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7.1 Limits and Colimits

Let $\operatorname{\mathcal{K}}$ and $\operatorname{\mathcal{C}}$ be categories. For every object $X \in \operatorname{\mathcal{C}}$, let $\underline{X}$ denote the constant functor from $\operatorname{\mathcal{K}}$ to $\operatorname{\mathcal{C}}$, carrying each object of $\operatorname{\mathcal{K}}$ to $X$ and each morphism of $\operatorname{\mathcal{K}}$ to the identity morphism $\operatorname{id}_{X}$. If $U: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ is an arbitrary functor, then a limit of $U$ is an object of $\operatorname{\mathcal{C}}$ which represents the functor $X \mapsto \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}},\operatorname{\mathcal{C}})}( \underline{X}, U )$. This can be formulated more precisely as follows:

Definition 7.1.0.1. Let $U: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. Let $Y$ be an object of $\operatorname{\mathcal{C}}$ and let $\alpha : \underline{Y} \rightarrow U$ be a natural transformation of functors. We say that the natural transformation $\alpha $ exhibits $Y$ as a limit of $U$ if the following condition is satisfied:

$(\ast )$

For every object $X \in \operatorname{\mathcal{C}}$, composition with $\alpha $ induces a bijection from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to the set $\operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{K}},\operatorname{\mathcal{C}})}( \underline{X}, U )$ of natural transformations from $\underline{X}$ to $U$.

We say that a natural transformation $\beta : U \rightarrow \underline{Y}$ exhibits $Y$ as a colimit of $U$ if the following dual condition is satisfied:

$(\ast ')$

For every object $Z \in \operatorname{\mathcal{C}}$, composition with $\beta $ induces a bijection from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ to the set $\operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{K}},\operatorname{\mathcal{C}})}( U, \underline{Z} )$ of natural transformations of $U$ to $\underline{Z}$.

Our goal in the section is to generalize the notions of limit and colimit to the setting of $\infty $-categories. Let us begin by discussing an easy special case.

Definition 7.1.0.2. Let $\operatorname{\mathcal{C}}$ be a category. An initial object of $\operatorname{\mathcal{C}}$ is an object $Y \in \operatorname{\mathcal{C}}$ with the property that, for every object $Z \in \operatorname{\mathcal{C}}$, there is a unique morphism $Y \rightarrow Z$: that is, the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ has exactly one element. A final object of $\operatorname{\mathcal{C}}$ is an object $Y \in \operatorname{\mathcal{C}}$ with the property that, for every object $X \in \operatorname{\mathcal{C}}$, the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ has exactly one element.

Remark 7.1.0.3. Let $\operatorname{\mathcal{C}}$ be a category. Then an object $Y \in \operatorname{\mathcal{C}}$ is initial if and only if it is a colimit of the unique diagram $\emptyset \rightarrow \operatorname{\mathcal{C}}$; here $\emptyset $ denotes the category with no objects. Similarly, an object $Y \in \operatorname{\mathcal{C}}$ is final if and only if it is a limit of the unique diagram $\emptyset \rightarrow \operatorname{\mathcal{C}}$.

Definition 7.1.0.2 has an obvious $\infty $-categorical counterpart. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, we say that an object $Y \in \operatorname{\mathcal{C}}$ is initial if, for every object $Z \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is contractible (Definition 7.1.1.1). Similarly, we say that $Y \in \operatorname{\mathcal{C}}$ is final if, for every object $X \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is contractible. These conditions have a number of equivalent formulations, which we discuss and compare in §7.1.1

To define $\infty $-categorical notions of limit and colimit for general diagrams, it will be convenient to consider a reformulation of Definition 7.1.0.1.

Exercise 7.1.0.4. Let $U: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $\operatorname{\mathcal{C}}_{/U}$ be the slice category of Construction 4.3.1.8. By virtue of Remark 4.3.1.11, the objects of $\operatorname{\mathcal{C}}_{/U}$ can be identified with pairs $(Y, \alpha )$, where $Y$ is an object of $\operatorname{\mathcal{C}}$ and $\alpha : \underline{Y} \rightarrow U$ is a natural transformation of functors. Show that $\alpha $ exhibits $Y$ as a limit of $U$ (in the sense of Definition 7.1.0.1) if and only if the pair $(Y, \alpha )$ is a final object of the category $\operatorname{\mathcal{C}}_{/U}$. Similarly, show that a natural transformation $\beta : U \rightarrow \underline{Y}$ exhibits $Y$ as a colimit of $U$ if and only if the pair $(Y,\beta )$ determines an initial object of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{U/}$.

In the $\infty $-categorical setting, we will take Exercise 7.1.0.4 as our starting point. Given an $\infty $-category $\operatorname{\mathcal{C}}$ and a diagram $U: K \rightarrow \operatorname{\mathcal{C}}$, we define a limit of $U$ to be an object $\varprojlim (U) \in \operatorname{\mathcal{C}}$ which can be written as the image of a final object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/U}$ (Definition 7.1.3.1). It is not difficult to see that if such an object exists, then it is uniquely determined up to isomorphism (Notation 7.1.3.8). In §7.1.3, we show that if every diagram $U: K \rightarrow \operatorname{\mathcal{C}}$ admits a limit, then the construction $U \mapsto \varprojlim (U)$ can be regarded as a functor of $\infty $-categories $\varprojlim : \operatorname{Fun}(K,\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ which is characterized (up to isomorphism) by the requirement that it is right adjoint to the diagonal functor

\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \quad \quad X \mapsto \underline{X} \]

(Proposition 7.1.3.10). To prove this, we use a general criterion for existence of right adjoints in terms of representable right fibrations (Corollary 7.1.2.14), which we formulate and prove in §7.1.2.

Let us now regard the simplicial set $K$ as fixed. In §7.1.4, we study the dependence of $K$-indexed limits (and colimits) on the ambient $\infty $-category in which they are formed. We say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $K$-indexed limits if, for every diagram $U: K \rightarrow \operatorname{\mathcal{C}}$, the induced functor $\operatorname{\mathcal{C}}_{/U} \rightarrow \operatorname{\mathcal{D}}_{/(F \circ U)}$ carries final objects of $\operatorname{\mathcal{C}}_{/U}$ to final objects of $\operatorname{\mathcal{D}}_{ /(F \circ U)}$ (Definition 7.1.4.1). We illustrate this concept with a few examples:

  • If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories, then it preserves $K$-indexed limits (and colimits) for every simplicial set $K$ (Proposition 7.1.4.4).

  • Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $K$-indexed limits, and let $f: A \rightarrow \operatorname{\mathcal{C}}$ be any morphism of simplicial sets. Then the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ also admits $K$-indexed limits, and the projection map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed limits (Corollary 7.1.4.17).

  • Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right fibration of $\infty $-categories, and suppose that $\operatorname{\mathcal{D}}$ admits $K$-indexed limits. If $K$ is weakly contractible, then the $\infty $-category $\operatorname{\mathcal{C}}$ also admits $K$-indexed limits, and the right fibration $F$ preserves $K$-indexed limits (Corollary 7.1.5.12). This is a consequence of a transitivity statement for relative limits (Proposition 7.1.5.9), which we formulate in §7.1.5.

  • Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. For every simplicial set $K$, the functor $G$ preserves $K$-indexed limits, and the functor $F$ preserves $K$-indexed colimits (Corollary 7.1.4.19).

Structure

  • Subsection 7.1.1: Initial and Final Objects of $\infty $-Categories
  • Subsection 7.1.2: Representable Fibrations
  • Subsection 7.1.3: Limits and Colimits
  • Subsection 7.1.4: Preservation of Limits and Colimits
  • Subsection 7.1.5: Relative Limits and Colimits