specifically [16:02] and sieves [16:02] in presheaf categories [16:03] which trun out to be incredibly useful [16:03] in a number of ways [16:04] i wasn't kidding about the physics either, i am gonna talk about how sieves are important to quantum mechanics near the end [16:04] anyhow i wrote some notes [16:04] on the web [16:04] http://home.earthlink.net/~ssabean/topos3.pdf [16:04] or ps [16:05] or dvi [16:05] or tex [16:05] the latex is amslatex and requires paul taylor's diagrams package [16:05] which is free [16:06] ok so first let's rehearse what i already did [16:06] I gave a history of sheaves and gave 14 topoi [16:06] btw the document i refer to in those notes is in progress ;) [16:07] http://home.earthlink.net/~ssabean/topos.pdf for details in the mean time [16:07] i showed how each of them admits all finite limits [16:08] since each has a terminal object and all pullbacks [16:08] then we needed subobjects [16:09] so I modeled their construction on subobjects in Sets [16:09] namely a subobject is the pullback of a monic true along a characteristic function [16:10] actually equivalence class of them [16:10] so some cool things happen [16:10] like subobjects in the category of representations of a fixed group [16:11] just subsets that are closed under the action of the group [16:11] which is convenient [16:11] this time [16:11] i'll talk about subobjects in functor categories [16:12] i.e., subfunctors [16:12] after which i'll define sieves [16:13] and then show that a sieve is a subfunctor of a representable functor [16:13] which in turn will give the subobject classifier of the functor category [16:14] then a few words about how two brits have used this notion in quantum physics [16:15] so let C be a small category [16:15] C^op the dual [16:15] we want to talk about the category of all functors C^op --> Sets [16:16] this guy is also known as the category of presheaves on C [16:16] for good reason - the connection to sheaf theory will be made another time [16:17] the objects are all functors C^op --> Stes [16:17] Sets [16:17] and the arrows are all natural transformations of those functors [16:18] so I define the concept of a subfunctor [16:19] as just a restriction on the image sets and functions [16:19] of a functor [16:19] but that doesn't gel with the idea i gave last time that subobjects are equivalence classes on monics [16:20] think about Sets here [16:20] a subset is an equivalence class of injections [16:21] in fact [16:21] a subfunctor induces a monic arrow [16:21] i.e., a monic natural transformation [16:21] in fact, a whole equivalence class of them [16:22] i do that in some detail in the notes [16:22] so i won't repeat it here [16:22] read if you like [16:24] it is pretty easy to see the diagram commutes because of the subsets and inclusion maps [16:24] and that the natural transformation is monic follows from the fact that monics are taken pointwise [16:25] just like limits [16:25] so each subfunctor gives rise to a subobject [16:25] and the converse holds too [16:26] each subobject gives rise to a subfunctor [16:27] again this just comes down to injections on sets [16:28] so the next little section just says if we have a subobject classifier in our functor category Sets^C^op [16:29] then it must classify the subobjects of each representable presheaf [16:29] recall the yoneda embedding [16:29] btw [16:29] if anyone is still following this stuff, i suggest you learn the yoneda embedding well [16:29] because it will come uo again and again [16:29] up [16:30] the idea of the yoneda embedding is it embeds C in Sets^C^op [16:30] fully and faithfully [16:31] let y be the yoneda embedding [16:31] y(C) = Hom(-, C) [16:31] the set of all arrows with codomain C [16:32] note that this is indeed a set [16:32] since C is a small, and therefore locally small category [16:33] so Y(C) is an object of Sets^C^op [16:34] and what i am saying is, a candidate subobject classifier must give us the subobjects of y(C) for every C [16:35] last time a gave a theorem [16:35] it says Sub(X) ~= Hom(X,Omega) [16:36] the subobjects of X are isomorphic as a category the arrows from X to the truth object [16:37] indeed naturally so [16:37] so the subobjects of a representable presheaf are the natural transformations of it into Omega [16:38] Furthermore, using the yoneda lemma [16:39] the subobjects of a representable presheaf are in 1-1 correspondence with the set Omega(C) [16:39] remember Omega is a functor [16:40] this is where the concept of a sieve becomes useful [16:41] a sieve on an object C [16:41] is a set of arrows [16:41] all of which have codomain C [16:42] such that if S is a sieve and F in S and fh is defined then fh in S [16:43] intuitively we might picture a sieve as a path through the category that is allowed to get to the object C [16:43] element of a sieve there sorry [16:43] intuitively we might picture an element of a sieve as a path through the category that is allowed to get to the object C [16:43] that's what I meant to say [16:44] so then i give some elementary examples [16:44] a sieve, for example, on a group is a right coset [16:45] sieves on posets are interesting [16:45] and aid the intuition a little i feel [16:46] if x gets through the sieve, then so does anything smaller [16:47] the whole reason for bringing up sieves then [16:47] is that there we can identify sieves on an object C [16:48] with subfunctors of the representable functor on C [16:49] at the end there i was pressed for time so i'll treat it in detail next time [16:49] but [16:49] we can define a truth object Omega in Sets^C^op as follows [16:50] ! [16:50] yeh mt-i [16:50] isn't a sieve on a group necessarily the whole group? [16:50] if f is in S, then so is f(f^-1 x) ? [16:51] a sieve would be xG, G is a group, x an element, right? [16:52] ok nevermind [16:52] ok [16:52] bak to Omega [16:52] Omega(C) = the set of all sieves on C [16:53] Omega(g) for g:C-->C' is just a map of sets given by pullback [16:55] a maximal sieve on C is the set of _all_ arrows to C [16:55] i say a few words about how maximal sieves can be patched together to obtain a natural transformation 1-->Omega [16:56] but i definitely want to talk about that a bit more [16:57] but it turns out that every subobject in Sets^C^op is the pullback of true:1-->Omega along the characteristic function phi_C i define in the notes [16:57] which is what you want [16:58] ok so i'll round this up with the dreaded physics discussion [16:58] in 1998, chris isham at imperial and jeremy butterfield at oxford wrote a paper [17:00] in it they devised a way to assign "generalized valuations" to physical quantities in quentum mechanics [17:00] quantum [17:01] they did this by looking at sieves on operators in the spectral presheaf of an algebra of self adjoint operators on a Hilbert space [17:02] the spectral presheaf is the functor that takes operators to their spectra [17:02] as stes [17:02] sets [17:03] 4 years later, in 2002, isham now realizes that these valuations can be recognized as subobjects of the spectral presheaf [17:03] anyhow [17:03] i am about done [17:03] questions comments or flames [17:03] <^LoNeR^> ! [17:03] my fingers are tired [17:04] ^LoNeR^ [17:04] yeh [17:04] <^LoNeR^> Regarding mt-i's question [17:04] <^LoNeR^> if G is the group on which the sieves are defined and x is an element of the group aren't all cosets xG =G? [17:05] <^LoNeR^> Or am I misunderstanding something? [17:05] well, maybe it's more interesting in a groupoid? [17:06] now you're right i screwed up, i meant xH, H < G [17:06] geez [17:06] <^LoNeR^> ok, now that makes more sense :) [17:06] rotten tomato, anyone? [17:06] <^LoNeR^> so I had missed something [17:06] no you are correct [17:06] Out of curiosity, what's the source category of the spectral presheaf? [17:07] * mt-i is here to ask dumb questions :-) [17:07] bounded self adjoint operators over a complex Hilbert space [17:07] but isham recently showed any von Neumann algebra will do [17:07] and in a lot of ways, is preferable [17:07] is that a category? [17:08] yep [17:08] ok :-) [17:08] i know you must be curious about the arrows [17:09] the arrows are all Borel maps such that B = f(A) [17:10] where f(A) is defined the usual way by integration over the spectrum [17:10] wow [17:10] yeh wow [17:11] anyhow, thanks all for letting me blast category theory at you again [17:11] <^LoNeR^> Thank you for another great talk R[[x]]