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One day while doing a computation on the board in front of my students,
I accidentally wrote 1 + 1 = 1. (No idea why.)


Student: Um, don't you mean 1 + 1 = 2?

Me (embarrassed): Oh right, thanks.
[Erases mistake. Pauses.]
Wait. Is there a universe in which 1 + 1 = 1?

Class: ... [Stares blankly]

Me: That's not such a strange question. Don't you know that YOU live in a world where 11 + 1 = 0?

Class:... [Jaws drop.]

Me: Yeah! Think about it! 12 = 0 and 13 = 1 and 14 = 2 and 15 = 3... sound familiar?

Class: Military time!

Me: Exactly! [Points to clock on the wall and starts impromptu lesson on modular arithmetic.]


I hope at least one student was convinced that math is cool.

'Cause I'm convinced that making mistakes isn't always a bad thing!

Limits and Colimits, Part 2 (Definitions)

Welcome back to our mini-series on categorical limits and colimits! In Part 1 we gave an intuitive answer to the question, "What are limits and colimits?" As we saw then, there are two main ways that mathematicians construct new objects from a collection of given objects: 1) take a "sub-collection," contingent on some condition or 2) "glue" things together. The first construction is usually a limit, the second is usually a colimit. Of course, this might've left the reader wondering, "Okay... but what are we taking the (co)limit of?" The answer? A diagram. And as we saw a couple of weeks ago, a diagram is really a functor.

We are now ready to give the formal definitions (along with more intuition). First, here's a bit of setup.

The Setup

In what follows, let $\mathsf{C}$ denote any category and let $\mathsf{I}$ be an indexing category. Note that given any object $X$ in $\mathsf{C}$, there is a constant functor - let's also call it $X$ - from $\mathsf{I}$ to $\mathsf{C}$. This functor sends every object to $X$ and every morphism to the identity of $X$.
Therefore, given any functor (ahem, diagram) $F:\mathsf{I}\to\mathsf{C}$, we can make sense of a natural transformation between $X$ and $F$. Such a natural transformation consists of a collection of morphisms between $X$ and the objects in the diagram $F$. Moreover, these morphisms must commute with all the morphisms that appear in diagram. If the arrows point from $X$ to the diagram $F$, then the setup is called a cone over $F$, as we previously discussed here. If, on the other hand, the arrows point from the diagram $F$ to $X$, then it's called a cone under $F$ (or sometimes a cocone).
Notice that a cone is comprised of two things: an object AND a collection of arrows to or from it. Now here's the punchline:

The limit of a diagram $F$ is a special cone over $F$.
The colimit of $F$ is a special cone under $F$.

Let's take a look at the formal definitions. I'll give a lite version first, followed by the full version.

Definitions (Lite Version)

Definition (limit): The limit of a diagram $F$ is the "shallowest" cone over $F$.

By "shallowest" (not a technical term) I mean in the sense of the picture to the right. There may be many cones -- many objects with maps pointing down to the diagram $F$ (depicted as a blob) -- over $F$, but the limit is the cone that is as close as possible to the diagram $F$. Perhaps this is why "limit" is a good choice of terminology. You might imagine all the cones over $F$ as cascading down to the limit.

If we let gravity pull all the arrows down, then we obtain the dual notion: a colimit.

Definition (colimit): The colimit of a diagram $F$ is the "shallowest" cone under $F$.

Again, by "shallowest" (not a technical term) I mean in the sense of the picture on the left. There may be many cones under $F$, but the colimit is the one that's closest to the diagram. It's the shallowest.

Okay, this is all very handwavy and not very informative. To capture the mathematics behind "shallowest," we'll use a universal property. I'll comment on intuition below.

Definition: Limit (Full Version)

The definitions themselves can be stated very succinctly. But like little onions, they have several layers, which we will peel away slowly to minimize the shedding of tears.

Definition (1): The limit of a diagram $F:\mathsf{I}\to\mathsf{C}$ the universal cone over $F$.


Let's unwind this a bit...

Definition (2): The limit of a diagram $F:\mathsf{I}\to\mathsf{C}$ is an object $\text{lim }F$ in $\mathsf{C}$ together with a natural transformation $\eta:\text{lim }F\Rightarrow F$ with the following property: for any object $X$ and for any natural transformation $\alpha\colon X\Rightarrow F$, there is a unique morphism $f\colon X\to \text{lim }F$ such that $\alpha=\eta\circ f$.

Let's unwind this a bit more...

Definition (3): The limit of a diagram $F:\mathsf{I}\to\mathsf{C}$ is an object $\text{lim }F$ in $\mathsf{C}$ together with morphisms $\eta_A:\text{lim }F\to A$, for each $A$ in the diagram, satisfying $\eta_B=\phi_{AB}\circ \eta_A$ for every morphism $\phi_{AB}\colon A\to B$ in the diagram. Morever, these maps have the following property: for any object $X$ and for any collection of morphisms $\alpha_A:X\to A$ satisfying $\alpha_B=\phi_{AB}\circ \alpha_A$, there exists a unique morphism $f\colon X\to\text{lim }F$ such that $$\alpha_A=\eta_A\circ f \qquad\text{for all objects $A$ in the diagram.}$$
three copy.jpg
In summary, for all objects $X$ in $\mathsf{C}$:

Definition: Colimit (Full Version)

Definition (1): The colimit of a diagram $F:\mathsf{I}\to\mathsf{C}$ the universal cone under $F$.


Let's unwind this a bit...

Definition (2): The colimit of a diagram $F:\mathsf{I}\to\mathsf{C}$ is an object $\text{colim }F$ in $\mathsf{C}$ together with a natural transformation $\epsilon:F\Rightarrow \text{colim }F$ with the following property: for any object $X$ and for any natural transformation $\beta\colon F\Rightarrow X$, there is a unique morphism $g\colon \text{colim }F\to F$ such that $\beta= g\circ \epsilon$.

Let's unwind this a bit more...

Definition (3): The colimit of a diagram $F:\mathsf{I}\to\mathsf{C}$ is an object $\text{colim }F$ in $\mathsf{C}$ together with morphisms $\epsilon_A:A\to\text{colim }F$, for each $A$ in the diagram, satisfying $\epsilon_A=\epsilon_B\circ \phi_{AB}$ for every morphism $\phi_{AB}\colon A\to B$ in the diagram. Morever, these maps have the following property: for any object $X$ and for any collection of morphisms $\beta_A: A\to X$ satisfying $\beta_A=\beta_B\circ \phi_{AB}$, there exists a unique morphism $g\colon \text{colim }F\to X$ such that $$\beta_A= g \circ \epsilon_A\qquad\text{for all objects $A$ in the diagram.}$$
In summary, for all objects $X$ in $\mathsf{C}$:

a little intuition  +   a little exercise

I once heard (or read?) Eugenia Cheng refer to a universal property as a way to describe a special role than an object - or in our case, a cone - plays. I like that analogy, and it's exactly what's going on with limits. (Similar sentiments hold for colimits.) Let me elaborate:

Out of all the cones over a diagram $F$, there is exactly one that plays the role of limit, namely the pair $(\text{lim}F,\eta)$. Of course you might come across another cone $(X,\alpha)$ that plays a very similar role. Perhaps $\alpha$ behaves very similarly to $\eta.$ BUT -- and this is the punchline -- this behavior is no coincidence! The natural transformation $\alpha$ "behaves" like $\eta$ because it is built up from $\eta$! More precisely, it has $\eta$ as a factor: $\alpha=\eta\circ f$ for some unique morphism $f$!

By way of analogy, think of the role that the number 2 plays among the integers. Out of all the integers, we might say that 2 is the quintessential candidate for "an integer which possesses the quality of 'two-ness,'" that is, of being even. Of course, there are other integers $a$ that play a similar role. In particular, if $a$ is an even integer, then it also possesses the quality of "two-ness." But this is no coinicidence! An even integer is even because it is built up from 2! More precisely, it has 2 as a factor: $a=2k$, for some unique integer $k$!

These two equations $a=2k$ and $\alpha=\eta\circ f$ are analogous. In fact, they're more than analogous....



Let $\mathsf{C}$ be the category $2\mathbb{Z}$ of even integers. A morphism $n\to m$ in this category is an integer $k$ such that $n=mk$. For example, 3 defines an arrow $6\overset{3}{\longrightarrow} 2$ because $6=2\times 3$. On the other hand, there is no arrow $8\longrightarrow 6$.

Show that $2$ is the limit of a particular diagram in $2\mathbb{Z}$.

That is, come up with an indexing category $\mathsf{I}$ and a functor $\mathsf{I}\to2\mathbb{Z}$ and a natural transformation $\eta$, and show that $2$ and $\eta$ satisfy the universal property above. Hint: Despite all the fancy jargon, it's really simple! (Next question: does this diagram have a colimit? If so, what is it?)
I'll close with one final thought. Once we get used to the ideas/definitions above, we discover that limits and colimits have very familiar names, depending on the shape of the indexing category $\mathsf{I}$!
In the next two posts, I'll justify some of these claims by giving explicit examples of limits and colimts in the category $\mathsf{Set}$.

Until then!

coming soon!

coming soon!



Brouwer's Fixed Point Theorem (Proof)


Today I'd like to talk about Brouwer's Fixed Point Theorem. Literally! It's the subject of this week's episode on PBS Infinite Series. Brouwer's Fixed Point Theorem is a result from topology that says no matter how you stretch, twist, morph, or deform a disc (so long as you don't tear it), there's always one point that ends up in its original location.

Brouwer's Fixed Point Theorem
Every continuous function from a disk
to itself has a fixed point.


As you can see in the video, I chose to focus on a proof of the theorem, rather than elaborating on its meaning or its applications. The mathematics behind the proof are just as fascinating as the theorem itself! This proof uses a "portal" between topology and algebra -- a fancier version of the interplay between geometry and algebra that one learns in high school mathematics. In today's blog post, I'd like to present the proof again and fill in some details that I left out of the video. In particular, I'd like to be more explicit about the "portal," which as I mention, is really a functor!

What IS a functor, exactly? It's very much like a bridge between different realms (or categories) of mathematics. I've written an introduction to functors here. Incidentally, Example #2 in that link is precisely the functor featured in the video! It's called 

the fundamental group

and is denoted $\pi_1\colon \mathsf{Top}\to\mathsf{Group}$. As I explain there, $\pi_1$ assigns a group $\pi_1(X)$ to each topological space $X$. The elements of $\pi_1(X)$ are really (homotopy classes of) maps of the circle into $X$. For now, it's enough to think of $\pi_1(X)$ as a "hole-detector," which keeps track of loops in $X$. You can find much more detail in a book on algebraic topology (such as Allen Hatcher's or Peter May's) or this friendly primer on Jeremy Kun's blog. We'll only need to consider when $X$ is a disk $D^2$ or a circle $S^1$. In that case

$\pi_1(D^2)\cong 0$     and    $\pi_1(S^1)\cong \mathbb{Z}.$

Intuitively, $\pi_1(D^2)\cong 0$ because every loop in a disc can be wiggled down to a point. Topologically, there are no fundamental loops in a disc. On the other hand, $\pi_1(S^1)\cong \mathbb{Z}$ because, well, a circle IS just a loop, which we can further loop around itself zero, one, two, three,... times clockwise (the positive integers) or counterclockwise (the negative integers). For a rigorous proof of this isormorphism, check out the books referred to above or take a look at this mini-series on Math3ma. You can also find the proof in this tweet, which was later expanded into this blog post.
In addition to assigning groups to topological spaces, $\pi_1$ also preserves the relationships between them. In other words, it assigns a group homomorphism $\pi_1(f)\colon \pi_1(X)\to \pi_1(Y)$ to every continuous function $f\colon X\to Y$ with the properties that $$\pi_1(f\circ g)=\pi_1(f)\circ\pi_1(g) \qquad\text{and}\qquad \pi_1(\text{id}_{X})=\text{id}_{\pi_1(X)}.$$ In other words, functors preserve compositions and sends the identity map of $X$ to the identity map of its fundamental group. This is known as functoriality and is the “key feature” alluded to in the video when I said
Any scenario that we can construct between a circle and a disc, must correspond to an identical scenario between the integers and the number 0. In short, whatever happens in the land of topology should be mirrored in the land of algebra.
This interplay between topology and algebra is heart of this proof of Brouwer's Fixed Point Theorem, which we are now ready to state.

The Proof

If Brouwer's Fixed Point Theorem is not true, then there is a continuous function $g\colon D^2\to D^2$ so that $x\neq g(x)$ for all $x\in D^2$. This allows us to construct a function $h$ from $D^2$ to its boundary $S^1$ by drawing a ray from $g(x)$ to $x$. This ray intersects with a point, which we can label $h(x)$, on the boundary circle of the disc. Th assignment $x\mapsto h(x)$ is continuous: a small change in $x$ corresponds to a small change in $h(x)$. Moreover, $h(x)=x$ for every $x\in S^1$.
Let $i\colon S^1\to D^2$ denote the inclusion of the circle into the disc. That is, $i(x)=x$ for all $x\in S^1$. Then $h(i(x))=x$ for all $x\in S^1$ and so $$h\circ i = \text{id}_{S^1}.$$
Applying $\pi_1$, we obtain a composition of group homomorphisms $$\pi_1(h)\circ\pi_1(i)=\text{id}_{\mathbb{Z}}.$$ (Here is where we're using functoriality: $\pi_1(h\circ i) = \pi_1(h)\circ\pi_1(i)$ and $\pi_1(\text{id}_{S^1}) = \text{id}_{\pi_1(S^1)}=\text{id}_\mathbb{Z}$.) Notice that $\pi_1(i)\colon \mathbb{Z}\to 0$ is the constant map at 0; it assigns every integer to 0. On the other hand, $\pi_1(h)$ is injective: it assigns 0 to exactly one integer (0, in fact). YET their composition must assign each integer to itself! That is, the composition $\pi_1(h)\circ\pi_1(i)$ must be equal to $\text{id}_\mathbb{Z}$, the identity on $\mathbb{Z}$.

But this is impossible!
Conclusion? The function $h$ cannot exist, and so there must be at least one point $x\in D^2$ so that $g(x)=x$. In other words, Brouwer's Fixed Point Theorem is true.

The Takeaway

The takeaway is that

we cannot map the set of all integers to zero and then reverse the process.

We don't even need group theory to see this. It follows from the definition of a function! And thanks to the correspondence between topology and algebra, namely:

#1) a disc is the topologist's version of the number zero, and
#2) the boundary of a disk (i.e. a circle) is the topologist's version of the set of integers,

we can obtain an analogous, topological statement:

we cannot map the boundary of a disc to the disc and then reverse the process,

at least, not without tearing the disc (i.e. without violating continuity). If however Brouwer's Fixed Point Theorem is not true, then we CAN perform such a reversal using the map, h.

That is the contradiction.

And this (as my advisor likes to say) is algebraic topology par excellence.

A Diagram is a Functor


Last week was the start of a mini-series on limits and colimits in category theory. We began by answering a few basic questions, including, "What ARE (co)limits?" In short, they are a way to construct new mathematical objects from old ones. For more on this non-technical answer, be sure to check out Limits and Colimits, Part 1. Towards the end of that post, I mentioned that (co)limits aren't really related to limits of sequences in topology and analysis (but see here). There is, however, one similarity. In analysis, we ask for the limit of a sequence. In category theory, we also ask for the (co)limit OF something. But if that "something" is not a sequence, then what is it?

Answer: a diagram.

We've talked about diagrams before: for a quick refresher, check out this post. Today I'd like to give you a different way to think about diagrams - namely, as functors!  In other words, I hope to convince you that

a diagram is a functor.

Once we adopt this viewpoint, we'll be ready to look at the formal definition of limits and colimits. Now, how can we view diagrams as functors? Suppose $F:\mathsf{I}\to\mathsf{C}$ is a functor between categories $\mathsf{I}$ and $\mathsf{C}$. We'll call $\mathsf{I}$ an indexing category, and for the sake of illustration let's suppose it's a simple one:
I've labeled the objects in $\mathsf{I}$ with colors and there is an identity arrow for each object, though I haven't drawn them. Let's also suppose that the horizontal arrow arrow is the composition of the two diagonal arrows.

So what's a functor $F$ out of this category?

It's simply a choice of three objects and three arrows in $\mathsf{C}$.
Here $F($$\bullet$$)=A$ and $F($$\bullet$$)=B$ and $F($$\bullet$$)=C$, and the image of the three arrows in $\mathsf{I}$ are the arrows $f,g$ and $h$ in $\mathsf{C}$ where $f=h\circ g$. So you see? That's all there is to it! The image of $F$ is no more and no less than a diagram in $\mathsf{C}$. We might even call it an "$\mathsf{I}$-shaped diagram" since different shapes for $\mathsf{I}$ lend to different shapes of diagrams. For example,

In short, a diagram is a functor.



By the way...

This idea of identifying a map with its image is nothing new. After all, a sequence of real numbers is technically a function $x:\mathbb{N}\to\mathbb{R}$, though we usually write $x_n$ for the image $x(n)$ and think of the sequence as the collection $\{x_n\}_{n\in\mathbb{N}}$ rather than the function $x$ itself.

Likewise the formal definition of a path in a topological space $X$ is: "a continuous function from the closed unit interval into $X$," i.e. $p:[0,1]\to X$. But when we think about paths, we often have the image $p(I)\subset X$ of $p$ in mind.
And in differential geometry, a vector field on a differentiable manifold $M$ is a section of the tangent bundle, i.e. a map $\phi$ from $M$ into its tangent bundle $TM$ such that the composition of $\phi$ with the projection $TM\to M$ is the identity on $M$. Of course that was a mouthful, and so we often just think of a vector field as a collection of tangent vectors - one attached to each point on the manifold. That is, we identify $\phi$ with its image.

These examples are all similar to the statement, "a diagram is a functor."


Back to (co)limits...

Now that we can view diagrams as functors, we can make sense of maps between diagramas, i.e. natural transformations between functors. As we'll see next time, the (co)limit of a diagram $F$ is a particular natural transformation between $F$ and another diagram of a particular shape. What's neat is that if $F$ is shaped like one of those diagrams drawn in the table above, then the (co)limit is given a familiar name, like intersection, union, Cartesian product, kernel, direct sum, and quotient!

We'll explore all the details in the coming weeks.

Limits and Colimits, Part 1 (Introduction)

Limits and Colimits, Part 1 (Introduction)

I'd like to embark on yet another mini-series here on the blog. The topic this time? Limits and colimits in category theory! But even if you're not familiar with category theory, I do hope you'll keep reading. Today's post is just an informal, non-technical introduction. And regardless of your categorical background, you've certainly come across many examples of limits and colimits, perhaps without knowing it! They appear everywhere - in topology, set theory, group theory, ring theory, linear algebra, differential geometry, number theory, algebraic geometry. The list goes on. But before diving in, I'd like to start off by answering a few basic questions.

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Topology vs. "A Topology" (cont.)

Topology vs. "A Topology" (cont.)

This blog post is a continuation of today's episode on PBS Infinite Series, "Topology vs. 'a' Topology." My hope is that this episode and post will be helpful to anyone who's heard of topology and thought, "Hey! This sounds cool!" then picked up a book (or asked Google) to learn more, only to find those formidable three axioms of 'a topology' that, admittedly, do not sound cool.

But it turns out those axioms are what's "under the hood" of the whole topological business! So without further ado, let's pick up where we left off in the video.

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Multiplying Non-Numbers

Multiplying Non-Numbers

In last last week's episode of PBS Infinite Series, we talked about different flavors of multiplication (like associativity and commutativity) to think about when multiplying things that aren't numbers. My examples of multiplying non-numbers were vectors and matrices, which come from the land of algebra. Today I'd like to highlight another example:

We can multiply shapes!

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What is an Operad? Part 2

What is an Operad? Part 2

Last week we introduced the definition of an operad: it's a sequence of sets or vector spaces or topological spaces or most anything you like (whose elements we think of as abstract operations), together with composition maps and a way to permute the inputs using symmetric groups. We also defined an algebra over an operad, which a way to realize each abstract operation as an actualoperation. Now it's time for some examples!

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What is an Operad? Part 1

What is an Operad? Part 1

If you browse through the research of your local algebraist, homotopy theorist, algebraic topologist or―well, anyone whose research involves an operation of some type, you might come across the word "operad." But what are operads? And what are they good for? Loosely speaking, operads―which come in a wide variety of types―keep track of various "flavors" of operations.

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The Yoneda Embedding

The Yoneda Embedding

Last week we began a discussion about the Yoneda lemma. Though rather than stating the lemma (sans motivation)we took a leisurely stroll through an implication of its corollaries - the Yoneda perspective, as we called it: An object is completely determined by its relationships to other objects, i.e. by what the object "looks like" from the vantage point of each object
in the category.

But this left us wondering, What are the mathematics behind this idea? And what are the actual corollaries? In this post, we'll work to discover the answers.

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