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.==