Category Archives: category theory

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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Turning definitions into mathematical objects

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When G&G was moved to this current location, most of the links were trashed, so I have been repairing them a bit at a time. There are still some broken links from 2009 and before but I am working on them.  Honest.

G&G contained a series of posts about turning definitions into mathematical objects, mostly written in 2009. Not only were their links broken (and they used many links to each other), but two of the articles were trashed.  I have now removed them from this website. They are all still at the old website: http://sixwingedseraph.wordpress.com/ and as far as I know all the links to each other work.

When I have time I will combine them into one long article.  Until then, the old website will remain.

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Showing categorical diagrams in 3D

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook algebra1.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

In Graph-Based Logic and Sketches, Atish Bagchi and I needed to construct a lot of cones based on fairly complicated diagrams. We generally show the base diagram and left the reader to imagine the cone. This post is an experiment in presenting such a diagram in 3D, with its cone and other constructions based on it.

To understand this post, you need a basic understanding of categories, functors and limit cones (see References).

The notebook and CDF files that generate this display may be downloaded from here:

These files may be used and modified as you wish according to the Creative Commons rule listed under “Permissions” (at the top of the window).

The sketch for categories: composition

A finite-limit sketch (FL sketch) is a category with finite limits given by specifying certain  nodes and arrows, commutative diagrams using these nodes and arrows, and limit cones based on diagrams using the given nodes and arrows.  A model of an FL sketch is a finite-limit-preserving functor from the FL sketch into some category \mathcal{C}.  Detailed descriptions of FL-sketches are  in References [1], [2] and [3] (below).

Categories themselves may be sketched by FL-sketches. Here I will present the part of the sketch that constructs (in a model) the object of composites of two arrows.  This is the specification for composite:

  1. The composite of two arrows f:A\to B and g:B'\to C is defined if and only if B=B'.
  2. The composite is denoted by gf.
  3. The domain of gf is A and the codomain is C.

We start with a diagram in the FL sketch for categories that gives the data corresponding to two arrows that may be composed.  This diagram involves nodes ob and ar, which in a model become the object of objects and the object of arrows of the category object in \mathcal{C}.  (Suppose \mathcal{C} is the category of sets; then the model is simply a small category.  The node ob goes to the set of objects of the small category and ar goes to the set of arrows.)  The arrows labeled dom and cod take (in a model) an arrow to its domain and codomain respectively. Here is the diagram:

You can move the diagram around in three dimensions to see it from different perspectives. (Of course it isn’t really in three dimensions. Your eyes-to-brain module reconstructs the illusion of three dimensions when you twirl the diagram around.)

Note that this is a diagram, not a directed graph (digraph). (In the paper, Atish and I, like most category theorists, say “graph” instead of “digraph”.) It has an underlying digraph (see Chapter 2 of Graph-Based Logic and Sketches), but the labeling of several different nodes of the underlying digraph by the name of the same node of the sketch is meaningful. 

Here, the key fact is that in the diagram there are two arrows, one labeled dom and the other cod, to the same node labeled ob, and two other arrows to two different nodes labeled ob. 

Now click c1.

This shows a cone over the diagram.  One of the nodes in the sketch must be cp (in other words given beforehand; that is, we are specifying not only that the blue stuff is a limit cone but that the limit is the node cp.)   In a model, this cone must become a limit cone.  It follows from the properties of limits that the elements of cp in the model in Sets are pairs of arrows with the property that one has a codomain that is the same as the domain of the other.  The label “cp” stands for “compatible pairs”.

Now click c2.

The green stuff is a diagram showing two arrows from the node labeled ar to the left and right nodes labeled ob in the original black diagram.  This is not a cone; it is just a diagram.  In a model, any arrow in the vertex must have domain the same as the domain of one of the arrows in the compatible pair, and codomain the same as the codomain of the other arrow of the pair.  Thus in the model, an arrow living in the set labeled with “ar” in green must satisfy requirement 3 in the specification for composition given above.

Note that the requirement that the green diagram be commutative in a model is vacuous, so it doesn’t matter whether we specify it specifically as a diagram in the sketch or not.

Now click c3.

The arrow labeled comp must be specified as an arrow in the sketch.  We want its value to be the composite of an element of cp in a model, in other words a compatible pair of arrows.  At this point that will not necessarily be true.  But all can be saved:

Now click c4.

We must specify that the diagram given by the thick arrows must be a diagram of the sketch.  The fact that it must become commutative in a model means exactly that the red arrow comp from cp to ar takes a compatible pair to an arrow that satisfies requirements 1–3 of the specification of composite shown above.

References

  1. Peter T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Volume 2 (Oxford Logic Guides 44), by Oxford University Press, ISBN 978-0198524960.
  2. Michael Barr and Charles Wells, Category theory for computing science (1999).    (This is the easiest to start with but it doesn’t get very far.)
  3. Michael Barr and Charles Wells, Toposes, Triples and Theories (2005).  Reprints in Theory and Applications of Categories 1.

 
\mathcal{C}

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abstractmath.org, category theory, computer science, language of math, math, representations, understanding math

Operation as metaphor in math

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Operation: Is it just a name or is there a metaphor behind it?

A function of the form {f:S\times S\rightarrow S} may be called a binary operation on {S}. The main point to notice is that it takes pairs of elements of {S} to the same set {S}.

A binary operation is a special case of n-ary operation for any natural number {n}, which is a function of the form {f:S^n\rightarrow S}. A {1}-ary (unary) operation on {S} is a function from a set to itself (such as the map that takes an element of a group to its inverse), and a {0}-ary (nullary) operation on {S} is a constant.

It is useful at times to consider multisorted algebra, where a binary operation can be a function {f:S_1\times S_2\rightarrow S_3} where the {S_i} are possibly different sets. Then a unary operation is simply a function.

Calling a function a multisorted unary operation suggest a different way of thinking about it, but as far as I can tell the difference is only that the author is thinking of algebraic operations as examples. This does not seem to be a different metaphor the way “function as map” and “function as transformation” are different metaphors. Am I missing something?

In the 1960’s some mathematicians (not algebraists) were taken aback by the idea that addition of real numbers (for example) is a function. I observed this personally. I don’t think any mathematician would react this way today.

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Multivalued Functions

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Multivalued functions

I am reconstructing the abstractmath website and am currently working on the part on functions. This has generated some bloggable blustering.

The phrase multivalued function refers to an object that is like a function {f:S\rightarrow T} except that for {s\in S}, {f(s)} may denote more than one value. Multivalued functions arose in considering complex functions such as {\sqrt{z}}. Another example: the indefinite integral is a multivalued operator.

It is useful to think of a multivalued function as a function although it violates one of the requirements of being a function (being single-valued).

A multivalued function {f:S\rightarrow T} can be modeled as a function with domain {S} and codomain the set of all subsets of {T}. The two meanings are equivalent in a strong sense (naturally equivalent). Even so, it seems to me that they represent two different ways of thinking about multivalued functions.: “The value may be any of these things…” as opposed to “The value is this whole set of things.”) The “value may be any of these…” idea has a perfectly good mathematical model: a relation (set of ordered pairs) from {S} to {T} which is the inverse of a surjective function.

Phrases such as “multivalued function” and “partial function” upset some uptight types who say things like, “But a multivalued function is not a function!”. A stepmother is not a mother, either.

I fulminated at length about this in the Handbook article on radial category. (This is conceptual category in the sense of Lakoff, Women, fire and dangerous things, University of Chicago, 1986.). The Handbook is on line, but it downloads very slowly, so I have extracted the particular page on radial categories here.

Functions generate a radial category of concepts in mathematics. There are lots of other concepts in math that have generated radial categories. Think of “incomplete proof” or “left identity”. Radial categories are a basic mechanism of the way we think and function in the world. They should not be banished from mathematics.

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Function as map

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This is a first draft of an article to eventually appear in abstractmath.

Images and metaphors

To explain a math concept, you need to explain how mathematicians think about the concept. This is what in abstractmath I call the images and metaphors carried by the concept. Of course you have to give the precise definition of the concept and basic theorems about it. But without the images and metaphors most students, not to mention mathematicians from a different field, will find it hard to prove much more than some immediate consequences of the definition. Nor will they have much sense of the place of the concept in math and applications.

Teachers will often explain the images and metaphors with handwaving and pictures in a fairly vague way. That is good to start with, but it’s important to get more precise about the images and metaphors. That’s because images and metaphors are often not quite a good fit for the concept — they may suggest things that are false and not suggest things that are true. For example, if a set is a container, why isn’t the element-of relation transitive? (A coin in a coinpurse in your pocket is a coin in your pocket.)

“A metaphor is a useful way to think about something, but it is not the same thing as the same thing.” (I think I stole that from the Economist.) Here, I am going to get precise with the notion that a function is a map. I am acting like a mathematician in “getting precise”, but I am getting precise about a metaphor, not about a mathematical object.

A function is a map

A map (ordinary paper map) of Minnesota has the property that each point on the paper represents a point in the state of Minnesota. This map can be represented as a mathematical function from a subset of a 2-sphere to {{\mathbb R}^2}. The function is a mathematical idealization of the relation between the state and the piece of paper, analogous to the mathematical description of the flight of a rocket ship as a function from {{\mathbb R}} to {{\mathbb R}^3}.

The Minnesota map-as-function is probably continuous and differentiable, and as is well known it can be angle preserving or area preserving but not both.

So you can say there is a point on the paper that represents the location of the statue of Paul Bunyan in Bemidji. There is a set of points that represents the part of the Mississippi River that lies in Minnesota. And so on.

A function has an image. If you think about it you will realize that the image is just a certain portion of the piece of paper. Knowing that a particular point on the paper is in the image of the function is not the information contained in what we call “this map of Minnesota”.

This yields what I consider a basic insight about function-as-map:  The map contains the information about the preimage of each point on the paper map. So:

The map in the sense of a “map of Minnesota” is represented by the whole function, not merely by the image.

I think that is the essence of the metaphor that a function is a map. And I don’t think newbies in abstractmath always understand that relationship.

A morphism is a map

The preceding discussion doesn’t really represent how we think of a paper map of Minnesota. We don’t think in terms of points at all. What we see are marks on the map showing where some particular things are. If it is a road map it has marks showing a lot of roads, a lot of towns, and maybe county boundaries. If it is a topographical map it will show level curves showing elevation. So a paper map of a state should be represented by a structure preserving map, a morphism. Road maps preserve some structure, topographical maps preserve other structure.

The things we call “maps” in math are usually morphisms. For example, you could say that every simple closed curve in the plane is an equivalence class of maps from the unit circle to the plane. Here equivalence class meaning forget the parametrization.

The very fact that I have to mention forgetting the parametrization is that the commonest mathematical way to talk about morphisms is as point-to-point maps with certain properties. But we think about a simple closed curve in the plane as just a distorted circle. The point-to-point correspondence doesn’t matter. So this example is really talking about a morphism as a shape-preserving map. Mathematicians introduced points into talking about preserving shapes in the nineteenth century and we are so used to doing that that we think we have to have points for all maps.

Not that points aren’t useful. But I am analyzing the metaphor here, not the technical side of the math.

Groups are functors

People who don’t do category theory think the idea of a mathematical structure as a functor is weird. From the point of view of the preceding discussion, a particular group is a functor from the generic group to some category. (The target category is Set if the group is discrete, Top if it is a topological group, and so on.)

The generic group is a group in a category called its theory or sketch that is just big enough to let it be a group. If the theory is the category with finite products that is just big enough then it is the Lawvere theory of the group. If it is a topos that is just big enough then it is the classifying topos of groups. The theory in this sense is equivalent to some theory in the sense of string-based logic, for example the signature-with-axioms (equational theory) or the first order theory of groups. Johnstone’s Elephant book is the best place to find the translation between these ideas.

A particular group is represented by a finite-limit-preserving functor on the algebraic theory, or by a logical functor on the classifying topos, and so on; constructions which bring with them the right concept of group homomorphisms as well (they will be any natural transformations).

The way we talk about groups mimics the way we talk about maps. We look at the symmetric group on five letters and say its multiplication is noncommutative. “Its multiplication” tells us that when we talk about this group we are talking about the functor, not just the values of the functor on objects. We use the same symbols of juxtaposition for multiplication in any group, “{1}” or “{e}” for the identity, “{a^{-1}}” for the inverse of {a}, and so on. That is because we are really talking about the multiplication, identity and inverse function in the generic group — they really are the same for all groups. That is because a group is not its underlying set, it is a functor. Just like the map of Minnesota “is” the whole function from the state to the paper, not just the image of the function.

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Where does the generic triangle live?

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I recently posted the following question on MathOverflow.  It is a revision of an earlier question.

Motivation 1

There is such a thing as a generic group. In category theory this is done by constructing “theory” of the group, which is a category in a certain doctrine. Functors (in that doctrine) to Set, or more generally to any topos, are groups. The barest such theory (as usually seen) is the Lawverean algebraic theory of groups. This theory is a category containing an object and operations making it a group object in that category, and the theory is the smallest such category that contains all finite limits. There are fancier ones; the fanciest is the classifying topos for groups, which is in some sense the initial topos-with-group object. Since in a topos, you have full-scale first order intuitionistic logic, the classifying topos for groups allows you to reason about the generic group inside the classifying topos and the theorems you prove will be true for all groups. (This is only an approximation of the actual situation.) In particular you can’t prove it is abelian and you can’t prove it isn’t; the logic clearly does not have excluded middle.

Motivation 2

You can prove that a triangle that has two angles that are equal must be isosceles (has two sides that are equal). You can do this with Pappus’ proof: Look at the triangle, flip it over the perpendicular from the odd angle to the other side, look at it again, and the side-angle-side theorem shows you that the “two” triangles are congruent, so two sides much be equal. This appears to me to be true without requiring the parallel postulate. So the theorem and the proof must be true not only in Euclidean 2-space but in any surface of constant curvature. (Here I am getting into territory I know very little about, so this particular motivation may be totally misguided.)

The Question

So what I want is a classifying space of some sort that contains the generic triangle in such a way that maps of the correct sort to any surface of constant curvature are triangles, and so that Pappus’ proof can be carried out in the classifying space. The space doesn’t have to be a topos or a category at all. I have no clue as to what sort of structure it would be.

Note 1: Even the Lawvere theory of groups has its own internal logic — in this case equational logic. You certainly cannot prove the generic group is or is not abelian with equational logic.

Note 2: It does not seem reasonable to me that Pappus’ proof would work in a surface with variable curvature. But maybe there is some trick to define “angle mod curvature” that would make it true.

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Just-in-time foundations

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Introduction

In MathOverflow, statements similar to the following two occurred in comments:

  1. Sets and functions do not form a category
  2. Categories and functors do not form a category.

I cannot find either one of them now, but I want to talk about them anyway.

If you look at the definition of categories in various works (for example references [1] through [3] below) you find that the objects and arrows of a category must each form a “collection” or “class” together with certain operations.   The authors all describe the connection with Grothendieck’s concept of “universe” and define “large categories” and “small categories” in the usual way.  So Statement 1 above is simply wrong.

Statement 2 is more problematic.  The trouble is that if the word “categories” includes large categories then the objects do not form a set even in the second universe.  You have to go to the third universe.

Now there is a way to define categories where this issue does not come up.  It allows us to think about categories without having a particular system such as ZF and universes in mind.

A syntactic definition of category

A category consists of objects and arrows, together with four methods of construction M1 – M4 satisfying laws L1 -L7.  I treat “object” and “arrow” as predicates:  object[f] means f is an object and arrow[a] means a is an arrow.  “=” means equals in the mathematical sense.

M1 Source If arrow[f], object[f.source].
M2 Target If arrow [f], object[f.target].
M3 Identity If object[a],  arrow[a.identity].
M4 Comp If arrow[g] and arrow[f] and  f.target = g.source, then arrow[(g,f).comp].
L1. If object[a],  a.identity.source = a.
L2. If object[a], a.identity.target = a.
L3. If arrow[g] and arrow[f] and  f.target = g.source, then (g,f).comp.source = f.source.
L4. If arrow[g] and arrow[f] and  f.target = g.source, then (g,f).comp.target = g.target.
L5. If object[a] and arrow[f] and f.source = a, then (f, a.identity) = f.
L6.  If object[a] and arrow[g] and g.target = a, then (a.identity, g) = g.
L7.  If arrow[h] and arrow[g] and arrow[f] and h.source= g.target and g.source = f.target, then (h,(g,f).comp = ((h,g).comp, f.comp).
Remarks on this definition
1. I have deliberately made this definition look like a specification in an object oriented program (see [6]), although the syntax is not the same as any particular oo language.  It is as rigorous a mathematical definition as you could want, and it could presumably be compiled in some oo language, except that I don’t know if oo languages allow the conditional definition of a method as given in M4.
2.  I could have given the definition in mathematical English, for example “If f is an arrow then the source of f is an object”.  My point in providing the impenetrable definition above is to make a connection (admittedly incompletely established) with a part of math (the theory of oo languages) that is definitely rigorous but is not logic.  An informal definition in math English of course could also be transformed rigorously into first order logic.
3.  This definition is exactly equivalent to the FL sketch for categories given in my post [5].  That sketch has models in many categories, not just Set, as well as its generic model living in the corresponding FL-cattheory (or in the classifying topos it generates).
4.  Saunders Mac Lane defined metacategory in precisely this way in [1].  That was of course before anyone every heard of oo languages.  I think he should have made that the definition of category.

Just-in-time foundations

Mathematicians work inside the categories Set (sets and functions) and Cat (categories and functors) all the time, including functors to or from Cat or Set. When they consider a category, the use theorems that follow from the definition above.  They do not have to have foundations in mind.

Once in awhile, they are frustrated because they cannot talk about the set of objects of some category.  For example, Freyd’s solution set condition is required to prove the existence of a left adjoint because of that problem.  The ss condition is a work-around for a familiar obstruction to an easy way to prove something.  I can imagine coming up with such a work-around without ever giving a passing thought to foundations, in particular without thinking of universes.

When you work with a mathematical object, the syntax of the definitions and theorems give you all you need to justify the claim that something is a theorem.  You absolutely need models of the theory to think up and understand proofs, but the models could be sets or classes with structure, or functors (as in sketch theory), or you may work with generic models which may require you to use intuitionistic reasoning.  You don’t have to have any particular kind of model in mind when you work in Set or Cat.

When you do run into something like the impossibility of forming the set of objects of some category (which happens in any model theory environment that uses classical rather than intuitionistic reasonins) then you may want to consider an approach through some theory of foundations.  That is what most mathematicians do: they use just-in-time foundations. For example, in a particular application you may be happy to work in a topos with a set-of-all-objects, particularly if you are a certain type of computer scientists who lives in Pittsburgh.  You may be happy to explicitly consider universes, although I am not aware of any category-theoretical results that do explicitly mention universes.

But my point is that most mathematicians think about foundations only when they need to, and most mathematicians never need to think about foundations in their work. Moral: Don’t think in terms of foundations unless you have to.

This point of view is related to the recent discussions of pragmatic foundations [7] [8].

Side remark

The situation that you can’t always construct a set of somethings is analogous to the problem that you have in working with real numbers:  You can’t name most real numbers. This may get in the way of some analyst wanting to do something, I don’t know.  But in any branch of math, there are obstructions to things you want to do that really do get in your way.  For example, in beginning linear algebra, it may have occurred to you, to your annoyance, that if you have the basis of a subspace you can extend it to the basis for the whole space, but if you have a basis of the whole space, and a subspace, the basis may not contain a basis of the subspace.

References and links

  1. Saunders Mac Lane, Categories for the working mathematician. Springer-Verlag, 1971.
  2. Wikipedia article on category theory
  3. Michael Barr and Charles Wells, Category Theory for Computing Science, Third Edition (1999). Les Publications CRM, Montreal (publication PM023).
  4. Discussion of functions in abstractmath.org.
  5. Definitions into Mathematical Objects 7.
  6. Object oriented programming in Wikipedia.
  7. M. Gelfand, We Do Not Choose Mathematics as Our Profession, It Chooses Us: Interview with Yuri Manin.
  8. Discussion in n-category cafe.
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Addenda to the 1993 Sketches paper

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I have uploaded here a version of my 1993 sketches paper with an addendum listing a few relevant papers written since then.  I have not kept up with the field well enough to contemplate a complete revision of the 1993 paper.

I recommend that more people update their papers this way.  I did it by making a new PDF file with the added references and then using Acrobat to combine it with the old paper into one file.  That way I didn’t have to re-TeX the old paper, which is a good thing, since I don’t know where some of the .sty files are.

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Composites of functions

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In my post on automatic spelling reform, I mentioned the various attempts at spelling reform that have resulted in both the old and new systems being used, which only makes things worse.  This happens in Christian denominations, too.  Someone (Martin Luther, John Wesley) tries to reform things; result: two denominations.   But a lot of the time the reform effort simply disappears.  The Chicago Tribune tried for years to get us to write “thru” and “tho” —  and failed.  Nynorsk (really a language reform rather than a spelling reform) is down to 18% of the population and the result of allowing Nynorsk forms to be used in the standard language have mostly been nil.  (See Note 1.)

In my early years as a mathematician I wrote a bunch of papers writing functions on the right (including the one mentioned in the last post).  I was inspired by some algebraists and particularly by Beck’s Thesis (available online via TAC), which I thought was exceptionally well-written.  This makes function composition read left to right and makes the pronunciation of commutative diagrams get along with notation, so when you see the diagram below you naturally write h = fg instead of h = gf. Composite

Sadly, I gave all that up before 1980 (I just looked at some of my old papers to check).  People kept complaining.  I even completely rewrote one long paper (Reference [3]) changing from right hand to left hand (just like Samoa).  I did this in Zürich when I had the gout, and I was happy to do it because it was very complicated and I had a chance to check for errors.

Well, I adapted.  I have learned to read the arrows backward (g then f in the diagram above).  Some French category theorists write the diagram backward, thus:

CompositeOp

But I was co-authoring books on category theory in those days and didn’t think people would accept it. Not to mention Mike Barr (not that he is not a people, oh, never mind).

Nevertheless, we should have gone the other way.  We should have adopted the Dvorak keyboard and Betamax, too.

Notes

[1] A lifelong Norwegian friend of ours said that when her children say “boka” instead of “boken” it sound like hillbilly talk does to Americans.  I kind of regretted this, since I grew up in north Georgia and have been a kind of hillbilly-wannabe (mostly because of the music); I don’t share that negative reaction to hillbillies.  On the other hand, you can fageddabout “ho” for “hun”.

References

[1] Charles Wells, Automorphisms of group extensions, Trans. Amer. Math. Soc, 155 (1970), 189-194.

[2] John Martino and Stewart Priddy, Group extensions and automorphism group rings. Homology, Homotopy and Applications 5 (2003), 53-70.

[3] Charles Wells, Wreath product decomposition of categories 1, Acta Sci. Math. Szeged 52 (1988), 307 – 319.

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