A Mind for Madness

Musings on art, philosophy, mathematics, and physics


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Theological Points of Tron: Legacy

Last night I saw Tron: Legacy and I thought I’d share some of my thoughts on it. The movie certainly wants you to draw a parallel between it and certain aspects of the Bible, so we’ll start by looking at exactly how it does this. Then we’ll move on to more speculative symbolism I saw that lead me to believe it was anti-Christian or at least wanted people to think about whether certain non-traditional theological positions were viable. I will not attempt to hide spoilers, so don’t read any further if you don’t want the ending ruined.

The movie had an almost face-palm level of explicit Biblical parallels. A man creates a program “in his image”. I wanted to laugh that they kept using that phrase. The program was perfect at first, but then it started thinking on its own and got away from him. It started doing evil. Then this man’s son has to come and save everyone. Many of the programs inside of Tron use the phrase “our creator” to refer to the father.

Another story told in the movie is how this girl was about to be destroyed by the evil guys only to find “the father” standing over her after she blacked out and he “saved her”. The father’s disk contains all the information about everything in the grid, so in a sense the father is omniscient. Clearly the father is a stand in for the Biblical God since he creates a whole universe and is omniscient (and by some of the actions he performs in the movie, he seems omnipotent as well). I don’t think the Christian parallels are all in my head.

There is some sketchy theology that occurs that might be in my head, though. The “real world” outside of the grid seems to be a symbol for heaven. If this is true, it presents a very interesting theological point. The evil guy wants to let everyone into heaven. It is only because the father is “selfish” that only some get to go there. So a point is made that not letting everyone into heaven even though it is in his power to do so is selfish. Of course, he isn’t allowing the people who follow the evil guy in and he does want to allow the “good creations” in.

The next non-traditional theological point is that the reason evil came into the world was not the evil guy’s fault since he was made in the father’s image. In fact, the evil guy is in some sense the father, so it is actually the father’s fault. The father recognizes this, and explicitly says it in the movie and apologizes to his creations for allowing it to happen. The creations took a life of their own and he lost control. So theologically this seems to be trying to explain the problem of evil by saying that the father is not actually omnipotent, and that the creations don’t have to follow his plan. Also, it puts the responsibility on the father and not on the creations. This is much closer to a deist position than a theist position.

Lastly, the father actually sacrifices himself to save everyone. Now when it is worded that way, it seems to follow traditional Christian doctrine, except that the way it is done in the movie seems to indicate that the father is literally gone after that. The way that God gives salvation is to remove himself from the equation (a phrase used many times during the movie). i.e. In terms of theology, the movie seems to want to reinterpret the meaning of the sacrifice as saying that God no longer exists. Maybe he did at one time, but not after the crucifixion.

This is why I think Tron: Legacy is explicitly anti-Christian. It makes the creator a helpless person that has to sit by and watch his creations destroy eachother. There is nothing he can do about the evil. Which of course gets around the problem of evil, but also puts God in a merely creator role. The theology puts God as sometimes loving, but mostly a selfish creator. On the other hand, it tries to rewrite the evils explicitly done by God in the Bible by making the evil guy do them. The theology says that God should apologize to us and not the other way around.

Interestingly, the purpose of the son going into Tron is to save the father (you know, the one responsible for the evil) and is not there for the purpose of saving the people committing the evil acts (again, not their fault according to the movie since they were designed in the image of an evil creator). The son is also told when he enters that his purpose is to “survive” which is quite the opposite of his purpose in the Bible.

Maybe I’m just seeing too much in this, but usually I can turn off my brain from doing this if it is just subtle symbolism. In this case, everything was so explicit I had to watch in horror as my brain kept trying to fit all the symbols into a theological viewpoint. I couldn’t turn it off because all the phrases they kept using were designed with the intention of evoking these thoughts.


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Stacks 2: An example

This will hopefully be a short, yet enlightening post in which the concept of a stack starts to make more sense than the abstract nonsense of the last few posts. Recall that we formed a category of line bundles on a manifold {L(X)} and had a natural forgetful functor: {L(X)\rightarrow \text{Top}(X)}.

If one is not writing a blog and wants to be much more careful, one should probably check that {L(X)} is indeed a category and the given functor is actually a functor. Most readers that have made it this far probably aren’t concerned with this part, though.

Is this fibered in groupoids? Well for checking what sorts of things lie over certain objects {U\in \text{Top}(X)}, the situation has been rigged so that the objects over {U} are precisely the line bundles on {U} as a manifold. The first type of “square” we have to be able to complete is as follows {\begin{matrix} & & (V, L_V) \\ \\ U & \hookrightarrow & V \end{matrix}}. Well, all we need to be able to do is find some {(U, L_U) \rightarrow (V, L_V)}. But by definition of our category this would consist of an iso {L_V|_U \rightarrow L_U}. These are line bundles, so we can always restrict to get another one, so just take {(U, L_V|_U)} lying over {U} to complete it.

What about the second diagram of being fibered in groupoids? The base is just {U\hookrightarrow V \hookrightarrow W}. Now suppose we have line bundles over these {L_U, L_V,} and {L_W} and isomorphisms {L_W|_V\rightarrow L_V} and {L_W|_U\rightarrow L_U}. This certainly tells us that there is an iso {L_V|_U\rightarrow L_U}, and uniqueness is just from the fact that it has to be the one that makes the composition what we said it had to be. You could think of this coming from the fact that {L_V|_U\rightarrow L_U} is unique up to automorphism of {L_U}, and we know which automorphism it from the other condition.

Now we check that Isom forms a sheaf. Let {U} be some open set. Let {L} and {S} be two line bundles over {U} (in this case, this literally just means line bundles on {U} as a topological space). Now we want to check that the presheaf (of sets) {\mathcal{F}(V)=\{L_V\stackrel{\sim}{\rightarrow} S_V\}} is actually a sheaf. This is a presheaf just because isomorphisms restrict. It is a sheaf because all the information is local. If you have isomorphisms defined on open subsets that agree on overlaps, then you can glue them to make an isomorphism on the union. These are just two basic properties of line bundles that most people have already seen. So Isom is a sheaf.

Lastly we need to check the stack condition. Maybe I should remark on the terminology here. A collection of objects and isos over a covering of an open set that satisfies the cocycle condition is called a descent datum. If the objects glue in the way of the stack condition, then that descent datum is said to be effective, so the stack condition is sometimes stated that every descent datum is effective.

Given a descent datum, the fact that you can glue to get an object over the whole open set is just a standard exercise or proven proposition in basically any text on manifolds. In fact all of the above things are true for any rank {r} vector bundle. So we actually get the stack of rank {r} vector bundles on {X} for any {r}. Since I’m not sure we’ll return to this example, we’ll just temporarily notate it {\text{Vect}^r(X)}, and hence {\text{Vect}^1(X)=L(X)}.

If you’ve been following along, it should be pretty clear how to translate all of this over to a stack on the Zariski site rather than on {\text{Top}(X)}, but we’ll make that more explicit next time and get some more examples.


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Music 2010

What a fantastic year for music. I was so disappointed last year. I didn’t like very much last year, but this year time after time I kept being pleasantly surprised. I couldn’t believe how high the quality was. Everything in the top ten this year is probably better than any of the top 3 from last year.

Without further delay here is my top 10:

1 Joanna Newsom – Have One On Me
2 Dillinger Escape Plan – Option Paralysis
3 Owen Pallett – Heartland
4 Sufjan Stevens – The Age of Adz and All Delighted People
5 The Whiskers – War of Currents
6 The Head and the Heart – Self-Titled
7 Dirty Projectors/Bjork – Mount Wittenberg Orca
8 Kayo Dot – Coyote
9 Land of Talk – Cloak and Cipher
10 Moonface – Dreamland EP

Honorable Mentions:
The Tallest Man on Earth
Xiu Xiu
Arcade Fire
Jonsi

The stuff that was pretty good, but just had a enough things I didn’t like that made it hard to listen all the way through:
The National
Corinne Baily Rae
Interpol
Antony and the Johnsons
Kanye West
Gayngs

The bottom 5 in order:
Stars
Band of Horses
No Age
Spoon
LCD Soundsystem

Now for a few words. I’ll start with the bad. Honestly, I’m really disappointed about Band of Horses. I really loved their previous release, but this one consisted of some songs that became annoying after a few listens or were just plain boring. There were a few gems on it, but overall I can’t listen to it. Spoon was horrifyingly bad. It is filled with cliche riffs and progressions. The lyrics are childish and petty. The repetition is annoying. I truly dreaded listening to it again to put into the list. Same goes for LCD Soundsystem. I liked their last album, but this one the repetition exceeded Philip Glass and wasn’t as interesting.

For the good, I’ve posted many times about Joanna Newsom. She is one of the greatest musicians alive right now. This album is so intricate and perfectly put together. The songs reference eachother and repeated listenings are constantly rewarded. This was the first year I’ve had to deal with single artists releasing more than one album. Sufjan Stevens and The Tallest Man on Earth both did this.

The Sufjan album is so good. If you’ve liked his stuff in the past, then don’t miss this. If you only want to get one of them, get The Age of Adz. It is full of heartwrenching ballads and electronic hip-hop and full orchestra arrangements and just him and his acoustic guitar. Basically it the most diverse album listed here.

Now for my favorite tracks:
1 Impossible Soul or All Delighted People – Sufjan Stevens
2 Tornado – Jonsi
3 California – Joanna Newsom
4 Marsh Blood – Whiskers
5 Lewis Takes off his Shirt – Owen Pallett
6 Gold Teeth on a Bum – Dillinger Escape Plan
7 Color Me Badd – Land of Tallk
8 Afraid of Everyone – The National
9 Down in the Valley – The Head and the Heart

If you don’t want to get a whole album listed above, and only want the best song on it here’s my best guesses. Impossible Soul is just an epic song. It is over 20 minutes if I remember correctly and pulls you through every style imaginable. It is something everyone should hear once. My jaw was dropped the whole time my first time through that he could actually transition through all these styles in a single coherent song.

Another mind blowing song is Tornado. I’ve never heard someone so perfectly make the form of the song mimic the content. Tornado refers to some internal struggle Jonsi has with himself. The lyrics refer to two competing parts of himself to get control. He uses two different time signatures that are both always present. When the one side is winning the one is more prevalent and the other is only background, but then the other comes to the front. It is truly mind blowing how well this gets pulled off and it isn’t even noticeable the first few times you listen. It is also just a really moving song overall with a great powerful climax.

Lastly, I listened to several more albums than listed above, but they were just blah and didn’t fit the above lists. I also didn’t get around to lots and lots of stuff that looked really good. As always if your list is similar to mine and you think I missed something awesome, please leave it in the comments. I’d love to hear more. Also, if you disagree with anything I’d like to hear that as well.


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Stacks 1

Today we’ll actually get towards a what a stack is. Last we talked about what it meant for a category to be fibered in groupoids over another. This was a very general definition for any two categories. Today we’ll actually need to make use of more topological or geometric notions. Let {X} be a topological space, then we can turn {X} into a category {\text{Top}(X)}, which is just the standard topological site. The objects are the open sets, and {Hom(U,V)=\left\{ \begin{array}{lr} \{U\hookrightarrow V\} & : U\subset V \\ \emptyset & : \text{else} \end{array} \right.}

See this post for more information on what that category is and why it is a site. From these earlier posts we also have a notion of what a sheaf on this site is. Since this is just a topological space and hence the motivating example for the sheaf (on a site) definition, you can just think in terms of what a sheaf on a topological space is if you wish.

Consider some category {\mathcal{C}} fibered in groupoids: {F: \mathcal{C}\rightarrow \text{Top}(X)}. Then we have a nice contravariant functor for any pair of objects lying over {U}, say {\eta} and {\eta '}. We’ll suggestively call the functor {\text{Isom}_{\eta, \eta '}: \text{Top}(U)\rightarrow \text{Sets}}. The functor is just the set of isomorphisms on the open set you apply it to. So {\text{Isom}_{\eta, \eta '}(V)=\{\eta |_V\stackrel{\sim}{\rightarrow} \eta '|_V\}}. If {\text{Isom}} is a sheaf for all pairs of objects and open sets, then sometimes we call {F: \mathcal{C}\rightarrow \text{Top}(X)} a pre-stack. You may want to take a moment to absorb this. There are tons of things going on in these words.

A stack is just a pre-stack that also satisfies a cocycle condition: Given any covering {\{U_i\}} of {U\in \text{Top}(X)} and objects lying over {U_i} say {\eta_i} and isomorphisms {\phi_{ij}: \eta_i|_{U_{ij}}\rightarrow \eta_j|_{U_{ij}}} satisfying the cocycle condition {\phi_{j,k}|_{U_{ijk}}\circ \phi_{i,j}|_{U_{ijk}}=\phi_{i,k}|_{U_{ijk}}} then there is an object {\eta} over {U} with isos {q_i: \eta|_{U_i}\rightarrow \eta_i} such that {\phi_{i,j}\circ q_i|_{U_{ij}}\simeq q_j|_{U_{ij}}}.

Don’t be scared off by this. If you’ve ever tried to do any gluing of objects in topology this should look familiar. It is just a purely formal way of saying that if you have locally defined things that are consistent on overlaps, then you can glue them together to get an object over the whole thing.

Another way to say these conditions is as follows. The first says that you can glue isos (Isom forms a sheaf, the non-trivial aspect of which is the gluing axiom). The second says that you can glue objects. If you want an exercise to see if you’ve parsed all this suppose {X} is a manifold. Define {L(X)} to be the category of line bundles on {X}. The objects of which are pairs {(U, \mathcal{L}_U)} an open set and a line bundle on {U}. The morphisms are “restriction”, {Hom((U, \mathcal{L}_U), (V, \mathcal{L}_V))} is empty if we don’t have an inclusion {U\hookrightarrow V}, and otherwise it consists of the inclusion itself along with the set of all isomorphisms {\mathcal{L}_V|_U\rightarrow \mathcal{L}_U}.

We have an obvious functor {L(X)\rightarrow \text{Top}(X)} by just forgetting the line bundle. Show this functor fibers the category in groupoids over {X}, the pre-stack condition holds, and the stack condition holds. My next post will be me going through these details to help show what is going on in these definitions. Then we’ll generalize from a stack on a topological space to a stack on a general site (particularly the Zariski site) along with more examples. Then we’ll move on to what a gerbe is and how it has anything to do with deformation theory.


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Categories Fibered in Groupoids

Sorry about the delay, I’ve been really busy with other things. Most (probably all) people have completely forgotten what I was talking about. Luckily you don’t need to in order to follow this post!

Today we’ll look at what it means for a category to be fibered in groupoids over another one. Suppose we have a (covariant) functor {F:\mathcal{C}\rightarrow\mathcal{D}}. I’ll refer to {\mathcal{D}} as the “base” category and {\mathcal{C}} as lying over {\mathcal{D}}. We’ll say {\mathcal{C}} is fibered in groupoids over {\mathcal{D}} (by {F}) if the functor satisfies two conditions.

First, suppose we have objects in the base with an arrow between them {A\rightarrow B} and an object lying over {B} (i.e. some object {Y\in \mathcal{C}} such that {F(Y)=B}). The condition is that whenever we have this situation we can “complete the square”. This means that we can find an object and an arrow {X\rightarrow Y} such that the arrow maps to the base arrow. So {F(X)=A} and {F(X\rightarrow Y)= A\rightarrow B}.

It might be good to visualize this in the following way: Given {\begin{matrix} & & Y \\ & & \\ A & \rightarrow & B \end{matrix}} you can always complete to {\begin{matrix} X & \rightarrow & Y \\ & & \\ A & \rightarrow & B\end{matrix}}

The second condition is that whenever you have objects and arrows {A\rightarrow B\rightarrow C} in the base category and you have {X, Y, Z} lying over {A, B, C} respectively with the two arrows {Y\rightarrow Z} lying over {B\rightarrow C} and {X\rightarrow Z} lying over the composite {A\rightarrow C}, there is a unique arrow {X\rightarrow Y} so that everying lies over {A\rightarrow B\rightarrow C}.

There is a nice way to visualize this as well, but I am still awful at making nice things in wordpress, so we’ll do it as follows, you have the following two pieces of information

{\begin{matrix} Y & \rightarrow & Z \\ & & \\ B & \rightarrow & C \end{matrix}} and {\begin{matrix} X & \rightarrow & \rightarrow & \rightarrow & Z \\ \\ A & \rightarrow & B & \rightarrow & C \end{matrix}}

the whole thing can be completed {\mathit{uniquely}} to {\begin{matrix} X & \rightarrow & Y & \rightarrow & Z \\ \\ A & \rightarrow & B & \rightarrow & C \end{matrix}}.

To get a feel for this, let’s look at a motivating example for the terminology. Consider any functor {F: \mathcal{C}\rightarrow \mathcal{D}}, then if we pick an object {D} in the base we can form the {\mathit{fiber \ category}} over {D}, which we’ll denote {\mathcal{C}_D}. This is just a category whose objects lie over {D} i.e. all {X} so that {F(X)=D}, and the morphisms are the ones the functor maps to the identity morphsim {id_D: D\rightarrow D}.

Claim: Any fiber category that is fibered in groupoids (via the functor that we are taking the fiber of) is a groupoid. By groupoid here we just mean that every morphism is an isomorphism.

Suppose there is a map between {Y} and {Z} in {\mathcal{C}_D}, say {Y\stackrel{f}{\rightarrow} Z}. Then by definition of the category {F(f)=id_D}. So we can build the situation of the second condition. The base is just {D\rightarrow D \rightarrow D} all the identity and hence the composition is the identity. We have {Y\rightarrow Z} lying over the second identity map, and we have the identity {Z\rightarrow Z} lying over the composition, so we get a unique map {Z\rightarrow Y} such that the composite {Z\rightarrow Y\rightarrow Z} is the identity. We can now repeat this process with this unique map {Z\rightarrow Y} to get {Y \rightarrow Z \rightarrow Y} which is the identity on {Y} and by uniqueness the completed map must be the original {f} and hence {f} is an isomorphism. Thus the fiber category is a groupoid since any morphism is an isomorphism.

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