Recall that for simplicial sets $X, Y$, we defined the join as $$ X \ast Y = \varinjlim_ {\Delta^n \to X, \Delta^m \to Y} \Delta^{m+n+1}. $$ We can also think of it as $$ (X \ast Y) _n = X_n \amalg \biggl( \coprod _ {i+j=n-1} (X_i \times Y_j) \biggr) \amalg Y_n. $$ Note that there are natural maps $$ X, Y \hookrightarrow X \ast Y, \quad X \ast Y \to \Delta^1. $$
Proposition 1. If $X, Y$ are $\infty$-categories, then $X \ast Y$ is also an $\infty$-category and moreover the mapping spaces can be described as $$ \operatorname{Hom} _ {X \ast Y}(x, y) = \begin{cases} \operatorname{Hom} _X(x, y) & \text{if } x, y \in X, \br \operatorname{Hom} _Y(x, y) & \text{if } x, y \in Y, \br \Delta^0 & \text{if } x \in X, y \in Y, \br \emptyset & \text{if } x \in Y, y \in X. \end{cases} $$
Definition 2. We define the left cone of $X$ as $X^\vartriangleleft = \Delta^0 \ast X$ and similarly the right cone of $X$ as $X^\vartriangleright = X \ast \Delta^0$.
Definition 3. For a map $p \colon K \to X$ of simplicial sets, we define simplicial sets $X_ {/p}$ and $X_ {p/}$ so that $$ \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(Y, X _ {/p}) \cong \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(Y \ast K, X) $$ and $$ \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(Y, X _ {p/}) \cong \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(K \ast Y, X). $$
There are these projection/forgetful maps $$ X_ {p/} \to X_ {p/} \ast K \to X, $$ which is always a right fibration when $X$ is an $\infty$-category, and also $$ X_ {/p} \to K \ast X_ {/p} \to X, $$ which is always a left fibration when $X$ is an $\infty$-category.
Initial and final objects#
We will give multiple definitions and show that they are equivalent.
Definition 4 (Definition 1). Let $\mathcal{C}$ be an $\infty$-category. An object $X \in \mathcal{C}$ is said to be final if the natural projection $$ \mathcal{C} _ {/X} \to \mathcal{C} $$ is a trivial Kan fibration.
Being a trivial Kan fibration just means that we can always lift $$ \begin{CD} \partial \Delta^n @>>> \mathcal{C} _ {/X} \br @VVV @VVV \br \Delta^n @>>> \mathcal{C}. \end{CD} $$ This would then be equivalent to lifting for $$ \begin{CD} \partial \Delta^{n+1} @>>> \mathcal{C} \br @VVV @VVV \br \Delta^{n+1} @>>> \ast \end{CD} $$ where the final vertex is mapped to $X$. (Here, we’re using that $\Delta^n \amalg_ {\partial \Delta^n} (\partial \Delta^n \ast \Delta^0)$ is just $\partial \Delta^{n+1}$.) So we can equivalently phrase this as the following.
Definition 5 (Definition 2). Let $\mathcal{C}$ be an $\infty$-category. An object $X \in \mathcal{C}$ is said to be final if any $\partial \Delta^n \to \mathcal{C}$ with $f(n) = X$ extends to $\Delta^n \to \mathcal{C}$.
Applying this to $\partial \Delta^1$, we see that every object maps to the final object, if it exists. But we want to say something like the mapping spaces to final objects are all contractible.
Definition 6 (Definition 3). Let $\mathcal{C}$ be an $\infty$-category. An object $X \in \mathcal{C}$ is said to be final if the mapping spaces $\operatorname{Hom} _ \mathcal{C}(Y, X)$ are contractible for all $Y \in \mathcal{C}$.
Why is this equivalent to the previous definition? Assume that $\mathcal{C} _ {/X} \to \mathcal{C}$ is a trivial Kan fibration. It turns out that, by definition, $$ \begin{CD} \operatorname{Hom} _ \mathcal{C}^R(Y, X) @>>> \mathcal{C} _ {/X} \br @VVV @VVV \br \Delta^0 @>{Y}>> \mathcal{C} \end{CD} $$ is a pullback, where $\operatorname{Hom}^R$ denotes the (right) pinched mapping space. Then we have this proposition.
Proposition 7. In any $\infty$-category $\mathcal{C}$ and objects $X, Y$, the mapping spaces are all naturally homotopy-equivalent: $$ \operatorname{Hom} _ \mathcal{C}^L(Y, X) \simeq \operatorname{Hom} _ \mathcal{C}(Y, X) \simeq \operatorname{Hom} _ \mathcal{C}^R(Y, X). $$
What about the other direction? We want to say that if all fibers are contractible, then it is in fact a trivial Kan fibration. This is another black box we will take for granted.
Proposition 8. If $\mathcal{C} \to \mathcal{D}$ is a right fibration of $\infty$-categories with contractible fibers, then it is a trivial Kan fibration.
Limits and colimits#
We can likewise give multiple equivalent definitions of colimits (and dually, limits).
Definition 9 (Definition 1). Let $p \colon K \to \mathcal{C}$ be a map of simplicial sets, where $\mathcal{C}$ is an $\infty$-category. The colimit of the diagram $p$ is defined as an initial object of $\mathcal{C} _ {p/}$.
Note that a point in $\mathcal{C} _ {p/}$ is the same, by adjunction, as a map $$ \bar{p} \colon K^{\vartriangleright} = K \ast \Delta^0 \to \mathcal{C}. $$
We can always define the “constant diagram” functor $$ \Delta \colon \mathcal{C} \to \operatorname{Fun}(K, \mathcal{C}). $$
Definition 10 (Definition 2). Let $p \colon K \to \mathcal{C}$ be a map of simplicial sets, where $\mathcal{C}$ is an $\infty$-category. The colimit of the diagram $p$ is defined as an object $Y \in \mathcal{C}$ together with a natural transformation $\eta \colon p \to \Delta(Y)$ such that the map $$ \operatorname{Hom} _ \mathcal{C}(Y, X) \to \operatorname{Hom} _ {\operatorname{Fun}(K, \mathcal{C})}(\Delta(Y), \Delta(X)) \to \operatorname{Hom} _ {\operatorname{Fun}(K, \mathcal{C})}(p, \Delta(X)) $$ is a homotopy equivalence.
Theorem 11. The two definitions are (essentially) equivalent.
Cofinal maps#
Definition 12. We say that a map $f \colon \mathcal{C} \to \mathcal{D}$ of simplicial sets is cofinal if for every $d \in \mathcal{D}$ the fiber $\mathcal{C} \times_ \mathcal{C} d$ (this should be some kind of homotopy fiber) is contractible.
Theorem 13 (Theorem 1-1). Let $A \hookrightarrow B$ be a monomorphism. Then $A \to B$ is cofinal if and only if $A \to B$ is right anodyne.
Theorem 14 (Theorem 1-2). If $\mathcal{C} \to \mathcal{D}$ is cofinal, then it can be written as $\mathcal{C} \to \mathcal{E} \to \mathcal{D}$ where the first map is right anodyne and the second map is a trivial Kan fibration.
Theorem 15 (Theorem 2). If $f \colon \mathcal{C} \to \mathcal{D}$ is cofinal and $p \colon \mathcal{D} \to \mathcal{E}$ is a diagram, then the colimit of $p$ is equivalent to the colimit of $p \circ f$.
Theorem 16 (Theorem 3). If $\mathcal{C}$ is a simplicial set, then there exists a $1$-category $I$ together with a cofinal map $NI \to \mathcal{C}$.
This allows us to calculate colimits using homotopy coherent diagrams.
Examples#
Given a diagram $$ \Delta^1 \times \Delta^1 \to \mathcal{C} $$ corresponding to a picture $$ \begin{CD} X @>>> Y \br @VVV @VVV \br Z @>>> W, \end{CD} $$ we say that this is a pullback if it is a limit diagram for $Z \to W \leftarrow Y$ (after identifying $\Delta^1 \times \Delta^1$ as a left cone for $\Delta^1 \amalg_ {\Delta^0} \Delta^1$) and a pushout if it is a colimit diagram for $Z \leftarrow X \to Y$.
Theorem 17. In the $\infty$-category $\mathcal{S}$ of spaces, pullbacks can be computed using homotopy pullbacks.
For example, $$ \begin{CD} \Omega X @>>> \ast \br @VVV @VVV \br \ast @>>> X \end{CD} $$ is a pullback.
Another example is geometric realization. There is a natural embedding $\mathsf{Set} \hookrightarrow \mathcal{S}$, and so if we have a simplicial set, we can compose them as $$ \Delta^\mathrm{op} \xrightarrow{X} \mathsf{Set} \hookrightarrow \mathcal{S}. $$ It turns out the colimit of this diagram can be computed by geometric realization $$ \lvert X \rvert \simeq \varinjlim X. $$
Adjunctions#
There are several equivalent definitions here as well.
Definition 18 (Definition 1). Two functors $L \colon \mathcal{C} \to \mathcal{D}$ and $R \colon \mathcal{D} \to \mathcal{C}$ between $\infty$ categories are said to be compatible up to homotopy if there are two natural transformations $$ \epsilon \colon \mathrm{id} _ \mathcal{C} \to R \circ L, \quad \delta \colon L \circ R \to \mathrm{id} _ \mathcal{D} $$ satisfying the following conditions:
- the composition $L = L \circ \mathrm{id} _ \mathcal{C} \to L \circ R \circ L \to \mathrm{id} _ \mathcal{D} \circ L = L$ is homotopic to the identity,
- similarly the composition $R = \mathrm{id} _ \mathcal{C} \circ R \to R \circ L \circ R \to R \circ \mathrm{id} _ \mathcal{D} = L$ is homotopic to the identity.
Definition 19 (Definition 2). Two functors $L \colon \mathcal{C} \to \mathcal{D}$ and $R \colon \mathcal{D} \to \mathcal{C}$ between $\infty$ categories are said to be compatible up to homotopy if there is a functorial homotopy equivalence $$ \operatorname{Hom} _ \mathcal{C}(R(x), y) \simeq \operatorname{Hom} _ \mathcal{D}(x, L(y)) $$ for all objects $x \in \mathcal{D}$ and $y \in \mathcal{C}$.
There is also another definition in terms of Grothendieck correspondences.
Theorem 20. The three definitions are “essentially” equivalent.