Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

4.3.1 Slices of Categories

We begin by discussing the slice construction in a special case.

Construction 4.3.1.1 (Slice Categories over Objects). Let $\operatorname{\mathcal{C}}$ be a category containing an object $S$. We define a category $\operatorname{\mathcal{C}}_{/S}$ as follows:

  • The objects of $\operatorname{\mathcal{C}}_{/S}$ are pairs $(X, f)$, where $X$ is an object of $\operatorname{\mathcal{C}}$ and $f: X \rightarrow S$ is a morphism in $\operatorname{\mathcal{C}}$.

  • If $(X,f)$ and $(Y, g)$ are objects of $\operatorname{\mathcal{C}}_{/S}$, then a morphism from $(X,f)$ to $(Y,g)$ in the category $\operatorname{\mathcal{C}}_{/S}$ is a morphism $u: X \rightarrow Y$ in the category $\operatorname{\mathcal{C}}$ satisfying $f = g \circ u$. In other words, morphisms from $(X,f)$ to $(Y,g)$ are given by commutative diagrams

    \[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{u} \ar [dr]_-{f} & & Y \ar [dl]^-{g} \\ & S & } \]

    in the category $\operatorname{\mathcal{C}}$.

  • Composition of morphisms in the category $\operatorname{\mathcal{C}}_{/S}$ is given by composition of morphisms in the category $\operatorname{\mathcal{C}}$.

We will refer to $\operatorname{\mathcal{C}}_{/S}$ as the slice category of $\operatorname{\mathcal{C}}$ over $S$.

Example 4.3.1.2. Let $\operatorname{Set}$ denote the category of sets, and let $S \in \operatorname{Set}$ be a set. Then the construction

\[ (f: X \rightarrow S) \mapsto \{ X_{s} = f^{-1} \{ s\} \} _{s \in S} \]

induces an equivalence of categories $\operatorname{Set}_{/S} \rightarrow \prod _{s \in S} \operatorname{Set}$.

Remark 4.3.1.3. Let $\operatorname{\mathcal{C}}$ be a category which admits finite limits and let $\ast $ denote a final object of $\operatorname{\mathcal{C}}$. For any object $S \in \operatorname{\mathcal{C}}$, one can adapt the construction of Example 4.3.1.2 to define a functor

\[ F: \operatorname{\mathcal{C}}_{/S} \rightarrow \prod _{s: \ast \rightarrow S} \operatorname{\mathcal{C}}\quad \quad F(X \rightarrow S) = \{ \ast \times _{S} X \} _{s: \ast \rightarrow S}. \]

Motivated by this observation, it is often useful to think of objects of the slice category $\operatorname{\mathcal{C}}_{/S}$ as “families” of objects of $\operatorname{\mathcal{C}}$ which are parametrized by $S$. Beware that the functor $F$ is usually not an equivalence of categories.

Variant 4.3.1.4 (Coslice Categories under Objects). Let $\operatorname{\mathcal{C}}$ be a category containing an object $S$. We define a category $\operatorname{\mathcal{C}}_{S/}$ as follows:

  • The objects of $\operatorname{\mathcal{C}}_{S/}$ are pairs $(X, f)$, where $X$ is an object of $\operatorname{\mathcal{C}}$ and $f: S \rightarrow X$ is a morphism in $\operatorname{\mathcal{C}}$.

  • If $(X,f)$ and $(Y, g)$ are objects of $\operatorname{\mathcal{C}}_{S/}$, then a morphism from $(X,f)$ to $(Y,g)$ in the category $\operatorname{\mathcal{C}}_{/S}$ is a morphism $u: X \rightarrow Y$ in the category $\operatorname{\mathcal{C}}$ satisfying $g = f \circ u$. In other words, morphisms from $(X,f)$ to $(Y,g)$ are given by commutative diagrams

    \[ \xymatrix@R =50pt@C=50pt{ & S \ar [dl]_-{f} \ar [dr]^-{g} & \\ X \ar [rr]^-{u} & & Y } \]

    in the category $\operatorname{\mathcal{C}}$.

  • Composition of morphisms in the category $\operatorname{\mathcal{C}}_{S/}$ is given by composition of morphisms in the category $\operatorname{\mathcal{C}}$.

We will refer to $\operatorname{\mathcal{C}}_{S/}$ as the coslice category of $\operatorname{\mathcal{C}}$ under $S$.

Remark 4.3.1.5. Variant 4.3.1.4 is formally dual to Construction 4.3.1.1. More precisely, if $S$ is an object of a category $\operatorname{\mathcal{C}}$, then we have a canonical isomorphism of categories

\[ (\operatorname{\mathcal{C}}_{/S})^{\operatorname{op}} \simeq (\operatorname{\mathcal{C}}^{\operatorname{op}})_{S/}, \]

where we view $S$ also as an object of the opposite category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

Remark 4.3.1.6 (Slice Categories as Comma Categories). Let $\operatorname{\mathcal{C}}$ be a category and let $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$ denote the arrow category of $\operatorname{\mathcal{C}}$, so that the elements $0,1 \in [1]$ determine evaluation functors

\[ \operatorname{ev}_0: \operatorname{Fun}([1], \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}\quad \quad \operatorname{ev}_1: \operatorname{Fun}([1], \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}. \]

For each object $S \in \operatorname{\mathcal{C}}$, the slice category $\operatorname{\mathcal{C}}_{/S}$ can be identified with the fiber of the evaluation functor $\operatorname{ev}_{1}$ over $S$, and the coslice category $\operatorname{\mathcal{C}}_{S/}$ can be identified with the fiber of the evaluation functor $\operatorname{ev}_0$ over $S$. That is, we have pullback diagrams

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{S/} \ar [r] \ar [d] & \operatorname{Fun}([1], \operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{ev}_0 } & \operatorname{\mathcal{C}}_{/S} \ar [r] \ar [r] \ar [d] & \operatorname{Fun}([1], \operatorname{\mathcal{C}}) \ar [d] \\ \{ S\} \ar [r] & \operatorname{\mathcal{C}}& \{ S\} \ar [r] & \operatorname{\mathcal{C}}. } \]

In other words, we can identify $\operatorname{\mathcal{C}}_{/S}$ with the comma category $\operatorname{\mathcal{C}}\downarrow S$ of Notation 2.1.4.19 (here we identify the object $S$ with the constant functor $[0] \rightarrow \operatorname{\mathcal{C}}$ taking the value $S$), and $\operatorname{\mathcal{C}}_{S/}$ with the comma category $S \downarrow \operatorname{\mathcal{C}}$.

For many applications it is useful to consider a generalization of Construction 4.3.1.1, which associates a slice category $\operatorname{\mathcal{C}}_{/F}$ to an arbitrary diagram $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ (instead of a single object $S \in \operatorname{\mathcal{C}}$).

Construction 4.3.1.7 (Slice Categories over Diagrams). Let $\operatorname{\mathcal{K}}$ and $\operatorname{\mathcal{C}}$ be categories. For each object $C \in \operatorname{\mathcal{C}}$, we let $\underline{C}: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ denote the associated constant functor (carrying each object of $\operatorname{\mathcal{K}}$ to the object $C$ and each morphism of $\operatorname{\mathcal{K}}$ to the identity morphism $\operatorname{id}_{C}$). The construction $C \mapsto \underline{C}$ determines a functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})$.

For every functor $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$, we let $\operatorname{\mathcal{C}}_{/F}$ denote the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})} \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})_{/F}$, where $\operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})_{/F}$ is the slice category of Construction 4.3.1.1. Similarly, we let $\operatorname{\mathcal{C}}_{F/}$ denote the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})_{F/}$, where $\operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})_{F/}$ denotes the coslice category of Variant 4.3.1.4. We will refer to $\operatorname{\mathcal{C}}_{/F}$ as the slice category of $\operatorname{\mathcal{C}}$ over $F$, and to $\operatorname{\mathcal{C}}_{F/}$ as the coslice category of $\operatorname{\mathcal{C}}$ under $F$.

Remark 4.3.1.8. The slice and coslice constructions of Construction 4.3.1.7 are mutually dual. More precisely, if $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ is a functor between categories and $F^{\operatorname{op}}: \operatorname{\mathcal{K}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is the induced functor between opposite categories, then we have canonical isomorphisms

\[ (\operatorname{\mathcal{C}}_{/F})^{\operatorname{op}} \simeq (\operatorname{\mathcal{C}}^{\operatorname{op}})_{F^{\operatorname{op}}/} \quad \quad ( \operatorname{\mathcal{C}}_{F/} )^{\operatorname{op}} \simeq ( \operatorname{\mathcal{C}}^{\operatorname{op}} )_{ / F^{\operatorname{op}} }. \]

Example 4.3.1.9. Let $[0]$ denote the category having a single object and a single morphism. For any category $\operatorname{\mathcal{C}}$, the diagonal map

\[ \delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( [0], \operatorname{\mathcal{C}}) \quad \quad S \mapsto \underline{S} \]

is an isomorphism of categories. It follows that, for any object $S \in \operatorname{\mathcal{C}}$, we have canonical isomorphisms

\[ \operatorname{\mathcal{C}}_{/S} \simeq \operatorname{\mathcal{C}}_{/ \underline{S}} \quad \quad \operatorname{\mathcal{C}}_{S/} \simeq \operatorname{\mathcal{C}}_{ \underline{S} / }. \]

Consequently, we can view Construction 4.3.1.1 and Variant 4.3.1.4 as special cases of Construction 4.3.1.7.

Remark 4.3.1.10. Let $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. Using Remark 4.3.1.6, we see that the slice and coslice categories of Construction 4.3.1.7 are given by the formulae

\[ \operatorname{\mathcal{C}}_{/F} \simeq \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} \times \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \operatorname{Fun}( [1] \times \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 1\} \times \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \{ F\} , \]
\[ \operatorname{\mathcal{C}}_{F/} \simeq \{ F \} \times _{ \operatorname{Fun}( \{ 0\} \times \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \operatorname{Fun}( [1] \times \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 1\} \times \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}. \]

Remark 4.3.1.11. Let $\operatorname{\mathcal{C}}$ be a category and let $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a diagram in $\operatorname{\mathcal{C}}$. If $F$ admits a limit $S = \varprojlim _{I \in \operatorname{\mathcal{K}}} F(I)$, then the slice category $\operatorname{\mathcal{C}}_{/F}$ is isomorphic to $\operatorname{\mathcal{C}}_{/S}$. Similarly, if $F$ admits a colimit $S' = \varinjlim _{I \in \operatorname{\mathcal{K}}} F(I)$, then the coslice category $\operatorname{\mathcal{C}}_{F/}$ is isomorphic to $\operatorname{\mathcal{C}}_{S/}$. We will return to this point later (see Remark ).