Introduction

• Multiplication on the space of loops
• What are ∞-categories?
• How do ∞-categories make our lives easier?

Let’s say we care about algebraic topology. An important invariant, and in fact the first algebraic invariant we learn is the fundamental group. Let’s recall how it was defined.

Multiplication on the space of loops#

Given a sufficiently nice topological space $X$ together with a point $x \in X$, there is this thing called a loop space $$ \Omega X = { \text{continuous maps } f \colon [0,1] \to X \text{ with } f(0) = f(1) = x }. $$ Given two loops $f$ and $g$, we can compose them by setting $$ (f \ast g)(t) = \begin{cases} f(2t) & 0 \le t \le 1/2, \br g(2t-1) & 1/2 \le t \le 1. \end{cases} $$ This construction gives a multiplication map $$ m \colon \Omega X \times \Omega X \to \Omega X. $$

We would like to say that this multiplication map defines a group structure on $\Omega X$, we can’t quite say that. The annoying thing is that multiplication is not associative. If we look at the two loops $(f \ast g) \ast h$ and $f \ast (g \ast h)$, there is a difference in the time parametrization. So the diagram $$ \begin{CD} \Omega X \times \Omega X \times \Omega X @>{m \times \mathrm{id}}>> \Omega X \times \Omega X \br @V{\mathrm{id} \times m}VV @V{m}VV \br \Omega X \times \Omega X @>{m}>> \Omega X \end{CD} $$ is not commutative, but only commutative up to homotopy in the sense that there is a homotopy between the bottom left composition and the upper right composition.

The simplest way of dealing with this issue is to just pass to the path components and forget about all the interesting topological information encoded in $\Omega X$.

Proposition 1. The multiplication map $m$ induces a group structure on the set of path components of $\Omega X$.

That is what they taught us in our first course in topology. But we lose too much information by passing to path components.

So how can we make sense of $\Omega X$ as a group in a better way? The next naive thing to do is to define a “group up to homotopy” in the following way.

Definition 2. A “group up to homotopy” is a pointed topological space $G$ with a continuous map $m \colon G \times G \to G$ such that

• $m \circ (m \times \mathrm{id} _G)$ and $m \circ (\mathrm{id} _G \times m)$ are homotopic,
• and blah blah blah.

Is everything good now? Actually, there still is a problem. There are these two points in $\operatorname{Map}(G \times G \times G, G)$ that we know are in the same path component, but for certain constructions we actually need a specific path between the two points. So let’s modify the definition to:

Definition 3. A “group up to homotopy” is a pointed topological space $G$ with

• a continuous map $m \colon G \times G \to G$,
• a homotopy $\alpha$ between $m \circ (m \times \mathrm{id} _G)$ and $m \circ (\mathrm{id} _G \times m)$,

such that blah blah blah.

Are we done now? Unfortunately, no. The space of possible ternary multiplications is an interval, which is contractible, so this is good. But the space of possible quaternary multiplications turns out to be a circle. $$ \begin{CD} x \cdot (y \cdot (z \cdot w)) @= (x \cdot y) \cdot (z \cdot w) @= ((x \cdot y) \cdot z) \cdot w \br @| @. @| \br x \cdot ((y \cdot z) \cdot w) @= = @= (x \cdot (y \cdot z)) \cdot w \end{CD} $$ To make this contractible as well, we need to fill in this $S^1$ and make it into $D^2$. Once we do that, we see that the space of $G^5 \to G$ forms an $S^2$, so we fill that into a $D^3$. This goes on to infinity, and we finally arrive at following definition.

Definition 4. A “group up to homotopy” is a pointed topological space $G$ with

• a continuous map $m_2 \colon G \times G \to G$,
• a homotopy $m_3 \colon D^1 \times G \times G \times G \to G$, which at the boundary of $D^1$ is the two different ways of composing $m_2$,
• a homotopy $m_4 \colon D^2 \times G^4 \to G$, which at the boundary of $D^2$ is the different ways of composing $m_3$ and $m_2$,
• a homotopy $m_5 \colon D^3 \times G^5 \to G$, which at the boundary of $D^3$, blah blah,
• and so on,

such that blah blah the point is a unit and every element has an inverse blah blah blah.

Admittedly, it is a horrible thing to work with, but it really is the correct definition. The space $\Omega X$ naturally has all these higher multiplication maps $m_k$, so it can be interpreted as a “group up to homotopy.”

Theorem 5 (Stasheff). If $G$ is a “group up to homotopy,” then there is a pointed space $X$ such that $G \simeq \Omega X$ in a way that respects all the higher multiplication maps up to homotopy.

What are ∞-categories?#

The main takeaway from the previous discussion is that if shouldn’t say “homotopic” and instead keep track of the actual homotopy with possibly higher homotopy coherence data.

An $\infty$-category is roughly a category $\mathcal{C}$ such that, when $x, y \in \mathcal{C}$ are two objects, the morphisms from $x$ to $y$ $$ \operatorname{Map}(x, y) $$ is a space up to homotopy, instead of just a set.

Example 6. There is the $\infty$-category of all (nice) topological spaces. For two topological spaces $X$ and $Y$, there is a natural topological space $\operatorname{Map}(X, Y)$ that is the space of all continuous maps from $X$ to $Y$.

Okay, but what do we mean by “up to homotopy” here? Well, since this is a category, there should be a composition map $$ \operatorname{Map}(x, y) \times \operatorname{Map}(y, z) \to \operatorname{Map}(x, z). $$

In a category, composition is supposed to be associative. But everything is up to homotopy, so it wouldn’t be associative on the nose. Instead, there will be a homotopy between $$ \begin{CD} \operatorname{Map}(x, y) \times \operatorname{Map}(y,z) \times \operatorname{Map}(z,w) @>>> \operatorname{Map}(x,y) \times \operatorname{Map}(y,w) \br @VVV @VVV \br \operatorname{Map}(x,z) \times \operatorname{Map}(z,w) @>>> \operatorname{Map}(x,w). \end{CD} $$ After that, there will be higher compositions maps, with certain coherence conditions.

This is what an $\infty$-category is supposed to be. Basically,

• for every pair of objects $x, y$, one should be able to extract a homotopy type $\operatorname{Map}(x, y)$, and
• there should be composition maps that are homotopy coherent.

But what a complicated thing this is.

The good news is that a lot of category theorists have thought about packaging all this data in the neatest way possible. They have come up with many different ways of describing the entire package, called different models of $\infty$-categoris. But the de facto standard is quasicategories defined and developed by Joyal and Lurie.

Definition 7. A quasicategory is a simplicial set that (over $\Delta^0$) satisfies the left lifting property with respect to inner horn inclusions $\Lambda_i^k \hookrightarrow \Delta^k$ for all $0 \lt i \lt k$.

We will see what these words mean next week, and discuss why this is a reasonable definition the week after that.

How do ∞-categories make our lives easier?#

Working with $\infty$-categories allows us to keep track of more data, and ultimately do more things. Sometimes constructions that seem ad hoc turn out to have a natural interpretation in terms of $\infty$-categories.

Here is an example that I find neat. There is $\infty$-category version of a colimit, and in particular, a pushout.

Proposition 8. If $f \colon X \to Y$ is a map of topological spaces, then in the category of spaces, the diagram $$ \begin{CD} X @>f>> Y \br @VVV @VVV \br \ast @>>> C_f \end{CD} $$ is a pushout diagram, where $C_f$ is the cone of $f$ defined by $$ C_f = ((X \times [0, 1]) \amalg Y) / (x,0) \sim (x^\prime,0), (x,1) \sim f(x). $$

Corollary 9. The cone of $Y \to C_f$ is homotopic to $\Sigma X = C_ {X \to \ast}$.

Corollary 10. If $X \to Y \to Z$ are topological spaces with continuous maps, there is a natural induced map $C_ {X \to Y} \to C_ {X \to Z}$, and moreover the cone of this map is homotopic to $C_ {Y \to Z}$.