This is a thing I do. I like to learn things by finding the thing I want to learn, and then work my way down as I encounter unknowns, exposing layers like an epistemological onion — a truth vegetable.
The current outermost layer is Homotopy Type Theory which is put forth by the HoTT Book. Per usual, as I dig down beneath the tunic I’m reminded that, for all intents and purposes, I know nothing and I spiral out of control wiki link after wiki link, my only respite a term I actually understand. It could be argued that this is called “learning”, but that label is dubious.
Digressions aside, Homotopy is, in my best attempt at lucidity, a continuous function that can map between two functions, say \(f,g: A \rightarrow B\) mapping from the same space to another same space. If this function exists, and we’ll call it \(H\), then those two maps are homotopic. Formally, it looks like this:
\[\ f, g: A \rightarrow B \\ H: A \times [0, 1] \rightarrow B \\ H(x, 0) = f(x) \\ H(x, 1) = g(x) \]
To put this in English: We’ve convered the first line. That’s our two maps: \(f\) and \(g\). The second line is illustrating the what \(H\) maps: the product of \(X\) and the unit interval to \(B\). A product in this case means and. As a quick explanation of product in this context, we can refer to some Haskell:
-- This is a sum type, i.e. an or. Meaning a Maybe' can be either a Just' a or -- Nothing. data Maybe' a = Just' a | Nothing' -- While this is a product, an and. The tuple is the product of something of -- type a and something of type b. data Tuple' a b = (a, b)
The unit interval however, is another scale to the onion.
The unit interval is the set of all the real numbers between \(0\) and \(1\). We use the bounds of this interval to be equal to our two maps given some \(x\). This means that our \(H\), our homotopy, is a function which through a transfinite series of inputs can “bend” \(f\) into \(g\).
That, in a nutshell is homotopy. But, as I delved deeper, like anything worth learning, that nascent intuition was simply a prerequisite.
Univalence Axiom: \((A = B) \approx (A \approx B)\).
The book says this about Univalence:
Thinking of types as spaces, […] the points of which are spaces[. T]o understand its identity type, we must ask, what is a path \(p : A \leadsto B\) between spaces in \(U\)? The univalence axiom says that such paths correspond to homotopy equivalences \(A \approx B\)[.] (Univalent Foundations of Mathematics, 4)
It goes on to say that Univalence is informally an equivalence, and more specifcally homotopy equivalence. Homotopy equivalence is given by an isomorphism between two spaces where the following holds:
Given spaces \(X\) and \(Y\) and maps \(f: X \rightarrow; Y\) and \(g: Y \rightarrow X\), the composition \(f \circ g\) is homotopic to \(Id_X\) and the composition \(g \circ f\) is homotopic to \(Id_Y\).
To put this is another way, and how I actually wrote this down in my notebook when I was searching for the intuition: Univalence is a topological bijection and that bijection is a valid isomorphism in the domain of pure topology. There’s a inside joke about topologists: Toplogists can’t tell the difference between a donut and a coffee mug. This is precisely why! The torus (donut) is homotopy equivalent, i.e. isomorphic, to a coffee mug.
So, when we talk about univalence in the context of Homotopy, we’re talking about a form of equality, an isomorphism.
This concludes the initial desiderata for grappling with Homotopy Type Theory, or at least its introduction. There’s more to come, for the both of us.