Subsection 5.2.6 (01Q7): Elements of Set-Valued Functors

5.2.6 Elements of Set-Valued Functors

Throughout this section, we let $\operatorname{Set}$ denote the category of sets.

Construction 5.2.6.1 (The Category of Elements). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a functor. We define a category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ as follows:

The objects of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ are pairs $(C, x)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $x$ is an element of the set $\mathscr {F}(C)$.
If $(C,x)$ and $(C', x')$ are objects of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$, then a morphism from $(C,x)$ to $(C',x')$ in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is a morphism $f: C \rightarrow C'$ in the category $\operatorname{\mathcal{C}}$ for which the induced map $\mathscr {F}(f): \mathscr {F}(C) \rightarrow \mathscr {F}(C')$ carries $x$ to $x'$.
Composition of morphisms in $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is given by composition of morphisms in $\operatorname{\mathcal{C}}$.

We will refer to $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ as the category of elements of the functor $\mathscr {F}$. Note that the construction $(C,x) \mapsto C$ determines a functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$, which we will refer to as the forgetful functor.

Variant 5.2.6.2. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ be a functor. We define a category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ as follows:

The objects of $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ are pairs $(C, x)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $x$ is an element of the set $\mathscr {F}(C)$.
If $(C,x)$ and $(C', x')$ are objects of $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$, then a morphism from $(C,x)$ to $(C',x')$ in the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ is a morphism $f: C \rightarrow C'$ in the category $\operatorname{\mathcal{C}}$ for which the induced map $\mathscr {F}(f): \mathscr {F}(C') \rightarrow \mathscr {F}(C)$ carries $x'$ to $x$.
Composition of morphisms in $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ is given by composition of morphisms in $\operatorname{\mathcal{C}}$.

We will refer to $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ as the category of elements of the functor $\mathscr {F}$. Note that the construction $(C,x) \mapsto C$ determines a functor $U: \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$, which we will refer to as the forgetful functor.

Remark 5.2.6.3. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a functor. Then we have a canonical isomorphism of categories

\[ (\int _{\operatorname{\mathcal{C}}} \mathscr {F})^{\operatorname{op}} \simeq (\int ^{\operatorname{\mathcal{C}}^{\operatorname{op}}} \mathscr {F}), \]

where $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is the category of elements introduced in Construction 5.2.6.1 and $\int ^{\operatorname{\mathcal{C}}^{\operatorname{op}}} \mathscr {F}$ is the category of elements introduced in Variant 5.2.6.2.

Example 5.2.6.4. Let $X: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ be a simplicial set. Then $\int ^{\operatorname{{\bf \Delta }}} X$ is the category of simplices $\operatorname{{\bf \Delta }}_{X}$ introduced in Construction 1.1.3.9.

Example 5.2.6.5. Let $\operatorname{\mathcal{C}}$ be a category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ denote the functor corepresented by $X$ (given on objects by the formula $h^{X}(Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$). Then the category of elements $\int _{\operatorname{\mathcal{C}}} h^{X}$ can be identified with the coslice category $\operatorname{\mathcal{C}}_{X/}$ of Variant 4.3.1.4. Similarly, if $h_{X}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is the functor represented by $X$ (given on objects by $h_{X}(Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$), then the category of elements $\int ^{\operatorname{\mathcal{C}}} h_{X}$ can be identified with the slice category $\operatorname{\mathcal{C}}_{/X}$.

Remark 5.2.6.6. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a functor. Then the category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ fits into a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d] & \operatorname{Set}_{\ast } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{Set}. } \]

Here $\operatorname{Set}_{\ast }$ denotes the category of pointed sets (see Example 4.2.3.3).

Remark 5.2.6.7. Let $\operatorname{\mathcal{C}}$ be a small category, let $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$ be the category of set-valued functors on $\operatorname{\mathcal{C}}^{\operatorname{op}}$, and let $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$ be the Yoneda embedding (so that $h$ carries each object $C \in \operatorname{\mathcal{C}}$ to the representable functor $h_{C} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \bullet , C)$). For any object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$, the category of elements $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ fits into a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \ar [d] \ar [r] & \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})_{/\mathscr {F}} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{h} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}). } \]

This is essentially a reformulation of Yoneda's lemma (see Corollary 8.4.2.7 for an $\infty $-categorical counterpart).

We now show that, up to isomorphism, every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ can be recovered from the category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ (together with the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$). Let $\operatorname{Cat}$ denote the category of (small) categories.

Proposition 5.2.6.8. Let $\operatorname{\mathcal{C}}$ be a small category. Then:

Construction 5.2.6.1 determines a fully faithful functor

\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{/\operatorname{\mathcal{C}}} \quad \quad \mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}. \]
Variant 5.2.6.2 determines a fully faithful functor

\[ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{/\operatorname{\mathcal{C}}} \quad \quad \mathscr {F} \mapsto \int ^{\operatorname{\mathcal{C}}} \mathscr {F}. \]

Proof. We will prove the first assertion; the second follows by a similar argument. Let $\mathscr {F}$ and $\mathscr {G}$ be functors from $\operatorname{\mathcal{C}}$ to the category of sets, and let $T: (\int _{\operatorname{\mathcal{C}}} \mathscr {F}) \rightarrow (\int _{\operatorname{\mathcal{C}}} \mathscr {G})$ be a functor for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [rr]^-{T} \ar [dr] & & \int _{\operatorname{\mathcal{C}}} \mathscr {G} \ar [dl] \\ & \operatorname{\mathcal{C}}& } \]

is strictly commutative, where the vertical maps are the forgetful functors. We wish to show that there is a unique natural transformation of functors

\[ f: \mathscr {F} \rightarrow \mathscr {G} \quad \quad \{ f_ C: \mathscr {F}(C) \rightarrow \mathscr {G}(C) \} _{C \in \operatorname{\mathcal{C}}} \]

for which the functor $T$ is given on objects by the construction $T(C,x) = (C, f_ C(x) )$. Note that this requirement uniquely determines the function $f_ C: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ for each object $C \in \operatorname{\mathcal{C}}$. We must show that the resulting collection $\{ f_ C \} _{C \in \operatorname{\mathcal{C}}}$ is a natural transformation: that is, for every morphism $u: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the diagram of sets

\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}(C) \ar [r]^-{f_ C} \ar [d]^-{\mathscr {F}(u)} & \mathscr {G}(C) \ar [d]^-{ \mathscr {G}(u)} \\ \mathscr {F}(D) \ar [r]^-{ f_ D) } & \mathscr {G}(D) } \]

is commutative. Fix an element $x \in \mathscr {F}(C)$, so that $u$ can be regarded as a morphism from $(C, x)$ to $(D,\mathscr {F}(u)(x) )$ in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Applying the functor $T$, we deduce that $u$ can also be regarded as a morphism from $(C, f_ C(x))$ to $(D, f_ D( \mathscr {F}(u)(x) ) )$ in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {G}$. It follows that $\mathscr {G}(u)( f_ C(x) ) = f_ D( \mathscr {F}(u)(x) )$, as desired. $\square$

Remark 5.2.6.9. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a functor, and let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the category of elements of $\mathscr {F}$ (Construction 5.2.6.1). Then the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left covering functor, in the sense of Definition 4.2.3.1. This follows from the pullback diagram

of Remark 5.2.6.6, together with Remark 4.2.3.6 and Example 4.2.3.3. We will see in §5.2.7 that the converse is also true: for every left covering functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ and isomorphism $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is compatible with the functor $U$ (Corollary 5.2.7.5). By virtue of Proposition 5.2.6.8, the functor $\mathscr {F}$ is unique up to canonical isomorphism.