## The Summer 2017 Mathematics A To Z: Functor

Gaurish gives me another topic for today. I’m now no longer sure whether Gaurish hopes me to become a topology blogger or a category theory blogger. I have the last laugh, though. I’ve wanted to get better-versed in both fields and there’s nothing like explaining something to learn about it.

# Functor.

So, category theory. It’s a foundational field. It talks about stuff that’s terribly abstract. This means it’s powerful, but it can be hard to think of interesting examples. I’ll try, though.

It starts with categories. These have three parts. The first part is a set of things. (There always is.) The second part is a collection of matches between pairs of things in the set. They’re called morphisms. The third part is a rule that lets us combine two morphisms into a new, third one. That is. Suppose ‘a’, ‘b’, and ‘c’ are things in the set. Then there’s a morphism that matches $a \rightarrow b$, and a morphism that matches $b \rightarrow c$. And we can combine them into another morphism that matches $a \rightarrow c$. So we have a set of things, and a set of things we can do with those things. And the set of things we can do is itself a group.

This describes a lot of stuff. Group theory fits seamlessly into this description. Most of what we do with numbers is a kind of group theory. Vector spaces do too. Most of what we do with analysis has vector spaces underneath it. Topology does too. Most of what we do with geometry is an expression of topology. So you see why category theory is so foundational.

Functors enter our picture when we have two categories. Or more. They’re about the ways we can match up categories. But let’s start with two categories. One of them I’ll name ‘C’, and the other, ‘D’. A functor has to match everything that’s in the set of ‘C’ to something that’s in the set of ‘D’.

And it does more. It has to match every morphism between things in ‘C’ to some other morphism, between corresponding things in ‘D’. It’s got to do it in a way that satisfies that combining, too. That is, suppose that ‘f’ and ‘g’ are morphisms for ‘C’. And that ‘f’ and ‘g’ combine to make ‘h’. Then, the functor has to match ‘f’ and ‘g’ and ‘h’ to some morphisms for ‘D’. The combination of whatever ‘f’ matches to and whatever ‘g’ matches to has to be whatever ‘h’ matches to.

This might sound to you like a homomorphism. If it does, I admire your memory or mathematical prowess. Functors are about matching one thing to another in a way that preserves structure. Structure is the way that sets of things can interact. We naturally look for stuff made up of different things that have the same structure. Yes, functors are themselves a category. That is, you can make a brand-new category whose set of things are the functors between two other categories. This is a good spot to pause while the dizziness passes.

There are two kingdoms of functor. You tell them apart by what they do with the morphisms. Here again I’m going to need my categories ‘C’ and ‘D’. I need a morphism for ‘C’. I’ll call that ‘f’. ‘f’ has to match something in the set of ‘C’ to something in the set of ‘C’. Let me call the first something ‘a’, and the second something ‘b’. That’s all right so far? Thank you.

Let me call my functor ‘F’. ‘F’ matches all the elements in ‘C’ to elements in ‘D’. And it matches all the morphisms on the elements in ‘C’ to morphisms on the elmenets in ‘D’. So if I write ‘F(a)’, what I mean is look at the element ‘a’ in the set for ‘C’. Then look at what element in the set for ‘D’ the functor matches with ‘a’. If I write ‘F(b)’, what I mean is look at the element ‘b’ in the set for ‘C’. Then pick out whatever element in the set for ‘D’ gets matched to ‘b’. If I write ‘F(f)’, what I mean is to look at the morphism ‘f’ between elements in ‘C’. Then pick out whatever morphism between elements in ‘D’ that that gets matched with.

Here’s where I’m going with this. Suppose my morphism ‘f’ matches ‘a’ to ‘b’. Does the functor of that morphism, ‘F(f)’, match ‘F(a)’ to ‘F(b)’? Of course, you say, what else could it do? And the answer is: why couldn’t it match ‘F(b)’ to ‘F(a)’?

No, it doesn’t break everything. Not if you’re consistent about swapping the order of the matchings. The normal everyday order, the one you’d thought couldn’t have an alternative, is a “covariant functor”. The crosswise order, this second thought, is a “contravariant functor”. Covariant and contravariant are distinctions that weave through much of mathematics. They particularly appear through tensors and the geometry they imply. In that introduction they tend to be difficult, even mean, creations, since in regular old Euclidean space they don’t mean anything different. They’re different for non-Euclidean spaces, and that’s important and valuable. The covariant versus contravariant difference is easier to grasp here.

Functors work their way into computer science. The avenue here is in functional programming. That’s a method of programming in which instead of the normal long list of commands, you write a single line of code that holds like fourteen “->” symbols that makes the computer stop and catch fire when it encounters a bug. The advantage is that when you have the code debugged it’s quite speedy and memory-efficient. The disadvantage is if you have to alter the function later, it’s easiest to throw everything out and start from scratch, beginning from vacuum-tube-based computing machines. But it works well while it does. You just have to get the hang of it.

• #### gaurish 9:55 am on Saturday, 12 August, 2017 Permalink | Reply

Can you suggest a nice introductory book on category theory for beginners? What I understand is that they generalize the notions defined concretely in algebra (which were motivated by arithmetic), but I lack any concrete understanding.

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• #### mathtuition88 2:56 pm on Sunday, 13 August, 2017 Permalink | Reply

“Categories for the Working Mathematician” by Mac Lane is good and foundational (recommended for serious readers). Another book “Cakes, Custard and Category Theory” by Eugenia Cheng is accessible even to laymen.

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• #### Joseph Nebus 5:08 pm on Sunday, 13 August, 2017 Permalink | Reply

I’m grateful to MathTuition88 for the suggestion. I’m afraid I’m poorly-enough read in category theory I don’t have any good idea where beginners ought to start.

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• #### elkement (Elke Stangl) 1:59 pm on Friday, 18 August, 2017 Permalink | Reply

May I ask a computer science question ;-) ? I tried to understand how this functor from category theory would be mapped onto (Ha – another level of mapping!! ;-)) a functor in C++ but was not very successful. In this discussion https://stackoverflow.com/questions/356950/c-functors-and-their-uses somebody says that a functor in category theory ‘has nothing to do with the C++ concept of functor’.

Would you agree? Or if not, can you maybe explain how an ‘implementation’ of your functor example would look like in C++ (or some pseudo-code in some language…). Or keep that in mind for a future post if you ever want to return to that subject!

Anyway: I really enjoy this series!!

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## The End 2016 Mathematics A To Z: General Covariance

Today’s term is another request, and another of those that tests my ability to make something understandable. I’ll try anyway. The request comes from Elke Stangl, whose “Research Notes on Energy, Software, Life, the Universe, and Everything” blog I first ran across years ago, when she was explaining some dynamical systems work.

## General Covariance

So, tensors. They’re the things mathematicians get into when they figure vectors just aren’t hard enough. Physics majors learn about them too. Electrical engineers really get into them. Some material science types too.

You maybe notice something about those last three groups. They’re interested in subjects that are about space. Like, just, regions of the universe. Material scientists wonder how pressure exerted on something will get transmitted. The structure of what’s in the space matters here. Electrical engineers wonder how electric and magnetic fields send energy in different directions. And physicists — well, everybody who’s ever read a pop science treatment of general relativity knows. There’s something about the shape of space something something gravity something equivalent acceleration.

So this gets us to tensors. Tensors are this mathematical structure. They’re about how stuff that starts in one direction gets transmitted into other directions. You can see how that’s got to have something to do with transmitting pressure through objects. It’s probably not too much work to figure how that’s relevant to energy moving through space. That it has something to do with space as just volume is harder to imagine. But physics types have talked about it quite casually for over a century now. Science fiction writers have been enthusiastic about it almost that long. So it’s kind of like the Roman Empire. It’s an idea we hear about early and often enough we’re never really introduced to it. It’s never a big new idea we’re presented, the way, like, you get specifically told there was (say) a War of 1812. We just soak up a couple bits we overhear about the idea and carry on as best our lives allow.

But to think of space. Start from somewhere. Imagine moving a little bit in one direction. How far have you moved? If you started out in this one direction, did you somehow end up in a different one? Now imagine moving in a different direction. Now how far are you from where you started? How far is your direction from where you might have imagined you’d be? Our intuition is built around a Euclidean space, or one close enough to Euclidean. These directions and distances and combined movements work as they would on a sheet of paper, or in our living room. But there is a difference. Walk a kilometer due east and then one due north and you will not be in exactly the same spot as if you had walked a kilometer due north and then one due east. Tensors are efficient ways to describe those little differences. And they tell us something of the shape of the Earth from knowing these differences. And they do it using much of the form that matrices and vectors do, so they’re not so hard to learn as they might be.

That’s all prelude. Here’s the next piece. We go looking at transformations. We take a perfectly good coordinate system and a point in it. Now let the light of the full Moon shine upon it, so that it shifts to being a coordinate werewolf. Look around you. There’s a tensor that describes how your coordinates look here. What is it?

You might wonder why we care about transformations. What was wrong with the coordinates we started with? But that’s because mathematicians have lumped a lot of stuff into the same name of “transformation”. A transformation might be something as dull as “sliding things over a little bit”. Or “turning things a bit”. It might be “letting a second of time pass”. Or “following the flow of whatever’s moving”. Stuff we’d like to know for physics work.

“General covariance” is a term that comes up when thinking about transformations. Suppose we have a description of some physics problem. By this mostly we mean “something moving in space” or “a bit of light moving in space”. That’s because they’re good building blocks. A lot of what we might want to know can be understood as some mix of those two problems.

Put your description through the same transformation your coordinate system had. This will (most of the time) change the details of how your problem’s represented. But does it change the overall description? Is our old description no longer even meaningful?

I trust at this point you’ve nodded and thought something like “well, that makes sense”. Give it another thought. How could we not have a “generally covariant” description of something? Coordinate systems are our impositions on a problem. We create them to make our lives easier. They’re real things in exactly the same way that lines of longitude and latitude are real. If we increased the number describing the longitude of every point in the world by 14, we wouldn’t change anything real about where stuff was or how to navigate to it. We couldn’t.

Here I admit I’m stumped. I can’t think of a good example of a system that would look good but not be generally covariant. I’m forced to resort to metaphors and analogies that make this essay particularly unsuitable to use for your thesis defense.

So here’s the thing. Longitude is a completely arbitrary thing. Measuring where you are east or west of some prime meridian might be universal, or easy for anyone to tumble onto. But the prime meridian is a cultural choice. It’s changed before. It may change again. Indeed, Geographic Information Services people still work with many different prime meridians. Most of them are for specialized purposes. Stuff like mapping New Jersey in feet north and east of some reference, for which Greenwich would make the numbers too ugly. But if our planet is mapped in an alien’s records, that map has at its center some line almost surely not Greenwich.

But latitude? Latitude is, at least, less arbitrary. That we measure it from zero to ninety degrees, north or south, is a cultural choice. (Or from -90 to 90 degrees. Same thing.) But that there’s a north pole an a south pole? That’s true as long as the planet is rotating. And that’s forced on us. If we tried to describe the Earth as rotating on an axis between Paris and Mexico City, we would … be fighting an uphill struggle, at least. It’s hard to see any problem that might make easier, apart from getting between Paris and Mexico City.

In models of the laws of physics we don’t really care about the north or south pole. A planet might have them or might not. But it has got some privileged stuff that just has to be so. We can’t have stuff that makes the speed of light in a vacuum change. And we have to make sense of a block of space that hasn’t got anything in it, no matter, no light, no energy, no gravity. I think those are the important pieces actually. But I’ll defer, growling angrily, to an expert in general relativity or non-Euclidean coordinates if I’ve misunderstood.

It’s often put that “general covariance” is one of the requirements for a scheme to describe General Relativity. I shall risk sounding like I’m making a joke and say that depends on your perspective. One can use different philosophical bases for describing General Relativity. In some of them you can see general covariance as a result rather than use it as a basic assumption. Here’s a 1993 paper by Dr John D Norton that describes some of the different ways to understand the point of general covariance.

By the way the term “general covariance” comes from two pieces. The “covariance” is because it describes how changes in one coordinate system are reflected in another. It’s “general” because we talk about coordinate transformations without knowing much about them. That is, we’re talking about transformations in general, instead of some specific case that’s easy to work with. This is why the mathematics of this can be frightfully tricky; we don’t know much about the transformations we’re working with. For a parallel, it’s easy to tell someone how to divide 14 into 112. It’s harder to tell them how to divide absolutely any number into absolutely any other number.

Quite a bit of mathematical physics plays into geometry. Gravity physicists mostly see as a problem of geometry. People who like reading up on science take that as given too. But many problems can be understood as a point or a blob of points in some kind of space, and how that point moves or that blob evolves in time. We don’t see “general covariance” in these other fields exactly. But we do see things that resemble it. It’s an idea with considerable reach.

I’m not sure how I feel about this. For most of my essays I’ve kept away from equations, even for the Why Stuff Can Orbit sequence. But this is one of those subjects it’s hard to be exact about without equations. I might revisit this in a special all-symbols, calculus-included, edition. Depends what my schedule looks like.

• #### elkement (Elke Stangl) 7:03 pm on Wednesday, 16 November, 2016 Permalink | Reply

Thanks for accepting this challenge – I think you explained it as good as one possibly can without equations!!

I think for understanding General Relativity you have to revisit some ideas from ‘flat space’ tensor calculus you took for granted, like a vector being sort of an arrow that can be moved around carelessly in space or what a coordinate transformation actually means (when applied to curved space). It seems GR is introduced either very formally, not to raise any false intuition, explaining the abstract big machinery with differentiable manifolds and atlases etc. and adding the actual physics as late as possible, or by starting from flat space metrics, staying close to ‘tangible physics’ and adding unfamiliar stuff slowly.
Sometimes I wonder if one (when trying to explain this to a freshman) could skip the ‘flat space’ part and start with the seemingly abstract but more general foundations as those cover anything? Perhaps it would be easier and more efficient never to learn about Gauss and Stokes theorem first but start with integration on manifolds and present such theorems as special cases?

And thanks for the pointer to this very interesting paper!

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• #### Joseph Nebus 3:47 am on Sunday, 20 November, 2016 Permalink | Reply

Thanks so for the kind words. I worried through the writing of this that I was going too far wrong and I admit I’m still waiting for a real expert to come along and destroy my essay and my spirits. Another few weeks and I should be far enough from writing it that I can take being told all the ways I’m wrong, though.

You’re right about the ways General Relativity seems to be often taught. And I also wonder if it couldn’t be better-taught starting from a completely abstract base and then filling in why this matches the way the world looks. Something equivalent to introducing vectors as “things that are in a vector space, which are things with these properties” instead of as arrows in space. I suspect it might not be really doable, though, based on how many times I crashed against covariant versus contravariant indices and that’s incredibly small stuff.

But there are so many oddball-perspective physics books out there that someone must have tried it at least once. And many of them are really good at least in making stuff look different, if not better. I’m sorry not to be skilled enough in the field to give it a fair try. Maybe some semester I’ll go through a proper text on this and post the notes I make on it.

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• #### elkement (Elke Stangl) 10:35 am on Sunday, 20 November, 2016 Permalink | Reply

I’ve recently stumbled upon this GR course http://www.infocobuild.com/education/audio-video-courses/physics/gravity-and-light-2015-we-heraeus.html : this lecturer is really very careful in introducing the foundations in the most abstract way, just as you say, without any intuitive references. No physics until lecture 9 (to prove that Newtonian gravity can also be presented in a generally covariant way – very interesting to read the history of science paper you linked in relation to this, BWT), then more math only, until finally in lecture 13 we return to ‘our’ spacetime.

I am also learning GR as a hobbyist project as this was not a mandatory subject in my physics degree program (I specialized in condensed matter, lasers, optics, superconductors…), and I admit I use mainly freely available sources like such lectures or detailed lecture notes. I have sort of planned to post about my favorite resources and/or that learning experience, too, but given my typical blogging frequency compared to yours I suppose I can wait for your postings and just use those as a reference :-)

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• #### Joseph Nebus 11:02 pm on Friday, 25 November, 2016 Permalink | Reply

Ooh, that’s a great-looking series, though I’ve lacked the time to watch it yet. I’ve regretted not taking a proper course on general relativity. When I was an undergraduate my physics department did a lecture series on general relativity without advanced mathematics, but it conflicted with something on my schedule and I hoped they’d rerun the series another semester. Of course they didn’t at least during my time there.

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## Tensor.

The true but unenlightening answer first: a tensor is a regular, rectangular grid of numbers. The most common kind is a two-dimensional grid, so that it looks like a matrix, or like the times tables. It might be square, with as many rows as columns, or it might be rectangular.

It can also be one-dimensional, looking like a row or a column of numbers. Or it could be three-dimensional, rows and columns and whole levels of numbers. We don’t try to visualize that. It can be what we call zero-dimensional, in which case it just looks like a solitary number. It might be four- or more-dimensional, although I confess I’ve never heard of anyone who actually writes out such a thing. It’s just so hard to visualize.

You can add and subtract tensors if they’re of compatible sizes. You can also do something like multiplication. And this does mean that tensors of compatible sizes will form a ring. Of course, that doesn’t say why they’re interesting.

Tensors are useful because they can describe spatial relationships efficiently. The word comes from the same Latin root as “tension”, a hint about how we can imagine it. A common use of tensors is in describing the stress in an object. Applying stress in different directions to an object often produces different effects. The classic example there is a newspaper. Rip it in one direction and you get a smooth, clean tear. Rip it perpendicularly and you get a raggedy mess. The stress tensor represents this: it gives some idea of how a force put on the paper will create a tear.

Tensors show up a lot in physics, and so in mathematical physics. Technically they show up everywhere, since vectors and even plain old numbers (scalars, in the lingo) are kinds of tensors, but that’s not what I mean. Tensors can describe efficiently things whose magnitude and direction changes based on where something is and where it’s looking. So they are a great tool to use if one wants to represent stress, or how well magnetic fields pass through objects, or how electrical fields are distorted by the objects they move in. And they describe space, as well: general relativity is built on tensors. The mathematics of a tensor allow one to describe how space is shaped, based on how to measure the distance between two points in space.

My own mathematical education happened to be pretty tensor-light. I never happened to have courses that forced me to get good with them, and I confess to feeling intimidated when a mathematical argument gets deep into tensor mathematics. Joseph C Kolecki, with NASA’s Glenn (Lewis) Research Center, published in 2002 a nice little booklet “An Introduction to Tensors for Students of Physics and Engineering”. This I think nicely bridges some of the gap between mathematical structures like vectors and matrices, that mathematics and physics majors know well, and the kinds of tensors that get called tensors and that can be intimidating.

• #### elkement 8:09 am on Monday, 3 August, 2015 Permalink | Reply

I think the most important thing to learn is and ‘conceptual leap’ to think of a tensor not of a ‘matrix’ but of that abstract object of which the ‘matrix’ is just one possible representation, just as ‘visualizing’ a vector as an ‘arrow’ in some hyperspace instead of a bunch of numbers.

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• #### Joseph Nebus 4:46 am on Tuesday, 4 August, 2015 Permalink | Reply

I think you’re right that understanding a matrix is just one way to represent a tensor is the great conceptual leap. The trouble is having a good explanation for what the tensor is, separate from its representations. Although come to think of it I can’t remember when I managed the conceptual leap between a vector as a set of numbers to a vector being some direction and length in space, either.

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