Wherein I start with section 1. from the FAQ [here](https://ncatlab.org/nlab/show/homotopy+type+theory+FAQ) and try to improve it using different LLM models. Let's start with the orignal. The prompt I used was just "Please Re-write this so it’s more accessible to non-mathematicians." then the paragraph verbatim. ## Original Homotopy type theory is a theory of objects, which we call terms, collections of objects, which we call types, and equalities between any two objects in a collection, which we call identity types. One fundamental aspect of mathematics is that equalities between different collections of objects behave differently from each other. If we have a type where for all terms have exactly one identity between them, this is called a proposition or a truth value. The type with no terms is trivially a proposition, as every single one of the nonexistent terms have exactly one identity between them. However, not all types are propositions: take the type with two distinct terms a and b, there is no identity between a and b. The identity types between two terms here are propositions, and the types are called sets. Still, not all types are sets: take the type of sets with two terms, the first one with terms a and b, and the second one with terms c and d. There are two ways to say that these types are equal: by adding an identity between a and c and between b and d, or by adding an identity between a and d and between b and c. Thus, the identity types between the two sets is set-valued, rather than proposition-valued, and the type is a groupoid, rather than a set. In this way, we could say that, for example, the set of positive integers is equal to the set of negative integers, or that the set of complex numbers is equal to the set of ordered pairs of real numbers. One could continue the process, yielding 2-groupoids whose identity types are groupoids, 3-groupoids whose identity types are 2-groupoids, and so forth. And finally, one could have types whose identity types are general types: these types are also known as ∞-groupoids. ## gab.ai Imagine you’re organizing your bookshelf. You have “books” (which we’ll call “terms”), and you group them into “categories” like fiction, history, and science (we’ll call these groups “types”). Sometimes, you notice two books are essentially the same—maybe you have two copies of the same novel. That similarity is what we call an “identity type”—it’s how we say two things are equal within a category. Now, some categories are very straightforward. Imagine a category containing only “true” and “false.” Here, everything is clear-cut: two things are either identical or they’re not. This kind of category is called a “proposition” or a “truth value”—it’s like a simple yes/no question. But not all categories are that simple. Think of a category with two distinct items, like an apple and an orange. There’s no inherent “sameness” between them. In this case, the category itself is what we call a “set.” It gets more interesting. Consider a category containing different sets themselves. Imagine one set is {apple, orange} and another is {banana, pear}. There might be multiple valid ways to consider these sets “equal.” You could pair apple with banana and orange with pear, or you could pair apple with pear and orange with banana. Each pairing is a valid “equality,” so the category of these sets isn’t just a simple set—it’s richer, what we call a “groupoid.” This idea can keep going. You can have categories where the ways things are equal are themselves complex, leading to “2-groupoids,” “3-groupoids,” and so on. At the most general level, you have categories where the notion of equality is as rich and layered as the objects themselves—these are called “∞-groupoids.” In essence, homotopy type theory is a way to formalize how we think about sameness and structure across different levels of complexity, from simple true/false questions to the most intricate mathematical objects. ## claude Here's a more accessible version: **Homotopy Type Theory: A Theory of Objects and How They Can Be Equal** Homotopy type theory studies three main things: individual objects (which we call _terms_), collections of objects (called _types_), and the different ways objects can be considered equal to each other. The key insight is that "equality" itself comes in different flavors depending on what kind of collection you're working with. **Simple Collections (Propositions)** Some collections are very simple. Either they contain nothing at all, or if they do contain objects, any two objects in the collection are equal in exactly one obvious way. Think of a collection like "all ways to prove 2+2=4" - there might be different proofs, but they're all essentially the same thing. These simple collections are called _propositions_. **Everyday Collections (Sets)** Most collections we think about daily work like traditional sets. Take a collection of people: either two people are the same person (equal) or they're different people (not equal). There's no middle ground or multiple ways for two people to be "the same." These are called _sets_. **More Complex Collections (Groupoids)** But some collections are more interesting. Imagine you have a collection containing "all possible ways to arrange two books on a shelf." You might have: - Arrangement 1: Book A on left, Book B on right - Arrangement 2: Book C on left, Book D on right Now, you could say these arrangements are "the same type of thing" in two different ways: you could match A with C and B with D, or you could match A with D and B with C. So there are multiple meaningful ways these arrangements can be considered equal. Collections like this are called _groupoids_. **Even More Complex Collections** You can keep building up complexity. Just as groupoids have multiple ways for objects to be equal, you can have collections where those "ways of being equal" themselves have multiple ways of being equal. This creates _2-groupoids_, then _3-groupoids_, and so on. At the top of this hierarchy are _∞-groupoids_ - collections where equality has infinitely rich structure, with layers upon layers of different ways things can be considered the same. **Why This Matters** This framework gives us a precise way to talk about how different mathematical objects relate to each other, capturing not just whether things are equal, but the rich structure of _how_ they can be equal. It's particularly useful for understanding spaces, shapes, and transformations in mathematics. ## gemini Imagine you have different boxes for sorting things. In Homotopy Type Theory (HoTT), these boxes are called **types**, and the things you put inside them are called **terms**. The most important idea in HoTT is about how we say two things are the same. Instead of just saying "this equals that," we need to provide a **reason** or a **path** that proves they're the same. This collection of reasons or paths is called an **identity type**. The complexity of these "reasons for equality" determines what kind of collection you're dealing with. --- ### Level 0: Propositions (Simple Truths) A **proposition** is the simplest kind of type. Think of it as a statement that is either true or false. If a statement is true, there's only **one, straightforward reason** why it's true. For example, the statement "2+2=4" is a proposition. It's true, and there aren't different _flavors_ of this truth—it just _is_. A type is a proposition if, for any two items in it, there's at most one reason for them to be considered the same. It's like a light switch: on or off. 💡 --- ### Level 1: Sets (Normal Collections) A **set** is a collection of distinct items, like the numbers {1, 2, 3}. The reasons for equality are still simple propositions. For example, there's a simple reason why `3` is the same as `3`, but there's **no reason at all** why `3` would be the same as `2`. Most of the math we learn in school (like arithmetic and basic algebra) deals with sets. You're just checking if things are identical or not. --- ### Level 2: Groupoids (Collections with Interesting Equalities) This is where things get more interesting. 🧐 A **groupoid** is a type where the **reasons for equality have their own structure**. In other words, there can be multiple, distinct ways to prove two things are equivalent. Imagine you have two pairs of socks. - Pair A is {Red Sock, Blue Sock}. - Pair B is {Green Sock, Yellow Sock}. How can we say "Pair A is equivalent to Pair B" (since they're both just pairs)? We can show this by matching them up in two different ways: 1. **Reason 1:** Match Red ↔ Green and Blue ↔ Yellow. 2. **Reason 2:** Match Red ↔ Yellow and Blue ↔ Green. Since there are _two different reasons_ (or paths) to explain their equivalence, the collection of "pairs of socks" isn't a simple set. It's a groupoid. The "identity type" connecting Pair A and Pair B contains two distinct proofs. This allows us to make powerful statements, like saying the set of positive integers `{1, 2, 3, ...}` is equivalent to the set of negative integers `{-1, -2, -3, ...}` because we can create a clear mapping between them (`n ↔ -n`). --- ### To Infinity and Beyond! (∞-Groupoids) HoTT doesn't stop there. - You can have **2-groupoids**, where the reasons _between the reasons_ also have their own structure. - You can have **3-groupoids**, and so on. This process can continue forever, leading to **∞-groupoids**. These are types where the reasons for equality are just as rich and complex as the original items themselves. This is where the "homotopy" (which studies shapes) part of the name comes from: you can think of types as complex shapes and the identity paths as ways of moving around on their surfaces. ## chatgpt Homotopy type theory (HoTT) is a way of talking about three things: 1. **Objects** (which we call _terms_), 2. **Collections of objects** (which we call _types_), and 3. **Ways of saying two objects are the same** (which we call _identities_). A key idea in mathematics is that equality can behave differently depending on what kinds of objects you’re talking about. For example, suppose you have a type where, for every two objects in it, there is exactly one way to say they’re equal. Such a type is called a **proposition** or a **truth value**. The simplest case is the empty type (the collection with no objects at all). Since there are no objects to compare, it automatically satisfies the rule—trivially, every non-existent object is “equal to itself” in the only possible way. But not every type is like this. Imagine a type with just two objects, call them _a_ and _b_. There’s no way to declare _a_ and _b_ equal, since they’re distinct. Still, if you look at the identity of _a_ with itself, or _b_ with itself, each of those behaves like a proposition. Types like this are called **sets**. Yet, even sets are not the end of the story. Take a type that itself contains two sets: - one set with objects _a_ and _b_, - another set with objects _c_ and _d_. Now, how can we say these two sets are equal? There are two different valid ways: - Match _a_ with _c_ and _b_ with _d_, or - Match _a_ with _d_ and _b_ with _c_. So here, the _identity_ between the two sets isn’t just a single proposition—it’s itself set-like, because there are multiple distinct equalities. A type like this is called a **groupoid**, which is a richer structure than a set. In everyday terms, this kind of reasoning lets us say, for example, that the set of positive integers is “the same as” the set of negative integers, or that the set of complex numbers is “the same as” the set of pairs of real numbers—there can be more than one valid way to match things up. And this process keeps going. You can have: - **2-groupoids**, where the equalities between equalities behave like groupoids, - **3-groupoids**, where equalities between those behave like 2-groupoids, - and so on without end. The most general case is when you allow this to continue indefinitely. Such types are called **∞-groupoids**. ## chatgpt2 ``` ChatGPT actually did ask for some clarification so I'll put it's second response here as well. ``` ### What is Homotopy Type Theory? _(For the person on the street)_ At its heart, mathematics is about _things_ and the _relationships_ between them. Homotopy type theory (HoTT) is a new way of organizing those things and relationships. Think of it like this: - You have **objects** (like apples, chairs, or numbers). - You have **collections of objects** (like “all apples,” “all chairs,” or “all numbers”). - And you have **ways of saying when two objects are the same** (like saying two apples are equally red, or two numbers are equal). Now, here’s the twist: in HoTT, the idea of “sameness” isn’t always simple. Sometimes there’s only one clear way to say two things are the same. Other times, there might be _several different but equally valid_ ways. For example: - If a collection is so simple that there’s at most one way to compare any two things in it, that’s like a **yes-or-no fact**. (“Is this collection empty?” Yes or no.) - If a collection has distinct objects, but each equality is straightforward, that’s like an ordinary **set** (like the set of numbers 1, 2, and 3). - But sometimes, there can be _multiple ways_ to match one collection with another. That’s like saying the set of positive numbers is “the same as” the set of negative numbers—but there are two different ways to line them up: (1 → –1, 2 → –2, …) or (1 → –2, 2 → –1, …). When that happens, sameness itself starts to feel richer—it’s not just “yes or no,” it’s more like a **web of possibilities**. And this idea doesn’t stop: you can talk about equalities between equalities, and equalities between _those_, and so on, forever. HoTT has a name for this tower of relationships: **∞-groupoids**. So, in plain terms: Homotopy type theory is a way of thinking about mathematics where _equality itself has layers_. Instead of just asking “are these two things the same?”, it asks “in how many ways can they be the same—and how do those ways relate to each other?”