Constructing ∞-categories

• The opposite ∞-category
• The functor category
• Lifting problems
• The homotopy coherent nerve
• Slice constructions

Last time we talked about $\infty$-categories and their homotopy categories.

The opposite ∞-category#

Definition 1. Given a $X_ \bullet \in \mathsf{Set} _ \Delta$, we can define its opposite $X_ \bullet^\mathrm{op} \in \mathsf{Set} _ \Delta$ as

• $X_n^\mathrm{op} = X_n$,
• $d_i^\mathrm{op} = d_ {n-i}$, $s_i^\mathrm{op} = s_ {n-i}$.

Applying this to an $\infty$-category gives its opposite $\infty$-category.

Proposition 2. If $X_ \bullet$ is an $\infty$-category, then $X_ \bullet^\mathrm{op}$ is an $\infty$-category.

The functor category#

For $\infty$-categories $\mathcal{C}$ and $\mathcal{D}$, a functor is just a map between the simplicial sets: $$ \operatorname{Fun}(\mathcal{C}, \mathcal{D}) _n = \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(\mathcal{C} \times \Delta^n, \mathcal{D}). $$

Theorem 3 (Theorem 1). If $\mathcal{D}$ is an $\infty$-category and $S$ is any simplicial set, then the simplicial set $\operatorname{Fun}(S, \mathcal{D})$ is an $\infty$-category.

Proving this requires a theorem of Joyal.

Theorem 4 (Joyal, Theorem 2). For a simplicial set $\mathcal{C}$, the following conditions are equivalent:

1. $\mathcal{C}$ is an $\infty$-category,
2. the map $$ \operatorname{Fun}(\Delta^2, \mathcal{C}) \to \operatorname{Fun}(\Lambda_1^2, \mathcal{C}) $$ is a trivial Kan fibration.

Definition 5. We say that a map $S_ \bullet \to T_ \bullet$ is a trivial Kan fibration when any diagram $$ \begin{CD} \partial \Delta^n @>>> S_ \bullet \br @VVV @VVV \br \Delta^n @>>> T_ \bullet \end{CD} $$ has a lifting.

Proposition 6 (Proposition 3). When $S_ \bullet \to T_ \bullet$ is a trivial Kan fibration and $A_ \bullet \hookrightarrow B_ \bullet$ is any monomorphism, then any diagram $$ \begin{CD} A_ \bullet @>>> S_ \bullet \br @VVV @VVV \br B_ \bullet @>>> T_ \bullet \end{CD} $$ has a lifting.

Using these facts, we can prove the theorem about mapping simplicial sets to $\infty$-categories being $\infty$-categories.

Lifting problems#

Let’s discuss the idea of the proof of Theorem 2. If we look at the second condition, we see that this is the lifting problem for the following diagram. $$ \begin{CD} \partial \Delta^n \times \Delta^2 \amalg_ {\partial \Delta^n \times \Lambda_1^2} \Delta^n \times \Lambda_1^2 @>>> \mathcal{D} \br @VVV @VVV \br \Delta^n \times \Delta^2 @>>> \ast \end{CD} $$

At this point, we need to study general extension problems. For $S$ a collection of morphisms, denote by $L(S)$ the set of morphisms having the left lifting property for all morphisms in $S$, and by $R(S)$ the set of morphisms having the right lifting property for all morphisms in $S$. What we want to show is that the left vertical arrow is inside $L(R(S))$, where $$ S = { \Lambda_i^n \hookrightarrow \Delta^n : 0 \lt i \lt n } $$ is the set of inner horn inclusions.

Definition 7. The maps in $L(R(S))$, where $S$ is the inner horn inclusions, are called inner anodyne.

Definition 8. We say that a collection of maps is weakly saturated if it is closed under

• pushouts,
• retrats,
• transfinite composition.

Proposition 9. $L(S)$ is always weakly saturated.

We know that $L(R(S))$ contains $S$, so what we do is combinatorially manipulate these inner horn inclusions, by taking pushouts, retracts, and transfinite compositions, to construct other maps.

Proposition 10 (Joyal). The class of inner anodyne maps is generated by (i.e., is the minimal weakly saturated family containing) any of the following collection of morphisms:

• ${ \Lambda_i^n \hookrightarrow \Delta^n : 0 \lt i \lt n }$,
• ${ (\Delta^m \times \Lambda_1^2) \amalg_ {\partial \Delta^m \times \Lambda_1^2} (\partial \Delta^m \times \Delta^2) \hookrightarrow \Delta^m \times \Delta^2 }$,
• ${ (B_ \bullet \times \Lambda_1^2) \amalg_ {A_ \bullet \times \Lambda_1^2} (A_ \bullet \times \Delta^2) \hookrightarrow B_ \bullet \times \Delta^2 : A_ \bullet \hookrightarrow B_ \bullet }$.

Proposition 3 then corresponds to the following fact.

Proposition 11. The set ${ \partial \Delta^n \hookrightarrow \Delta^n }$ generate the class of all monomorphisms.

Remark 12. The fibers of the fibration $$ \operatorname{Fun}(\Delta^2, \mathcal{C}) \to \operatorname{Fun}(\Lambda_1^2, \mathcal{C}) $$ will be trivial Kan complexes when $\mathcal{C}$ is an $\infty$-category. This can be thought of as the space of composing two given morphisms.

The homotopy coherent nerve#

Definition 13. A simplicial category is a category enriched in $\mathsf{Set} _ \Delta$. A simplicial functor between simplicial categories is a functor enriched in simplicial sets.

Given a $1$-category $\mathcal{C}$, we can regard it as a simplicial category by considering sets as discrete simplicial sets. A strict $2$-category can be regarded as a simplicial category by taking the nerves of mapping categories.

Example 14. Let $Q$ be a partially ordered set. We can define a strict $2$-category $\operatorname{Path} _ {[2]}(Q)$ with objects just elements of $Q$, and mapping category defined by $$ \operatorname{Path} _ {[2]}(Q)(x, y) = { x \lt x_1 \lt \dotsb \lt x_k \lt y }. $$ How is this a category? This set is ordered by the reverse inclusion.

Example 15. For $Q = [2]$, we see that $$ \operatorname{Path} _ {[2]}([2])(0, 2) = { 0 \lt 1 \lt 2, 0 \lt 2 } $$ and so this is just a $\Delta^1$. For $Q = [3]$, we can check that $$ \operatorname{Path} _ {[2]}([3])(0, 3) = \Delta^1 \times \Delta^1, $$ and in general it will be a hypercube.

So we have a mechanism for turning a poset into a strict $2$-category, and then a simplicial category. $$ [n] \mapsto \operatorname{Path} _ {[2]}([n]) \mapsto \operatorname{Path}[n]. $$ We can also do a Kan extension along $\Delta \hookrightarrow \mathsf{Set} _ \Delta$ to define a functor $$ \operatorname{Path}[-] \colon \mathsf{Set} _ \Delta \to \mathsf{Cat} _ \Delta, \quad \operatorname{Path}[X] = \varinjlim_ {\Delta^n \to X} \operatorname{Path}[\Delta^n]. $$

Definition 16. For $\mathcal{C} _ \bullet \in \mathsf{Cat} _ \Delta$ a simplicial category, we define its homotopy coherent nerve $N_ \bullet^\mathrm{hc}(\mathcal{C} _ \bullet)$ as $$ N_n^\mathrm{hc}(\mathcal{C}) = \operatorname{Hom} _ {\mathsf{Cat} _ \Delta}(\operatorname{Path}[n], \mathcal{C} _ \bullet). $$

For a simplicial category $\mathcal{C} _ \bullet$, we can explicitly describe low-dimensional simplices of $N_ \bullet^\mathrm{hc}(\mathcal{C} _ \bullet)$.

• $N_0^\mathrm{hc}(C_ \bullet)$ are just objects of $C_ \bullet$.
• $N_1^\mathrm{hc}(C_ \bullet)$ are just morphisms of $C_ \bullet$.
• $N_2^\mathrm{hc}(C_ \bullet)$ are the diagrams of $f \colon X \to Y$, $g \colon Y \to Z$, $h \colon X \to Z$, and a $1$-simplex from $g \circ f$ to $h$ in the mapping simplex $\operatorname{Hom}(X, Z)$.

Theorem 17. If $\mathcal{C} _ \bullet \in \mathsf{Cat} _ \Delta$ is locally Kan, i.e., all mapping simplicial sets are Kan complexes, then $N_ \bullet^\mathrm{hc}(\mathcal{C} _ \bullet)$ is an $\infty$-category.

Example 18. If we look at the full subcategory $\mathsf{Kan} \subset \mathsf{Set} _ \Delta$ of Kan complexes. This is naturally enriched in simplicial sets, and we know that mapping simplicial sets are Kan complexes. So we can apply the homotopy coherent nerve construction to get an $\infty$-category $\mathcal{S}$, which is also called the $\infty$-category of spaces.

Example 19. We can look at the full subcategory of $\infty$-categories in simplicial sets. Here, the mapping simplicial sets are not always Kan complexes, but we can look at the maximal Kan subcomplexes of the mapping simplicial sets. Taking the homotopy coherent nerve definfes the $\infty$-category of $\infty$-categories.

Slice constructions#

Definition 20. For $X, Y \in \mathsf{Set} _ \Delta$ two simplicial sets, we define the simplicial set $X \ast Y$ by $$ (X \ast Y) _ n = \coprod_ {i+j=n-1} (X_i \times Y_j) \amalg X_n \amalg Y_n. $$

Example 21. We have $\Delta^i \ast \Delta^j \cong \Delta^{i+j+1}$.

Definition 22. For a map $K \to X$ of simplicial sets, we define the simplicial set $X_ {/f}$ so that $$ \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(Y, X_ {/f}) \cong \operatorname{Hom} _ {(\mathsf{Set} _ \Delta) _ {K/}}(Y \ast K, X). $$

For example, if $K = \Delta^0$ with image $x \in X_0$, we the $0$-simplices of $X_ {/f}$ are $1$-simplices $y \to x$, and $1$-simplices are triangles, and so on.

Theorem 23. If $\mathcal{C}$ and $\mathcal{D}$ are $\infty$-categories, then $\mathcal{C} \ast \mathcal{D}$ is an $\infty$-category.

Theorem 24. If $K$ is a simplicial set mapping into a $\infty$-category $\mathcal{C}$, then $\mathcal{C} _ {f/}$ is an $\infty$-category and moreover $$ \mathcal{C} _ {f/} \to \mathcal{C} $$ is a left fibration. Similarly, $$ \mathcal{C} _ {/f} \to \mathcal{C} $$ is a right fibration between $\infty$-categories.