crumbs!

 

Physicist Freeman Dyson once observed that there are two types of mathematicians: birds -- those who fly high, enjoy the big picture, and look for unifying concepts -- and frogs -- those who dwell on the ground, find beauty in the scenery close by, and enjoy the details.

Of course, both vantage points are essential to mathematical progress, and
I often tend to think of myself as more of a bird.
(I'm, uh, bird-brained?)

Well one day I complained to my advisor, John Terilla -- you can see him doing cool math in that video above! -- about one of my classes which at the time felt a bit froggy:

 

Me: I'm having a hard time enjoying this class. I don't want to do a bunch of detailed computations on these objects. I'd rather spend my time learning category theory. I want to soar high and study the full landscape!

John: Well, yes, the sky is nice. But the ground -- and the ocean -- is nice, too. In fact, the best place to be is where the sky and the ocean meet, where the tips of the waves turn white.

That's the only place where you can go surfing.

That's where you want to be.


 

Snap, y'all.

I think I've a new respect for the little details.

Let's go surfing!

 

Group Elements, Categorically

On Monday we concluded our mini-series on basic category theory with a discussion on natural transformations and functors. This led us to make the simple observation that the elements of any set are really just functions from the single-point set {✳︎} to that set. But what if we replace "set" by "group"? Can we view group elements categorically as well?

The answer to that question is the topic for today's post, written by guest-author Arthur Parzygnat. Arthur is a mathematics postdoctoral fellow at the University of Connecticut, and, incidentally, was the first person to introduce me to categories as an undergraduate!

 

 

An element $x$ of a set $X$ can equivalently be described in terms of a function $x:\{*\}\to X$ from any one element set into $X.$ Similarly, a point $x$ in a topological space $(X,\tau)$ can be described in terms of a function $x:\{*\}\to(X,\tau)$ from the single point space into $(X,\tau).$ This same idea works in several categories of mathematical objects.

Definition 1. Let $\mathscr{C}$ be a category with a terminal object $T$ and let $X$ be an object of $\mathscr{C}.$ A point of $X$ is a morphism $x:T\to X.$

In many examples, such as the ones mentioned above, this produces the usual notion of a point. However, it fails with categories whose objects have additional algebraic structure and the morphisms respect this algebraic structure. For instance, in the category of vector spaces and linear transformations, the terminal object is a $0$-dimensional vector space, which we'll denote by $\mathbf{0}.$ If $V$ is a real vector space, there is only a single linear transformation $\mathbf{0}\to V$ since the zero vector must be preserved. On the other hand, linear transformations of the form $\mathbb{R}\to V$ do describe elements of $V$ (the image of the number 1). Furthermore, $\mathbb{R}$ is the monoidal unit for the tensor product of vector spaces.

Definition 2. Let $(\mathscr{C},\otimes,I)$ be a monoidal category (the other data are not explicitly written here) with monoidal unit $I.$ Let $X$ be an object of $\mathscr{C}.$ A point of $X$ is a morphism $x:I\to X.$

With this definition and an appropriate choice of monoidal structure, all the examples from above describe elements in the usual sense. Unfortunately, this still does not describe elements in all categories with algebraic structure. For example, in the category of groups, the terminal object and the monoidal unit for the direct product of groups are both the group $\{e\}$ with a single element, the identity. For any group $G,$ there is only a single morphism $\{e\}\to G$ sending the identity to the identity. Although group elements in $G$ can be described by a morphism $\mathbb{Z}\to G,$ $\mathbb{Z}$ is not a terminal object, nor is it a unit for the monoidal structure. There are a few possible ways to proceed.

  1. Find an appropriate monoidal structure on groups where $\mathbb{Z}$ plays the role of a unit.
  2. Generalize our definition of a point even further.
  3. Change the way we think about the totality of groups.

I have no idea how to do the first and if it's even possible. One possibility for the second option is known as the ``functor of points'' perspective, though we prefer another option. The third option is not only possible, but it offers an interesting perspective to how group elements are different from other types of elements. First note that every group can be viewed as a category with a single object. Normally, we think of the totality of groups as an ordinary category. However, viewing a group as a one object category, the totality of groups becomes a 2-category since the totality of categories has a canonical structure of a 2-category. Morphisms are functors, and the data of such a functor is equivalent to the data of a group homomorphism. A natural transformation

 
 
consists of a single element $h\in H$ satisfying $$ h\varphi(g)=\psi(g)h\qquad\forall\;g\in G. $$ In other words, the homomorphisms $\psi$ and $\varphi$ are conjugate by some element $h\in H.$ Composition of natural transformations corresponds to group multiplication.
 
 
In particular, all natural transformations between groups are invertible. Hence, the set of all natural transformations from a group homomorphism to itself forms a group. (Can you guess which subgroup it is?) Of course, unless you've specifically chosen your group homomorphisms, it's probably not likely that there exists a natural transformation between them. But, in some special cases, there are many such natural transformations. In fact, the group of all natural transformations
 
 

is canonically isomorphic to $G$ itself. Here $!:\{e\}\to G$ is the unique group homomorphism from the single element group to $G.$ Therefore, group elements are more appropriately associated with ``processes'' instead of static ``elements,'' which is appropriate anyway because we think of groups as symmetries of other mathematical objects. More precisely, we can make the following definition.

Definition 3. Let $\mathscr{C}$ be a 2-category with a terminal object $T.$ Let $C$ be an object of $\mathscr{C}$ and let $X:T\to C$ and $Y:T\to C$ be elements in $C.$ A process $f$ from $X$ to $Y$ in $C$ is a 2-morphism in $\mathscr{C}$ of the form

 
 

Example 1

Let $\mathscr{C}$ be the 2-category of groups viewed as one-object categories. Then $T=\{e\}$ is a single element group. Set $C:=G$ to be any group $G.$ Set $X:=\;!$ and $Y:=\;!$ to be the unique group homomorphisms from $\{e\}$ to $G.$ Finally, set $f:=g$ to be any element $g$ of $G.$ This shows that a group element is an example of a process.

Example 2

Let $\mathscr{C}$ be the 2-category of categories. Then $T$ is a 2-category with a single object, a single 1-morphism, and a single 2-morphism. Let $C$ be a category (such as the category of sets). Let $X$ and $Y$ be objects of $C$ (such as two sets). Let $f:X\to Y$ be a morphism in $C$ (a function in the case of sets). Then this is an example of a process, consistent with our usual notion of a process as a morphism between two objects in a category.
If we wanted to, we could also replace the 2-category $\mathscr{C}$ and terminal object $T$ in the definition of a process by a monoidal 2-category with a unit in a similar fashion to what was done to include vector spaces in the discussion of elements. I encourage you to come up with other examples in this case.

What is a Natural Transformation? Definition and Examples, Part 2

Continuing our list of examples of natural transformations, here is...

Example #2: double dual space

This is really the archetypical example of a natural transformation. You'll recall (or let's observe) that every finite dimensional vector space $V$ over a field $\mathbb{k}$ is isomorphic to both its dual space $V^*$ and to its double dual $V^{**}$.
In the first case, if $\{v_1,\ldots,v_n\}$ is a basis for $V$, then $\{v_1^*,\ldots,v_n^*\}$ is a basis for $V^*$ where for each $i$, the map $v_i^*:V\to\mathbb{k}$ is given by $$v_i^*(v_j)= \begin{cases} 1, &\text{if $i=j$};\\ 0 &\text{if $i\neq j$}. \end{cases}$$ Unfortunately, this isomorphism $V\overset{\cong}{\longrightarrow} V^*$ is not canonical. That is, a different choice of basis yields a different isomorphism. What's more, the isomorphism can't even materialize until we pick a basis.*

On the other hand, there is an isomorphism $V\overset{\cong}{\longrightarrow}V^{**}$ that requires no choice of basis: for each $v\in V$, let $\text{eval}_v:V^*\to\mathbb{k}$ be the evaluation map. That is, whenever $f:V\to \mathbb{k}$ is an element in $V^*$, define $\text{eval}_v(f):=f(v)$. Folks often refer to this isomorphism as natural. It's natural in the sense that it's there for the taking---it's patiently waiting to be acknowledged, irrespective of how we choose to "view" $V$ (i.e. irrespective of our choice of basis). This is evidenced in the fact that $\text{eval}$ does the same job on each vector space throughout entire category. One map to rule them all.**

For this reason, the totality of all the evaluation maps assembles into a natural transformation (a natural isomorphism, in fact) between two functors!

To see this, let $(-)^{**}:\mathsf{Vect}_{\mathbb{k}}\to\mathsf{Vect}_{\mathbb{k}}$ be the the double dual functor $(-)^{**}$ that sends a vector space $V$ to $V^{**}$ and that sends a linear map $V\overset{\phi}{\longrightarrow}W$ to $V^{**}\overset{\phi^{**}}{\longrightarrow} W^{**}$, where $\phi^{**}$ is precomposition with $\phi^{*}$ (which we've defined before). And let $\text{id}:\mathsf{Vect}_{\mathbb{k}}\to\mathsf{Vect}_{\mathbb{k}} $ be the identity functor.

Now let's check that $\text{eval}:\text{id}\Longrightarrow (-)^{**}$ is indeed a natural transformation.
 
 
By picking a $v\in V$ and chasing it around the diagram below, notice that the square commutes if and only if $\text{eval}_v\circ \phi^*=\text{eval}_{\phi(v)}$. (Here I'm using the fact that $\phi^{**}(\text{eval}_v)=\text{eval}_v\circ \phi^*$.)
 
 
Does this equality hold? Let's check! Suppose $f:W^{*}\to\mathbb{k}$ is an element of $W^{**}$. Then $$ \begin{align*} \text{eval}_v(\phi^*(f))&=\text{eval}_v(f\circ \phi)\\ &=(f\circ\phi)(v)\\ &= f(\phi(v))\\ &=\text{eval}_{\phi(v)}(f). \end{align*} $$ Voila! And because each $V\overset{\text{eval}}{\longrightarrow} V^{**}$ is an isomorphism, we've got ourselves a natural ismorphism $\text{id}\Longrightarrow (-)^{**}$.

As per our discussion last time, this suggests that $\text{id}$ and $(-)^{**}$ are really the same functor up to a change in perspective. Indeed, this interpretation pairs nicely with the observation that any vector $v\in V$ can either be viewed as, well, a vector, or it can be viewed as an assignment that sends a linear function $f$ to the value $f(v)$. In short, $V$ is genuinely and authentically just like its double dual.

They are - quite naturally - isomorphic.

Example #3: representability and Yoneda

In our earlier discussion on functors we noted that a functor $F:\mathsf{C}\to\mathsf{Set}$ is representable if, loosely speaking, there is an object $c\in\mathsf{C}$ so that for all objects $x$ in $\mathsf{C}$, the elements of $F(x)$ are "really" just maps $c\to x$ (or maps $x\to c$, if $F$ is contravariant). As an illustration, we noted that the functor $\mathscr{O}:\mathsf{Top}^{op}\to\mathsf{Set}$ that sends a topological space $X$ to its set $\mathscr{O}(X)$ of open subsets is represented by the Sierpinski space $S$ since $$\mathscr{O}(X)\cong \text{hom}_{\mathsf{Top}}(X,S)$$ where I'm using $\cong$ to denote a set bijection/isomorphism. So in other words, an open subset of $X$ is essentially the same thing as a continuous function $X\to S.$ (We discussed this in length here.)
Now it turns out that this $\cong$ is not just a typical, plain-vanilla isomorphism. It's natural! That is, the ensemble of isomorphisms $\mathscr{O}(X)\overset{\cong}{\longrightarrow}\text{hom}_{\mathsf{Top}}(X,S)$ (one for each $X$) assemble to form a natural isomorphism between the two functors $\mathscr{O}$ and $\text{hom}_{\mathsf{Top}}(-,S)$.***
In general, then, we say a functor $F:\mathsf{C}\to\mathsf{Set}$ is representable if there is an object $c\in\mathsf{C}$ so that $F$ is naturally isomorphic to the hom functor $\text{hom}_{\mathsf{C}}(c,-)$, i.e. if $$F(x)\cong\text{hom}_{\mathsf{C}}(c,x) \qquad \text{naturally, for all $x\in \mathsf{C}$}$$ (or if $F$ is contravariant, $F(x)\cong\text{hom}_{\mathsf{C}}(x,c)$).
 
 

Here's a very simple example. Suppose $A$ is any set and let $*$ denote the set with one element. Notice that a function from $*$ to $A$ has exactly one element in its image, i.e. the range of $*\to A$ is $\{a\}$ for some $a\in A$. This suggests that a map $*\to A$ is really just a choice of element in $A$! Intuitively then, the elements of $A$ are in bijection with functions $*\to A$, $$A\cong\text{hom}_{\mathsf{Set}}(*,A).$$ But more is true! The isomorphism $A\to \text{hom}_{\mathsf{Set}}(*,A)$ which sends $a\in A$ to the function, say, $\bar{a}:*\to A$, where $\bar{a}(*)=a$, is natural. That is, for any $A\overset{f}{\longrightarrow}B$, the following square commutes

 
 
Commutativity just says that given an element $a\in A$, we can think of the element $f(a)$ as a map $*\to B$ in one of two equivalent ways: either send $a$ to $f(a)$ via $f$ and then think of $f(a)$ as a map $*\to B$. OR first think of $a$ as a map $*\to A$, and then postcompose it with $f$.

In short, the identity functor $\text{id}:\mathsf{Set}\to\mathsf{Set}$ is represented by the one-point set $*$ since every function $*\to A$ is really just a choice of an element $a\in A$.

 

 
Representability is really the launching point for the Yoneda Lemma which is "arguably the most important result in category theory." We'll certainly chat about Yoneda in a future post.

To whet your appetite, I'll quickly say that one consequence of the Lemma is that we are prompted to think of an object $x$ -- no longer as an object, but now -- as a (representable) functor $\text{hom}(x,-)$, similar to how we may think of a point $a\in A$ as a map $*\to A$.

 
 
This perspective -- coupled with the idea that morphisms out of $x$ (i.e. the elements of $\text{hom}(x,-)$) are simply "the relationships of $x$ with other objects" -- motivates the categorical mantra that an object is completely determined by its relationships to other objects. As a wise person once said, "You tell me who your friends are, and I'll tell you who YOU are." The upshot is that this proverb holds in life as well as in category theory.

And that is The Most Obvious Secret of Mathematics!

 

 

*One can show that there is no "natural" isomorphism of a vector space with its dual. For instance, see p. 234 of Eilenberg and Mac Lane's 1945 paper, "The General Theory of Natural Equivalences."

** This sort of reminds me of the difference between pointwise and uniform convergence. A sequences of functions $\{f_n:X\to\mathbb{R}\}$ converges to a function $f$ pointwise if, from the vantage point of some $x\in X$, the $f_n$ are eventually within some $\epsilon$ of $f$. But that value of $\epsilon$ might be different at a different vantage point, i.e. at a different $x'\in X$. On the other hand, the sequence converges uniformly if there's an $\epsilon$ that does the job no matter where you stand, i.e. for all $x\in X$.

***Here, $\text{hom}_{\mathsf{Top}}(-,S):\mathsf{Top}^{op}\to\mathsf{Set}$ is the contravariant functor that sends a topological space $X$ to the set $\text{hom}_{\mathsf{Top}}(X,S)$ of continuous functions $X\to S$ and that sends a continuous function $X\overset{f}{\longrightarrow} Y$ to its pullback $\text{hom}_{\mathsf{Top}}(Y,S)\overset{f^*}{\longrightarrow}\text{hom}_{\mathsf{Top}}(X,S).$

What is a Natural Transformation? Definition and Examples

What is a Natural Transformation? Definition and Examples

I hope you have enjoyed our little series on basic category theory. (I know I have!) This week we'll close out by chatting about natural transformations which are, in short, a nice way of moving from one functor to another. If you're new to this mini-series, be sure to check out the very first post, What is Category Theory Anyway? as well as What is a Category? and last week's What is a Functor?

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What is a Functor? Definition and Examples, Part 1

What is a Functor? Definition and Examples, Part 1

Next up in our mini series on basic category theory: functors! We began this series by asking What is category theory, anyway? and last week walked through the precise definition of a category along with some examples. As we saw in example #3 in that post, a functor can be viewed an arrow/morphism between two categories.

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Introducing... crumbs!

Introducing... crumbs!

Hello friends! I've decided to launch a new series on the blog called crumbs! Every now and then, I'd like to share little stories -- crumbs, if you will -- from behind the scenes of Math3ma. To start us off, I posted (a slightly modified version of) the story below on January 23 on Facebook/Twitter/Instagram, so you may have seen this one already. Even so, I thought it'd be a good fit for the blog as well. I have a few more of these quick, soft-topic blurbs that I plan to share throughout the year. So stay tuned! I do hope you'll enjoy this newest addition to Math3ma.

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What is a Category? Definition and Examples

What is a Category? Definition and Examples

As promised, here is the first in our triad of posts on basic category theory definitions: categories, functors, and natural transformations. If you're just now tuning in and are wondering what is category theory, anyway? be sure to follow the link to find out!

A category 𝖢 consists of some data that satisfy certain properties...

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What is Category Theory Anyway?

What is Category Theory Anyway?

A quick browse through my Twitter or Instagram accounts, and you might guess that I've had category theory on my mind. You'd be right, too! So I have a few category-theory themed posts lined up for this semester, and to start off, I'd like to (attempt to) answer the question, What is category theory, anyway? for anyone who may not be familiar with the subject.

Now rather than give you a list of definitions--which are easy enough to find and may feel a bit unmotivated at first--I thought it would be nice to tell you what category theory is in the grand scheme of (mathematical) things. You see, it's very different than other branches of math....

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#TrustYourStruggle

#TrustYourStruggle

If you've been following this blog for a while, you'll know that I have strong opinions about the misconception that "math is only for the gifted." I've written about the importance of endurance and hard work several times. Naturally, these convictions carried over into my own classroom this past semester as I taught a group of college algebra students.

Whether they raised their hand during a lecture and gave a "wrong" answer, received a less-than-perfect score on an exam or quiz, or felt completely confused during a lesson, I tried to emphasize that things aren't always as bad as they seem...

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A Quotient of the General Linear Group, Intuitively

A Quotient of the General Linear Group, Intuitively

Over the past few weeks, we've been chatting about quotient groups in hopes of answering the question, "What's a quotient group, really?" In short, we noted that the quotient of a group G by a normal subgroup N is a means of organizing the group elements according to how they fail---or don't fail---to satisfy the property required to belong to N. The key point was that there's only one way to belong to N, but generally there may be several ways to fail to belong.

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