The language of ∞-categories

The language of ∞-groupoids#

Instead of sets, we want to work with ∞-groupoids. There is an adjunction $$ \lvert - \rvert \colon \mathsf{sSet} \Longleftrightarrow \mathsf{CGTop} \colon \operatorname{Sing} $$ where there are notions of weak equivalences on both sides. They are also both closed symmetric monoidal categories, where tensors are products, and the two functors are both symmetric monoidal. In turns out that once we localize at weak equivalences, the adjunction turns into an equivalence of categories $$ \mathcal{H} = \mathsf{sSet}[W^{-1}] \simeq \mathsf{CGTop}[W^{-1}]. $$ Another way of describing this category $\mathcal{H}$ is as the full subcategory $\mathsf{Kan}$ of Kan complexes modulo homotopy equivalence.

Definition 1. Two maps $f, g \colon K \to L$ of Kan complexes are homotopic if there exists an $F \colon \Delta^1 \times K \to L$ such that $F \vert_{\lbrace 0 \rbrace \times K} = f$ and $F \vert_{\lbrace 1 \rbrace \times K} = g$.

There is a good notion of homotopy limits and colimits in $\mathsf{sSet}$ and $\mathsf{CGTop}$. If we have a diagram $F \colon I \to \mathsf{sSet}$ or $I \to \mathsf{CGTop}$, we will have a natural equivalence $$ \underline{\Hom}(X, \operatorname{holim}_ I F) \simeq \operatorname{holim}_ I \underline{\Hom}(X, F(i)) $$ and so forth. We can define loop spaces as $$ \Omega X = \ast \times_X^h \ast. $$

There is also an adjunction $$ \pi_0 \colon \mathsf{sSet}, \mathsf{CGTop} \Longleftrightarrow \mathsf{Set} \colon \iota $$ and we can define higher homotopy groups as $$ \pi_i(X, \ast) = \pi_{i-1}(\Omega X, \ast). $$ If we have a fiber sequence, i.e., a homotopy pullback $$ \begin{CD} Z @>>> X \br @VVV @VVV \br \ast @>>> Y \end{CD} $$ then we get a long exact sequence of homotopy groups.

The language of ∞-categories#

We want to now talk about ∞-categories, where hom sets will be replaced by hom spaces. So in an ∞-category, we will have

objects $X, Y, Z, \dotsc$,
mapping spaces $\Hom_\mathcal{C}(X, Y)$,

satisfying certain axioms. Here is a naive definition that is actually hard to work with.

Definition 2. A topological category (respectively, simplicial category) is a category $\mathcal{C}$ enriched over $\mathsf{CGTop}$ (respectively, $\mathsf{sSet}$). Its homotopy category $\operatorname{h}\mathcal{C}$ is an (ordinary) category where
objects of $\operatorname{h}\mathcal{C}$ are the same as object of $\mathcal{C}$,
morphisms are $\Hom_{\operatorname{h}\mathcal{C}}(X, Y) = \pi_0 \Hom_\mathcal{C}(X, Y)$.
A functor $F \colon \mathcal{C} \to \mathcal{D}$ between topological (respectively, simplicial) categories is an enriched functor. We say that a functor $F \colon \mathcal{C} \to \mathcal{D}$ is
fully faithful if for all $X, Y \in \mathcal{C}$, the map $$ \Hom_\mathcal{C}(X, Y) \to \Hom_\mathcal{D}(F(X), F(Y)) $$ is a weak equivalence,
essentially surjective if the induced functor $\operatorname{h}F \colon \operatorname{h}\mathcal{C} \to \operatorname{h}\mathcal{D}$ is essentially surjective,
an equivalence if it is both fully faithful and essentially surjective.

Now we can develop of model category of topological or simplicial categories, with equivalences as weak equivalences, and working appropriately, we will get a good notion of ∞-categories. In particular, topological categories and simplicial categories give the same theory. But this turns out to be difficult to work with, and we will instead use quasi-categories.

To motivate the definition, we consider the adjunction $$ \mathsf{sSet} \Leftrightarrow \mathsf{Cat} \colon N_\bullet $$ where the functor $\mathsf{sSet} \to \mathsf{Cat}$ preserves all colimits and sends $[n]$ to the partially ordered set $0 \to 1 \to \dotsb \to n$ regarded as a category. The adjoint $N_\bullet \mathcal{C}$ is called the nerve of $\mathcal{C}$ and is defined by $$ N_n \mathcal{C} = \lbrace C_0 \to C_1 \to \dotsb \to C_n \rbrace. $$

Lemma 3. Let $S$ be a simplicial set. Then the following are equivalent:
$S \cong N_\bullet \mathcal{C}$ for some category $\mathcal{C}$,
for every $0 \lt i \lt n$ and $\Lambda_i^n \to S$, there exists a unique extension to $\Delta^n \to S$,
for every $n \ge 2$, the natural map $$ S_n \to S_1 \times_{S_0} S_1 \times_{S_0} \dotsb \times_{S_0} S_1 $$ induced by $[1] \cong \lbrace i, i+1 \rbrace \subset [n]$ is a bijection.

Remark 4. One model of ∞-category is a simplicial space satisfying the third condition, called the Segal condition, and an additional axiom. These are called complete Segal spaces.

We will use (2) to model ∞-categories.

Definition 5. A quasi-category is a simplicial set $S$ such that for every $0 \lt i \lt n$, every map $\Lambda_i^n \to S$ extends to $\Delta^n \to S$. A functor $F \colon \mathcal{C} \to \mathcal{D}$ between quasi-categories is a map between simplicial sets.

Example 6. Every Kan complex is a quasi-category.

For $\mathcal{C}$ and $\mathcal{D}$ quasi-categories, the mapping simplicial set $\mathsf{Fun}(\mathcal{C}, \mathcal{D})$ turns out to be a quasi-category. So we can define this to be the functor category. For $X, Y \in \mathcal{C}_ 0$, i.e., objects of $\mathcal{C}$, we can define $$ \operatorname{Maps}_ \mathcal{C}(X, Y) = \lbrace X \rbrace \times_\mathcal{C} \mathsf{Fun}([1], \mathcal{C}) \times_\mathcal{C} \lbrace Y \rbrace, $$ and this turns out to be a Kan complex.

Definition 7. A functor $F \colon \mathcal{C} \to \mathcal{D}$ between quasi-categories is
fully faithful when the induced $\operatorname{Maps}(X, Y) \to \operatorname{Maps}(F(X), F(Y))$ are equivalences,
essentially surjective when the induced $\operatorname{h}\mathcal{C} \to \operatorname{h}\mathcal{D}$ is essentially surjective,
an equivalence when it is fully faithful and essentially surjective.
Two functors $F, G \colon \mathcal{C} \to \mathcal{D}$ are naturally equivalent and write $F \simeq G$ when there exists a path connecting $F$ and $G$ inside $\mathsf{Fun}(\mathcal{C}, \mathcal{D})$.

Proposition 8. If $F \colon \mathcal{C} \to \mathcal{D}$ is an equivalence, there exists a $G \colon \mathcal{D} \to \mathcal{C}$ such that $G \circ F \simeq \id_\mathcal{C}$ and $F \circ G \simeq \id_\mathcal{D}$.

The simplicial nerve#

Given any simplicial category, we can get a simplicial set via the following construction. $$ \mathfrak{C} \colon \mathsf{sSet} \Leftrightarrow \mathsf{sCat} \colon N^{hc} $$ First we define $\mathfrak{C}(\Delta^n)$ as the simplicial category where

objects are $0, 1, \dotsc, n$,
maps from $i$ to $j$ is given by the nerve of the partially ordered set $P_{i,j}$ of subsets of $\lbrace i, j \rbrace$ ordered by inclusion.

This is really encapsulating the idea that if we have $i \to j$ and $j \to k$, their composition doesn’t need to be strictly $i \to k$; rather, there should be a morphism between the two. Using this, we can extend the functor $\mathfrak{C}$ to all simplicial sets. The adjoint $N^{hc}$ will then be described by $$ (N^{hc} \mathcal{C})_ n = \Hom_\mathsf{sCat}(\mathfrak{C}[\Delta^n], \mathcal{C}). $$

Theorem 9. Let $\mathcal{C}$ be a simplicial category such that for every $X, Y \in \mathcal{C}$, we have $\Hom_\mathcal{C}(X,Y) \in \mathsf{sSet}$ is a Kan complex. Then $N^{hc} \mathcal{C}$ is a quasi-category.

Example 10. Recall that $\mathsf{Kan}$ the category of Kan complexes is enriched over itself. So we can define $$ \mathsf{Spc} = \mathsf{Ani} = N^{hc} \mathsf{Kan}, $$ and similarly $$ \widehat{\mathsf{Cat}} = N^{hc} \mathsf{QCat} $$ where $\mathsf{QCat}$ is the simplicial category where objects are quasi-categories and morphisms are $\mathsf{Fun}(\mathcal{C}, \mathcal{D})^\simeq$ the maximal Kan complex of the functor category.

Limits and colimits#

Definition 11. Let $\mathcal{C}$ be a quasi-category. We say that $X \in \mathcal{C}$ is
initial when for all $Y \in \mathcal{C}$, the space $\operatorname{Maps}(X, Y)$ is contractible,
final when for all $Y \in \mathcal{C}$, the space $\operatorname{Maps}(Y, X)$ is contractible.

Definition 12. For a diagram $F \colon I \colon \mathcal{C}$, we define the over-category $$ \mathcal{C}_ {/F} = \mathcal{C} \times_{\mathsf{Fun}(I, \mathcal{C})} \mathsf{Fun}([1] \times I, \mathcal{C}) \times_{\mathsf{Fun}(I, \mathcal{C})} \lbrace F \rbrace, $$ and similarly the under-category $$ \mathsf{C}_ {F/} = \lbrace F \rbrace \times_{\mathsf{Fun}(I, \mathcal{C})} \mathsf{Fun}([1] \times I, \mathcal{C}) \times_{\mathsf{Fun}(I, \mathcal{C})} \mathcal{C}. $$

Definition 13. Let $\mathcal{C}$ be an ∞-category, and let $F \colon I \to \mathcal{C}$ be a diagram. A limit of $F$ is a final object of $\mathcal{C}_ {/F}$ and a colimit of $F$ is an initial object of $\mathcal{C}_ {F/}$.

Theorem 14. All (small) limits and colimits exist in $\mathsf{Spc}$ and also $\widehat{\mathsf{Cat}}$.

Limits and colimits in $\mathsf{Spc}$ really can be computed as homotopy limits and homotopy colimits in $\mathsf{Kan}$.

Cartesian and coCartesian fibrations#

Let’s to Cartesian fibrations.

Definition 15. Let $F \colon \mathcal{C} \to \mathcal{D}$ be a functor between quasi-categories. A $F$-Cartesian arrow is an arrow $f \colon C_1 \to C_2$ such that for all $C \in \mathcal{C}$, the map $$ \operatorname{Maps}(C, C_1) \to \operatorname{Maps}(C, C_2) \times_{\operatorname{Maps}(D, D_2)} \operatorname{Maps}(D, D_1) $$ is a homotopy equivalence, where $D = F(C)$ and $D_i = F(C_i)$.

Definition 16. A Cartesian fibration $F \colon \mathcal{C} \to \mathcal{D}$ is a map of quasi-categories such that every $C_2 \mapsto D_2$ and a map $D_1 \to D_2$, there is a $F$-Cartesian lift.

Then we will have some version of the Grothendieck fibration correspondence.