An Interlude -- Quasicategories (Homotopy Theories pt 3/4)

11 Jul 2022 - Tags: homotopy-theories

In the previous post, we defined $\infty$-categories as being categories “enriched in simplicial sets”. These are nice, and fairly quick to introduce, but if you start reading the $\infty$-category theoretic literature, you’ll quickly run into another definition: Quasicategories. In this post, we’ll give a quick introduction to quasicategories, and talk about how they’re related to the $\mathcal{S}$-enriched categories we’ve come to know and love.

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First, then, a reminder:

A Simplicial Set is a (contravariant) functor from the simplex category to $\mathsf{Set}$.

This seems strange at first, but it turns out that we can model many different kinds of behavior using simplicial sets.

Most notably, simplicial sets are a common generalization of both topological spaces (up to homotopy) and categories!

We’ve already seen how simplicial sets can represent topological spaces up to homotopy (this is given by a quillen equivalence of model structures on $\mathsf{Top}$ and $s\mathsf{Set}$). To a simplicial set, we associate its geometric realization, and to a topological space we associate its (singular) simplicial set $\text{Sing}_X$ where $\text{Sing}_X(\Delta^n)$ is the set of continuous maps from the topological $n$-simplex into $X$.

We know that $X$ is determined up to (weak) homotopy equivalence by $\text{Sing}_X$, so we previously defined a Simplicial Category to be a category enriched in simplicial sets. We wanted to intuitively view this as a category with a geometric space of arrows between any two objects, as this came up in our study of $\infty$-categories. This turns out to be a great definition for intuition, but it can be a bit difficult to work with.

I mentioned earlier that simplicial sets can directly generalize categories… Maybe there’s some way to define $\infty$-categories directly as simplicial sets as well?

The answer will be “yes”, but let’s start small. Is there a way for us to recognize when a simplicial set is $\text{Sing}_X$ for some topological space $X$?

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For instance, let’s look at $\Delta^2$. This is a solid triangle, and it has three horns – one for each vertex.

Concretely, $\Delta^2$ is given by

Then the $0$th horn of $\Delta^2$, denoted $\Lambda^2_0$, is what we get by removing the interior 2-cell, along with the 1-cell opposite $0$:

Analogously, we get $\Lambda^2_1$ by removing the interior 2-cell and the 1-cell opposite $1$:

and we get $\Lambda^2_2$ by removing the interior 2-cell and the 1-cell opposite $2$:

What about the horns of $\Delta^3$? Well now, we remove the interior 3-cell (the “volume” of the simplex) as well as the 2-cell opposite your favorite vertex. Concretely, we see $\Lambda^3_0$ is given by²

Similarly, $\Lambda^3_1$ is

$\Lambda^3_2$ is

and $\Lambda^3_3$ is

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Now then, we come to an important definition

In a commutative diagram, we ask that the following dotted morphism should always exist:

Why care about this? Because of the following major theorem:

So we see that we can completely recover the notion of topological space (up to homotopy) by looking at special simplicial sets… But wasn’t this all supposed to have something to do with category theory?

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Just like every topological space $X$ defines a simplicial set \(\text{Sing}_X\), every category also defines a simplicial set, called the Nerve of the category $\mathcal{C}$.

In general, the $n$-cells in the nerve $\mathcal{N}(\mathcal{C})$ will be given by the “paths” of length $n$ made of arrows in $\mathcal{C}$. That is

• The 0-cells will be objects of $\mathcal{C}$
• The 1-cells will be the arrows, $C_0 \to C_1$
• The 2-cells will be the paths of length 2, $C_0 \to C_1 \to C_2$
• The 3-cells will be the paths of length 3, $C_0 \to C_1 \to C_2 \to C_3$
• etc.

Concretely, let’s look at the following category (where $k = hf = hg$):

Then its nerve should have three 0-cells ($A$, $B$, and $C$), plus 1-cells for $f$, $g$, $h$, and the composite $k$ (notice this is only one 1-cell, since it’s a single arrow in $\mathcal{C}$). However, we add two 2-cells:

\[A \overset{f}{\to} B \overset{h}{\to} C\] \[A \overset{g}{\to} B \overset{h}{\to} C\]

since $k$ arises as a composite in two ways: $hf$ and $hg$.

Thus, the nerve of $\mathcal{C}$ is a cone

Perhaps a better way to visualize this is as a disk instead:

Of course, it’s easy to guess the next question. Can we tell which simplicial complexes arise as the nerve of some category?

Again, the answer is yes, and the answer will look shockingly similar to the case of topological spaces!

This should make sense as a definition, since in a category composition tells us that we can fill inner horns!

Indeed, consider the inner horn $\Lambda^2_1$:

If this diagram lives inside the nerve of a category $\mathcal{N}(\mathcal{C})$, then we can always fill the horn! Indeed, we have a 1-cell from $0 \to 2$ given by $gf$. We also have a 2-cell filling this triangle given by the path $0 \overset{f}{\to} 1 \overset{g}{\to} 2$.

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This all brings us to another major theorem:

Importantly, this means that every kan complex is a quasicategory! This tells us that quasicategories allow us to treat spaces and categories on equal footing⁴!

In particular, quasicategories give us a setting where we can “do homotopy theory” to categories, and if you remember back to the main post about $\infty$-categories, and how they solve the formal issues with model categories, that’s exactly what we were looking for!

This is great because it gives us a super concrete way of working with $\infty$ categories.

In this framework, our categories literally are geometric objects! The category of simplicial sets is a topos, so it has as many nice constructions as we could want, and many of these preserve kan complexes and quasicategories.

For instance, now functors are just simplicial maps, many limits and colimits of quasicategories can be computed as with geometric objects, exponentials give us functor quasicategories, and all of these work as well as we could hope.

Because of this concreteness, Jacob Lurie’s tomes on $\infty$-categories are primarily based on the language of quasicategories. But in the main post, we defined an $\infty$-category to be a category enriched in spaces…

How can we reconcile these viewpoints? Is there a way for us to apply the machinery proven in Lurie’s books to the hammock localization of a model category? Is there some way for us to use these nice geometric definitions for computations involving quasicategories, and leverage them to understand the simplicial categories we’ve been talking about? Why have we given two seemingly unrelated definitions of an $\infty$-category in the first place?

For answers to these questions and more, read on to the last post in this series⁵!

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