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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 $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ 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 $F$ 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 $\operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{K}},\operatorname{\mathcal{C}})}( \underline{X}, F )$ of natural transformations from $\underline{X}$ to $U$.

Our goal in the section is to generalize the notion of limit (and the dual notion of colimit) to the setting of $\infty $-categories. We begin in §7.1.1 by studying an $\infty $-categorical counterpart of Definition 7.1.0.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be an arbitrary simplicial set. If $F: K \rightarrow \operatorname{\mathcal{C}}$ is any diagram, then we say that a natural transformation $\alpha : \underline{Y} \rightarrow F$ exhibits $Y$ as a limit of $F$. For every object $X \in \operatorname{\mathcal{C}}$, composition with $\alpha $ induces a map of Kan complexes

\[ \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X , Y ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) }( \underline{X}, F ), \]

which is well-defined up to homotopy. We will say that $\alpha $ exhibits $Y$ as a limit of $F$ if this map is a homotopy equivalence for each $X \in \operatorname{\mathcal{C}}$ (Definition 7.1.1.1).

In §7.1.2, we study the special case where the simplicial set $K = \emptyset $ is empty. We say that an object $Y$ of an $\infty $-category $\operatorname{\mathcal{C}}$ is final if the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is contractible, for every object $X \in \operatorname{\mathcal{C}}$. (Definition 7.1.2.1). This can be regarded as a special case of the notion of limit: an object $Y \in \operatorname{\mathcal{C}}$ is final if and only if it is a limit of the unique diagram $\emptyset \rightarrow \operatorname{\mathcal{C}}$ (Remark 7.1.2.3). Conversely, if $K$ is an arbitrary simplicial set equipped with a diagram $F: K \rightarrow \operatorname{\mathcal{C}}$, we will show that a natural transformation $\alpha : \underline{Y} \rightarrow F$ exhibits $Y$ as a limit of $F$ if and only if it is final when viewed as an object of $\infty $-category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} $ (Proposition 7.1.4.1).

To make use of the preceding characterization, it is convenient to introduce some terminology. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories. We say that $U$ is representable if the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ has a final object, that $U$ is represented by an object $Y \in \operatorname{\mathcal{C}}$ if $Y$ is the image of a final object of $\widetilde{\operatorname{\mathcal{C}}}$ (in this case, the object $Y$ is uniquely determined up to isomorphism: see Proposition 7.1.3.5). This notion has many equivalent formulations, which we will study in §7.1.3. It follows that an object $Y \in \operatorname{\mathcal{C}}$ is a limit of a diagram $F: K \rightarrow \operatorname{\mathcal{C}}$ if and only if it represents the right fibration

\[ \pi : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} \rightarrow \operatorname{\mathcal{C}} \]

given by projection onto the first factor. Recall that $\pi $ is equivalent (but not isomorphic) to the right fibration $\operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}$ (Theorem 4.6.4.16). In §7.1.4, we use this observation to reformulate the notion of limit: an object $Y$ is a limit of a diagram $F: K \rightarrow \operatorname{\mathcal{C}}$ if there exists a diagram $\overline{F}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ which carries the cone point of $K^{\triangleleft }$ to the object $Y$ and which is final when viewed as an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/F}$ (Corollary 7.1.4.2). In this situation, we will refer to $\overline{F}$ as a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$ (Definintion 7.1.4.4).

In §7.1.5, we study the dependence of $K$-indexed limits on the ambient $\infty $-category in which they are formed. We say that a functor of $\infty $-categories $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $K$-indexed limits if, for every diagram $F: K \rightarrow \operatorname{\mathcal{C}}$, the induced functor $\operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{D}}_{/(G \circ F)}$ carries final objects of $\operatorname{\mathcal{C}}_{/F}$ to final objects of $\operatorname{\mathcal{D}}_{ /(G \circ F)}$ (Definition 7.1.5.1). We illustrate the concept in this section with a few elementary examples (and will encounter many others later in this book):

  • If $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories, then it preserves $K$-indexed limits for every simplicial set $K$ (Proposition 7.1.5.5).

  • 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 7.1.5.18).

  • Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ 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 7.1.7.18).

For many applications, it will be useful to consider a relative version of the theory of limit diagrams. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that an object $Y \in \operatorname{\mathcal{C}}$ is $U$-final if, for every object $X \in \operatorname{\mathcal{C}}$, the functor $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Y) )$ (Definition 7.1.6.1). We say that a diagram $\overline{F}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ with restriction $F = \overline{F}|_{K}$ is a $U$-limit diagram if it is $U_{/F}$-final when regarded as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/F}$, where $U_{/F}: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{D}}_{ / (U \circ F)}$ is the projection map (Definition 7.1.7.1). In the special case $\operatorname{\mathcal{D}}= \Delta ^0$, we recover the usual notions of final object and limit diagram, respectively (Examples 7.1.6.2 and 7.1.7.3). Moreover, most of the basic features of final objects and limit diagrams have counterparts in the relative setting, which we summarize in §7.1.6 and §7.1.7. Even if one is ultimately interested in the “absolute” theory, the language of relative limits is a useful tool: we illustrate this point in §7.1.8 by using the relative language to study limits in an $\infty $-category of the form $\operatorname{Fun}( B, \operatorname{\mathcal{C}})$ (our main result is that, under mild assumptions, such limits can be computed pointwise: see Proposition 7.1.8.1)

Remark 7.1.0.2. The preceding discussion has centered around the theory of limits. There is also a dual theory of colimits in the $\infty $-categorical setting, which can be obtained by passing to opposite $\infty $-categories. Every assertion concerning limits has a counterpart for colimits (and vice versa). We will often use this implicitly (for example, by stating a result only for colimits but later using the dual assertion for limits).

Structure

  • Subsection 7.1.1: Limits and Colimits in $\infty $-Categories
  • Subsection 7.1.2: Initial and Final Objects of $\infty $-Categories
  • Subsection 7.1.3: Representable Fibrations
  • Subsection 7.1.4: Limits and Colimit Diagrams
  • Subsection 7.1.5: Preservation of Limits and Colimits
  • Subsection 7.1.6: Relative Initial and Final Objects
  • Subsection 7.1.7: Relative Limits and Colimits
  • Subsection 7.1.8: Limits and Colimits of Functors