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category theory, exposition, math, understanding math

More about defining “category”

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In a recent post, I wrote about defining “category” in a way that (I hope) makes it accessible to undergraduate math majors at an early stage.  I have several more things to say about this.

Early intro to categories

The idea is to define a category as a directed graph equipped with an additional structure of composition of paths subject to some axioms.  By giving several small finite examples of categories drawn in that way that gives you an understanding of “category” that has several desirable properties:

  • You get the idea of what a category is in one lecture.
  • With the right choice of examples you get several fine points cleared up:
    • The composition is added structure.
    • A loop doesn’t have to be an identity.
    • Associativity is a genuine requirement —  it is not automatic.
  • You get immediate access to what is by far the most common notation used to work with a category — objects (nodes) and arrows.
  • You don’t have to cope with the difficult chunking required when the first examples given are sets-with-structure and structure-preserving functions.  It’s quite hard to focus on a couple of dots on the paper each representing a group or a topological space and arrows each representing a whole function (not the value of the function!).

Introduce more examples

Then the teacher can go on with the examples that motivated categories in the first place: the big deal categories such as sets, groups and topological spaces.   But they can be introduced using special cases so they don’t require much background.

  • Draw some finite sets and functions between them.  (As an exercise, get the students to find some finite sets and functions that make the picture a category with $f=kh$ as the composite and $f\neq g$.)
  • If the students have had calculus,  introduce them to the category whose objects are real finite nonempty intervals with continuous or differentiable mappings between them.  (Later you can prove that this category is a groupoid!)
  • Find all the groups on a two element set and figure out which maps preserve group multiplication.  (You don’t have to use the word “group” — you can simply show both of them and work out which maps preserve multiplication — and discover isomorphism!.)  This introduces the idea of the arrows being structure-preserving mape. You can get more complicated and use semigroups as well.  If the students know Mathematica you could even do magmas.  Well, maybe not.

All this sounds like a project you could do with high school students.

Large and small

If all this were just a high school (or intro-to-math-for-math-majors) project you wouldn’t have to talk about large vs. small.  However, I have some ideas about approaching this topic.

In the first place, you can define category, or any other mathematical object that might involve a proper class, using the syntactic approach I described in Just-in-time foundations.  You don’t say “A category consists of a set of objects and a set of arrows such that …”.  Instead you say something like “A category $\mathcal{C}$ has objects $A,\,B,\,C\ldots$ such that…”.

This can be understood as meaning “For any $A$, the statement $A$ is an object of  $\mathcal{C}$ is either true or false”, and so on.

This approach is used in the Wikibook on category theory.  (Note: this is a permanent link to the November 28 version of the section defining categories, which is mostly my work.  As always with Wikimedia things it may be entirely different when you read this.)

If I were dictator of the math world (not the same thing as dictator of MathWorld) I would want definitions written in this syntactic style.  The trouble is that mathematicians are now so used to mathematical objects having to be sets-with-structure that wording the definition as I did above may leave them feeling unmoored.  Yet the technique avoids having to mention large vs. small until a problem comes up. (In category theory it sometimes comes up when you want to quantify over all objects.)

The ideas outlined in this subsection could be a project for math majors.  You would have to introduce Russell’s Paradox.  But for an early-on intro to categories you could just use the syntactic wording and avoid large vs. small altogether.

 

http://en.wikibooks.org/w/index.php?title=Category_Theory/Categories&stableid=2221684

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category theory, exposition, math, understanding math

Defining “category”

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The concept of category is typically taught later in undergrad math than the concept of group is.  It is supposedly a more advanced concept.  Indeed, the typical examples of categories used in applications are more advanced than some of those in group theory (for example, symmetries of geometric shapes and operations on numbers).

Here are some thoughts on how categories could be taught as early as groups, if not earlier.

Nodes and arrows

Small finite categories can be pictured as a graph using nodes and arrows, together with a specification of the identity arrows and a definition of the composition.  (I am using the word “graph” the way category people use it:  a directed graph with possible multiple edges and loops.)

An example is the category pictured below with three objects and seven arrows. The composition is forced except for $kh$, which I hereby define to be $f$.

This way of picturing a category is  easy to grasp. The composite $kh$ visibly has to be either $f$ or $g$.  There is only one choice for the composite of any other composable pair.  Still, the choice of composite is not deducible directly by looking at the graph.

A first class in category theory using graphs as examples could start with this example, or the example in Note 1 below.  This example is nontrivial (never start any subject with trivial examples!) and easy to grasp, in this case using the extraordinary preprocessing your brain does with the input from your eyes.  The definition of category is complicated enough that you should probably present the graph and then give the definition while pointing to what each clause says about the graph.

Most abstract structures have several different ways of representing them. In contrast, when you discuss categorial concepts the standard object-and-arrow notation is the overwhelming favorite.  It reveals domains and codomains and composable pairs, in fact almost everything except which of several possible arrows the composite actually is.  If for example you try to define category using sets and functions as your running example, the student has to do a lot of on-the-go chunking — thinking of a set as a single object, of a set function (which may involve lots of complicated data) as a single chunk with a domain and a codomain, and so on.  But an example shown as a graph comes already chunked and in a picture that is guaranteed to be the most common kind of display they will see in discussions of categories.

After you do these examples, you can introduce trivial and simple graph examples in which the composition is entirely induced; for example these three:

(In case you are wondering, one of them is the empty category.)  I expect that you should also introduce another graph non-example in which associativity fails.

Multiplication tables

The multiplication table for a group is easy to understand, too, in the sense that it gives you a simple method of calculating the product of any two elements.  But it doesn’t provide a visual way to see the product as a category-as-graph does.  Of course, the graph representation works only for finite categories, just as the multiplication table works only for finite groups.

You can give a multiplication table for a small finite category, too, like the one below for the category above.  (“iA” means the identity arrow on A and composition, as usual in category theory, is right to left.) This is certainly more abstract than the graph picture, but it does hit you in the face with the fact that the multiplication is partial.

Notes

1. My suggested example of a category given as a graph shows clearly that you can define two different categorial structures on the graph.  One problem is that the two different structures are isomorphic categories.  In fact, if you engage the students in a discussion about these examples someone may notice that!  So you should probably also use the graph below,where you can define several different category structures that are not all isomorphic. 

2. Multiplication tables and categories-as-graphs-with-composition are extensional presentations.  This means they are presented with all their parts laid out in front of you.  Most groups and categories are given by definitions as accumulations of properties (see concept in the Handbook of Mathematical Discourse).  These definitions tend to make some requirements such as associativity obvious.

Students are sometimes bothered by extensional definitions.  “What are h and k (in the category above)?  What are a, b and c?” (in a group given as a set of letters and a multiplication table).

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Endograph and cograph of real functions

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This post is covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the work provided you follow the requirements of the license.

Introduction

In the article Functions: Images and Metaphors in abstractmath I list a bunch of different images or metaphors for thinking about functions. Some of these metaphors have realizations in pictures, such as a graph or a surface shown by level curves. Others have typographical representations, as formulas, algorithms or flowcharts (which are also pictorial). There are kinetic metaphors — the graph of {y=x^2} swoops up to the right.

Many of these same metaphors have realizations in actual mathematical representations.

Two images (mentioned only briefly in the abstractmath article) are the cograph and the endograph of a real function of one variable. Both of these are visualizations that correspond to mathematical representations. These representations have been used occasionally in texts, but are not used as much as the usual graph of a continuous function. I think they would be useful in teaching and perhaps even sometimes in research.

A rough and unfinished Mathematica notebook is available that contains code that generate graphs and cographs of real-valued functions. I used it to generate most of the examples in this post, and it contains many other examples. (Note [1].)

The endograph of a function

In principle, the endograph (Note [2]) of a function {f} has a dot for each element of the domain and of the codomain, and an arrow from {x} to {f(x)} for each {x} in the domain. For example, this is the endograph of the function {n\mapsto n^2+1 \pmod 11} from the set {\{0,1,\ldots,10\}} to itself:


“In principle” means that the entire endograph can be shown only for small finite functions. This is analogous to the way calculus books refer to a graph as “the graph of the squaring function” when in fact the infinite tails are cut off.

Real endographs

I expect to discuss finite endographs in another post. Here I will concentrate on endographs of continuous functions with domain and codomain that are connected subsets of the real numbers. I believe that they could be used to good effect in teaching math at the college level.

Here is the endograph of the function {y=x^2} on the reals:

I have displayed this endograph with the real line drawn in the usual way, with tick marks showing the location of the points on the part shown.

The distance function on the reals gives us a way of interpreting the spacing and location of the arrowheads. This means that information can be gleaned from the graph even though only a finite number of arrows are shown. For example you see immediately that the function has only nonnegative values and that its increase grows with {x}.(See note [3]).

I think it would be useful to show students endographs such as this and ask them specific questions about why the arrows do what they do.

For the one shown, you could ask these questions, probably for class discussion rather that on homework.

  • Explain why most of the arrows go to the right. (They go left only between 0 and 1 — and this graph has such a coarse point selection that it shows only two arrows doing that!)
  • Why do the arrows cross over each other? (Tricky question — they wouldn’t cross over if you drew the arrows with negative input below the line instead of above.)
  • What does it say about the function that every arrowhead except two has two curves going into it?

Real Cographs

The cograph (Note [4] of a real function has an arrow from input to output just as the endograph does, but the graph represents the domain and codomain as their disjoint union. In this post the domain is a horizontal representation of the real line and the codomain is another such representation below the domain. You may also represent them in other configurations (Note [5]).

Here is the cograph representation of the function {y=x^2}. Compare it with the endograph representation above.

Besides the question of most arrows going to the right, you could also ask what is the envelope curve on the left.

More examples

Absolute value function

Arctangent function

Notes

[1] This website contains other notebooks you might find useful. They are in Mathematica .nb, .nbp, or .cdf formats, and can be read, evaluated and modified if you have Mathematica 8.0. They can also be made to appear in your browser with Wolfram CDF Player, downloadable free from Wolfram site. The CDF player allows you to operate any interactive demos contained in the file, but you can’t evaluate or modify the file without Mathematica.

The notebooks are mostly raw code with few comments. They are covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the code following the requirements of the license. I am making the files available because I doubt that I will refine them into respectable CDF files any time soon.

[2] I call them “endographs” to avoid confusion with the usual graphs of functions — — drawings of (some of) the set of ordered pairs {x,f(x)} of the function.

[3] This is in contrast to a function defined on a discrete set, where the elements of the domain and codomain can be arranged in any old way. Then the significance of the resulting arrangement of the arrows lies entirely in which two dots they connect. Even then, some things can be seen immediately: Whether the function is a cycle, permutation, an involution, idempotent, and so on.

Of course, the placement of the arrows may tell you more if the finite sets are ordered in a natural way, as for example a function on the integers modulo some integer.

[4] The text [1] uses the cograph representation extensively. The word “cograph” is being used with its standard meaning in category theory. It is used by graph theorists with an entirely different meaning.

[5] It would also be possible to show the domain codomain in the usual {x-y} plane arrangement, with the domain the {x} axis and the codomain the {y} axis. I have not written the code for this yet.

References

[1] Sets for Mathematics, by F. William Lawvere and Robert Rosebrugh. Cambridge University Press, 2003.

[2] Martin Flashman’s website contains many exampls of cographs of functions, which he calls mapping diagrams.

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