Eq

What is Mathematics if not defining new things that are equal to something alse but with some added structure. Seeing how it has inadvertently imbued itself into the very nature of Mathematics, I want to take a moment to reflect on it.

What does “being the same” mean?

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This goes way beyond Mathematics, and we’ll see why at the end of this essay. The matter at hand is that there are certain things that we say are equal and other things that aren’t. Sheafification of G viewers: avert your eyes for I am about to assume the law of the excluded middle. We can describe this as a property a mathematical object can hold which can inhabit two values: True and False. From this, lets observe set equality. We say that two sets $A$ and $B$ are equal ($A = B$) when $A \subseteq B$ and $B \subseteq A$. This is very nice but it’s not an exactly satisfactory response. Subsets talk about the elements of $A$ and the elements of $B$, and whether they are subsets of each other just moves the question of “is $A$ equal to $B$” to a question of “are the elements of $A$ equal to the elements of $B$”, roughly speaking. We have a more elegant solution to this: let’s talk about set cardinality. We know that whenever $|A| = |B|$ we can find two functions to relabel all of the elements of $A$ to elements of $B$ without losing any information, back and forth. This equality of cardinalities avoids shifting the problem, we fixed the equality to be this: cardinality.

A friend said to me a really long time ago while we were taking Calculus I:

A set is just its cardinality.

This is what he was talking about.

Now, how does this look categorically? Well, $|A| = |B| \iff A \cong B$ or in other words their cardinalities are equal if there is a bijection in $\mathbf{Set}$ between $A$ and $B$. So it seems that, instead of talking about regular old sets $A$ and $B$, we could just talk about the cardinality of $A$ or $B$ and simply label the elements however we want. This is after all exactly what bijective functions are for.

I want to further expand on this idea about equality by looking at one of the most important results in Category Theory: The Yoneda Lemma.

Let $\mathcal{C}$ be a category¹, and let $F : \mathcal{C}^{\text{op}} \to \mathbf{Set}$ be a functor. Then, for every object $a \in \mathcal{C}$, there is a natural isomorphism²

$$\text{Nat}\left( \text{hom}_{\mathcal{C}}(-, a), F \right) \cong F(a).$$

Now, what the f**k does that mean Kobe Bryant? For now, let’s just take for granted that this is true (maybe in the future I’ll make a blog post explaining the intuition behind this and proving it). I’ll only be using the fact that this is true to get back to equality. Take an object $x \in \mathcal{C}$¹ and consider the hom-functor $\hom_\mathcal{C} (-, x)$. Well, that hom-functor is of the form $\hom_\mathcal{C}(-, x) : \mathcal{C}^{\text{op}} \to \mathbf{Set}$. So, what would happen if you were to apply the Yoneda Lemma to this? Well you get that for every object $a \in \mathcal{C}^{\text{op}}$ there is a natural isomorphism² of the form

$$\text{Nat}\left(\hom_\mathcal{C}(-, a), \hom_\mathcal{C}(-, x)\right) \cong \hom_\mathcal{C}(a, x).$$

That last expression is the set of morphisms from $a$ to $x$. But look at it closer, what would happen if $a \cong x$? You get that $\hom_\mathcal{C}(-, a) \cong \hom_\mathcal{C}(-, x)$! In other words, isomorphism in the category gets translated into an isomorphism between the morphisms going to $a$ and the morphisms going to $b$. This, I think, is one of the reasons why this is the most impressive results in Category Theory.

Instead of looking at the objects $a$ and $x$ you can instead look at the morphisms going into $a$ or the morphisms going into $x$ and the notion will be the same. This idea isn’t too strange; let’s see it from the perspective of an object $a$. The morphisms going into $a$ from another object $x$ will all look like a slightly blurry version of $x$. But then, do this for each painting coming from every other object in the category. Just looking at these, you’ll get a really good idea of how it feels like to be $a$. Now let’s try to answer the question “what is missing to feel entirely like $a$?” by assuming that there is some piece of information relevant to $a$ that somehow is not captured by these paintings. Then, necessarily because these paintings cannot capture that piece of information, then no other object in the category can know that $a$ has this piece of information. Since no other object in the category knows about this either, then it is also fine if we don’t know about this piece of information. This is lacking a lot of rigor, but this is my blog and not a white paper; on a white paper you can expect to see this discussed with proofs and rigor backing it up.

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By now, it must be clear that instead of looking at objects, we can just look at the morphisms going to an object. What other intuition can we extract from this? Well, if we can forget about the objects in the category, then the entire structure in a category is given by the relationships between the morphisms. In a further blogpost I will talk about relationships between morphisms, but for now I would like to make an emphasis on the fact that:

The category’s structure is exactly the morphisms.

Now let’s do something interesting and get rid of the structure. First, we originally got here by looking at equality, so why not do that again? What if every object in the category was equal? Well, this is a silly way of phrasing that, but we have two options (because of the law of the excluded middle):

• “every object” is really just a single object because it’s all “equal” and therefore the same object
• there aren’t any objects and therefore “every object” is actually talking about nothing and therefore the property is vacuously met

In the field of Category Theory, we call this a $(-1)$-groupoid. A groupoid is a category where all morphisms are isomorphisms. The $-1$ has a sort of complicated history, its name is historical and not indicative of some kind of intuition to be had with “negative one.” The important part here is that we got rid of all structure and ended up with two things: an empty groupoid and a groupoid with 1 object. I wonder where I’ve seen two things in close relation that are isomorphic and one represents “inhabitation” of something and the other represents the “lack of inhabitation” of that thing?

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I’ll drop the sarcasm now. The fact a $(-1)$-groupoid being defined like this suddenly hints at True and False is absolutely beautiful to me. We got structure by looking at what the category’s equality (isomorphism) would force on the objects. We get equality and inequality by getting rid of the structure. The Yoneda Lemma seems to strongly agree with all of this. The intuition I want readers of this essay to leave with is this:

Structure, equality and inequality are all inextricably intertwined with each other.