Contents

category theory

# Contents

## Definition

The slice category or over category $\mathbf{C}/c$ of a category $\mathbf{C}$ over an object $c \in \mathbf{C}$ has

• objects that are all arrows $f \in \mathbf{C}$ such that $cod(f) = c$, and

• morphisms $g: X \to X' \in \mathbf{C}$ from $f:X \to c$ to $f': X' \to c$ such that $f' \circ g = f$.

$C/c = \left\lbrace \array{ X &&\stackrel{g}{\to}&& X' \\ & {}_f \searrow && \swarrow_{f'} \\ && c } \right\rbrace$

The slice category is a special case of a comma category.

There is a forgetful functor $U_c: \mathbf{C}/c \to \mathbf{C}$ which maps an object $f:X \to c$ to its domain $X$ and a morphism $g: X \to X' \in \mathbf{C}/c$ (from $f:X \to c$ to $f': X' \to c$ such that $f' \circ g = f$) to the morphism $g: X \to X'$.

The dual notion is an under category.

## Examples

• If $\mathbf{C} = \mathbf{P}$ is a poset and $p \in \mathbf{P}$, then the slice category $\mathbf{P}/p$ is the down set $\downarrow (p)$ of elements $q \in \mathbf{P}$ with $q \leq p$.

• If $1$ is a terminal object in $\mathbf{C}$, then $\mathbf{C}/1$ is isomorphic to $\mathbf{C}$.

• For $X$ a topological space then the category of covering spaces over $X$ is a full subcategory of the slice category $Top_{/X}$ of the category of topological spaces.

• The fundamental theorem of topos theory states that the slice category over any object in a topos is itself a topos.

• For a monoidal category the slice category over any monoid object is monoidal.

For instance, the slice topos of a given topos over any monoid object is canonically a monoidal topos (see the Example there).

## Properties

If $C$ admits binary coproducts with the fixed object $c$, then the forgetful functor $C/c \to C$ is comonadic. See coreader comonad for more details.

### Relation to codomain fibration

The assignment of overcategories $C/c$ to objects $c \in C$ extends to a functor

$C/(-) : C \to Cat$

Under the Grothendieck construction this functor corresponds to the codomain fibration

$cod : [I,C] \to C$

from the arrow category of $C$. (Note that unless $C$ has pullbacks, this functor is not actually a fibration, though it is always an opfibration.)

###### Proposition

Let

$\mathcal{D} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}$

be a pair of adjoint functors (adjoint ∞-functors), where the category (∞-category) $\mathcal{C}$ has all pullbacks (homotopy pullbacks).

Then:

1. For every object $b \in \mathcal{C}$ there is induced a pair of adjoint functors between the slice categories (slice ∞-categories) of the form

(1)$\mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/b} \mathrlap{\,,}$

where:

• $L_{/b}$ is the evident induced functor (applying $L$ to the entire triangle diagrams in $\mathcal{C}$ which represent the morphisms in $\mathcal{C}_{/b}$);

• $R_{/b}$ is the composite

$R_{/b} \;\colon\; \mathcal{D}_{/{L(b)}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{(R \circ L(b))}} \overset{\;\;(\eta_{b})^*\;\;}{\longrightarrow} \mathcal{C}_{/b}$

of

1. the evident functor induced by $R$;

2. the (homotopy) pullback along the $(L \dashv R)$-unit at $b$ (i.e. the base change along $\eta_b$).

2. For every object $b \in \mathcal{D}$ there is induced a pair of adjoint functors between the slice categories of the form

(2)$\mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)} \mathrlap{\,,}$

where:

• $R_{/b}$ is the evident induced functor (applying $R$ to the entire triangle diagrams in $\mathcal{D}$ which represent the morphisms in $\mathcal{D}_{/b}$);

• $L_{/b}$ is the composite

$L_{/b} \;\colon\; \mathcal{D}_{/{R(b)}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{C}_{/{(L \circ R(b))}} \overset{\;\;(\epsilon_{b})_!\;\;}{\longrightarrow} \mathcal{C}_{/b}$

of

1. the evident functor induced by $L$;

2. the composition with the $(L \dashv R)$-counit at $b$ (i.e. the left base change along $\epsilon_b$).

The first statement appears, in the generality of (∞,1)-category theory, as HTT, prop. 5.2.5.1. For discussion in model category theory see at sliced Quillen adjunctions.
###### Proof

Recall that (this Prop.) the hom-isomorphism that defines an adjunction of functors (this Def.) is equivalently given in terms of composition with

• the adjunction unit $\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)$

• the adjunction counit $\;\;\epsilon_d \colon L \circ R(d) \xrightarrow{\;} d$

as follows:

Using this, consider the following transformations of morphisms in slice categories, for the first case:

(1a)

(2a)

(2b)

(1b)

Here:

• (1a) and (1b) are equivalent expressions of the same morphism $f$ in $\mathcal{D}_{/L(b)}$, by (at the top of the diagrams) the above expression of adjuncts between $\mathcal{C}$ and $\mathcal{D}$ and (at the bottom) by the triangle identity.

• (2a) and (2b) are equivalent expression of the same morphism $\tilde f$ in $\mathcal{C}_{/b}$, by the universal property of the pullback.

Hence:

• starting with a morphism as in (1a) and transforming it to $(2)$ and then to (1b) is the identity operation;

• starting with a morphism as in (2b) and transforming it to (1) and then to (2a) is the identity operation.

In conclusion, the transformations (1) $\leftrightarrow$ (2) consitute a hom-isomorphism that witnesses an adjunction of the first claimed form (1).

The second case follows analogously, but a little more directly since no pullback is involved:

(1a)

(2)

(1b)

In conclusion, the transformations (1) $\leftrightarrow$ (2) consitute a hom-isomorphism that witnesses an adjunction of the second claimed form (2).

###### Remark

The sliced adjunction (Prop. ) in the second form (2) is such that the sliced left adjoint sends slicing morphism $\tau$ to their adjuncts $\widetilde{\tau}$, in that (again by this Prop.):

$L_{/d} \, \left( \array{ c \\ \big\downarrow {}^{\mathrlap{\tau}} \\ R(b) } \right) \;\; = \;\; \left( \array{ L(c) \\ \big\downarrow {}^{\mathrlap{\widetilde{\tau}}} \\ b } \right) \;\;\; \in \; \mathcal{D}_{/b}$

The two adjunctions in admit the following joint generalisation, which is proven HTT, lem. 5.2.5.2. (Note that the statement there is even more general and here we only use the case where $K = \Delta^0$.)

###### Proposition

Let

$\mathcal{C} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D}$

be a pair of adjoint ∞-functors, where the ∞-category $\mathcal{C}$ has all homotopy pullbacks. Suppose further we are given objects $c \in \mathcal{C}$ and $d \in \mathcal{D}$ together with a morphism $\alpha: c \to R(d)$ and its adjunct $\beta:L(c) \to d$.

Then there is an induced a pair of adjoint ∞-functors between the slice ∞-categories of the form

(3)$\mathcal{C}_{/c} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D}_{/d} \mathrlap{\,,}$

where:

• $L_{/c}$ is the composite

$L_{/c} \;\colon\; \mathcal{C}_{/{c}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{D}_{/{L(c)}} \overset{\;\;\beta_!\;\;}{\longrightarrow} \mathcal{D}_{/d}$

of

1. the evident functor induced by $L$;

2. the composition with $\beta:L(c) \to d$ (i.e. the left base change along $\beta$).

• $R_{/d}$ is the composite

$R_{/d} \;\colon\; \mathcal{D}_{/{d}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{R(d)}} \overset{\;\;(\alpha^*\;\;}{\longrightarrow} \mathcal{C}_{/c}$

of

1. the evident functor induced by $R$;

2. the homotopy along $\alpha:c \to R(d)$ (i.e. the base change along $\alpha$).

### Presheaves on over-categories and over-categories of presheaves

Let $C$ be a category, $c$ an object of $C$ and let $C/c$ be the over category of $C$ over $c$. Write $PSh(C/c) = [(C/c)^{op}, Set]$ for the category of presheaves on $C/c$ and write $PSh(C)/Y(c)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(c)$ is the Yoneda embedding.

###### Proposition

There is an equivalence of categories

$e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c) \,.$
###### Proof

The functor $e$ takes $F \in PSh(C/c)$ to the presheaf $F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map $\eta_d: \sqcup_{f \in C(d,c)} F(f) \to C(d,c)$.

A weak inverse of $e$ is given by the functor

$\bar e : PSh(C)/Y(c) \to PSh(C/c)$

which sends $\eta : F' \to Y(C))$ to $F \in PSh(C/c)$ given by

$F : (f : d \to c) \mapsto F'(d)|_c \,,$

where $F'(d)|_c$ is the pullback

$\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.$
###### Example

Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphisms to $C$, i.e. suppose that it factors through the forgetful functor from the over category to $C$:

$F : (C/c)^{op} \to C^{op} \to Set \,.$

Then $F'(d) = \sqcup_{f \in C(d,c)} F(f) = \sqcup_{f \in C(d,c)} F(d) \simeq C(d,c) \times F(d)$ and hence $F ' = Y(c) \times F$ with respect to the closed monoidal structure on presheaves.

For the analogous statement in (∞,1)-category theory see at (∞,1)-category of (∞,1)-presheaves – Interaction with overcategories?.

### Limits and colimits

###### Proposition

A colimit in an over category is computed as a colimit in the underlying category.

Precisely: let $\mathcal{C}$ be a category, $t \in \mathcal{C}$ an object, and $\mathcal{C}/t$ the corresponding overcategory, and $p \colon \mathcal{C}/t \to \mathcal{C}$ the obvious projection.

Let $F \colon D \to \mathcal{C}/t$ be any functor. Then, if it exists, the colimit of $p \circ F$ in $\mathcal{C}$ is the image under $p$ of the colimit over $F$:

$p \big( \underset{\longrightarrow}{\lim} F \big) \;\simeq\; \underset{\longrightarrow}{\lim} (p \circ F)$

and $\underset{\longrightarrow}{\lim} F$ is uniquely characterized by $\underset{\longrightarrow}{\lim} (p \circ F)$ this way.

This statement, and its proof, is the formal dual to the corresponding statement for undercategories, see there.

###### Proposition

For $\mathcal{C}$ a category, $X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ a diagram, $\mathcal{C}_{/X}$ the comma category (the over-category if $\mathcal{D}$ is the point) and $F \;\colon\; K \to \mathcal{C}_{/X}$ a diagram in the comma category, then the limit $\underset{\leftarrow}{\lim} F$ in $\mathcal{C}_{/X}$ coincides with the limit $\underset{\leftarrow}{\lim} F/X$ in $\mathcal{C}$.

For a proof see at (∞,1)-limit here.

### Initial and terminal objects

As a special case of the above discussion of limits and colimits in a slice $\mathcal{C}_{/X}$ we obtain the following statement, which of course is also immediately checked explicitly.

###### Corollary
• If $\mathcal{C}$ has an initial object $\emptyset$, then $\mathcal{C}_{/X}$ has an initial object, given by $\langle \emptyset \to X\rangle$.

• The terminal object of $\mathcal{C}_{/X}$ is $\mathrm{id}_X$.

## References

Formalization in cubical Agda:

Last revised on September 11, 2024 at 13:05:52. See the history of this page for a list of all contributions to it.