7.3 Kan Extensions
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. In practice, it is often possible to reconstruct the functor $F$ (at least up to isomorphism) from its restriction to a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. To make this more precise, it will be convenient to introduce some terminology.
Definition 7.3.0.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. We say that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if, for every object $C \in \operatorname{\mathcal{C}}$, the collection of morphisms $\{ F(u): F(C_0) \rightarrow F(C) \} _{u: C_0 \rightarrow C}$ exhibits $F(C)$ as a colimit of the diagram
\[ ( \operatorname{\mathcal{C}}^{0} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} ) \rightarrow \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}. \]
The central features of Definition 7.3.0.1 can be summarized as follows:
Exercise 7.3.0.2 (Uniqueness of Kan Extensions). Let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors between categories, and suppose that $F$ is left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Show that the restriction map
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Natural transformations from $F$ to $G$} \} \ar [d] \\ \{ \textnormal{Natural transformations from $F|_{\operatorname{\mathcal{C}}^{0}}$ to $G|_{\operatorname{\mathcal{C}}^{0} }$} \} } \]
is a bijection. In particular, the functor $F$ can be recovered (up to canonical isomorphism) from the restriction $F|_{ \operatorname{\mathcal{C}}^{0} }$.
Exercise 7.3.0.3 (Existence of Kan Extensions). Let $\operatorname{\mathcal{C}}$ be a category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $F_0: \operatorname{\mathcal{C}}^{0} \rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Show that the following conditions are equivalent;
- $(1)$
There exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ and satisfies $F|_{ \operatorname{\mathcal{C}}^{0} } = F_0$.
- $(2)$
For every object $C \in \operatorname{\mathcal{C}}$, the diagram
7.12
\begin{equation} \label{equation:Kan-extension-existence} ( \operatorname{\mathcal{C}}^{0} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} ) \rightarrow \operatorname{\mathcal{C}}^{0} \xrightarrow { F_0 } \operatorname{\mathcal{D}}\end{equation}
has a colimit in $\operatorname{\mathcal{D}}$.
Stated more informally, if the diagram (7.12) has a colimit in $\operatorname{\mathcal{D}}$, then that colimit depends functorially on the object $C \in \operatorname{\mathcal{C}}$.
In this section, we adapt the theory of Kan extensions to the $\infty $-categorical setting. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. We will say that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if it satisfies an $\infty $-categorical analogue of the condition appearing in Definition 7.3.0.1, which we formulate in §7.3.2 (see Definition 7.3.2.1). Our main results are $\infty $-categorical counterparts of Exercises 7.3.0.2 and 7.3.0.3, which we prove in §7.3.6 and §7.3.5, respectively (see Corollary 7.3.6.9 and Corollary 7.3.5.8).
For many applications, it will be useful to consider a different generalization of Definition 7.3.0.1, where we replace the inclusion map $\operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ by an arbitrary functor $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$. Suppose we are given functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$, and $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$, together with a natural transformation $\beta : F_0 \rightarrow F \circ \delta $, as indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F} \ar@ {<=}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\beta } & \\ \operatorname{\mathcal{K}}\ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}. } \]
We will say that $\beta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ if, for every object $C \in \operatorname{\mathcal{C}}$, the collection of morphisms $\{ (F(u) \circ \beta _{X}): F_0(X) \rightarrow F(C) \} _{u: \delta (X) \rightarrow C}$ exhibits $F(C)$ as a colimit of the diagram $\operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{K}}\xrightarrow {F_0} \operatorname{\mathcal{D}}$. This notion also has an $\infty $-categorical generalization which we introduce in §7.3.1 (Variant 7.3.1.5), for which we have counterparts of Exercises 7.3.0.2 and 7.3.0.3 (see Propositions 7.3.6.1 and 7.3.5.1). In the special case where $\operatorname{\mathcal{K}}= \operatorname{\mathcal{C}}^{0}$ is a full subcategory of $\operatorname{\mathcal{C}}$ and $\delta $ is the inclusion map, the Kan extension condition guarantees that $\beta $ is an isomorphism, and therefore essentially reduces to the notion of Kan extension introduced in Definition 7.3.0.1 (see Corollary 7.3.2.7 for a precise statement). In §7.3.4 we study a different extreme, where the functor $\delta $ is assumed to be a cocartesian fibration: in this case, the left Kan extension $F$ of a functor $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$ along $\delta $ is given concretely by the formula
\[ F(C) = \varinjlim _{ \delta (X) = C}(F_0(X) ) \]
where the colimit is taken over the fiber $\operatorname{\mathcal{K}}_{C} = \operatorname{\mathcal{K}}\times _{ \operatorname{\mathcal{C}}} \{ C\} $ (see Proposition 7.3.4.1 and Corollary 7.3.4.2).
In §7.3.3 we consider another variant of Definition 7.3.0.1, where we replace colimits in $\operatorname{\mathcal{D}}$ by the more general notion of $U$-colimit for an auxiliary functor $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ (see §7.1.5). The extra generality afforded by the relative setting is quite convenient in practice: for example, in §7.3.6 we show that relative Kan extensions satisfy a universal property (Proposition 7.3.6.7, analogous to Exercise 7.3.0.2) which can be formally deduced from an existence criterion (Proposition 7.3.5.5, analogous to Exercise 7.3.0.3).
In §7.3.8, we study the transitivity properties of Kan extensions. Let $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and suppose we are given full subcategories $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$ such that $F = \overline{F}|_{\operatorname{\mathcal{C}}}$ is left Kan extended from $\operatorname{\mathcal{C}}_0$. We will show that $\overline{F}$ is left Kan extended from $\operatorname{\mathcal{C}}$ if and only if it is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (Corollary 7.3.8.8). Moreover, we prove analogous statements for relative left Kan extensions (Proposition 7.3.8.6) and for Kan extensions along more general functors (Proposition 7.3.8.18). In §7.3.9, we apply these ideas to give a characterization of $U$-colimit diagrams in the special case where $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a cocartesian fibration of $\infty $-categories.
Structure
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Subsection 7.3.1: Kan Extensions along General Functors
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Subsection 7.3.2: Kan Extensions along Inclusions
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Subsection 7.3.3: Relative Kan Extensions
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Subsection 7.3.4: Kan Extensions along Fibrations
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Subsection 7.3.5: Existence of Kan Extensions
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Subsection 7.3.6: The Universal Property of Kan Extensions
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Subsection 7.3.7: Kan Extensions in Functor $\infty $-Categories
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Subsection 7.3.8: Transitivity of Kan Extensions
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Subsection 7.3.9: Relative Colimits for Cocartesian Fibrations