nLab over category

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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 $f' \circ g = f$.

The slice category is a special case of a comma category. Specifically, if $F_c: \mathbf{*} \to \mathbf{C}$ is a constant functor to $c$, then $\mathbf{C}/c := (Id_{\mathbf{C}} \downarrow F_c)$.

The slice constructor $\mathbf{C}/(\cdot): \mathbf{C} \to Cat$ is functorial. A morphism $f: c \to d$ in $\mathbf{C}$ induces a functor between over-categories $F: \mathbf{C}/c \to \mathbf{C}/d$ by “composing” an object of $\mathbf{C}/c$ with $f$. Below is a commutative diagram illustrating the functoriality of $F$: \begin{tikzcd} \bullet && \bullet \ & {\bullet{c}} \ & {\bullet{d}} \arrow[{\phi = F(\phi)}{description}, from=1-1, to=1-3] \arrow[\alpha, from=1-1, to=2-2] \arrow[{F(\alpha)}, from=1-1, to=3-2] \arrow[\beta, from=1-3, to=2-2] \arrow[{F(\beta)}, from=1-3, to=3-2] \arrow[f, from=2-2, to=3-2] \end{tikzcd} The properties of the slice functor are explored further below.

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

Comonadicity

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

Under the Grothendieck construction this functor corresponds to the codomain fibration

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.)

Base change

(Main article: base change) This is another important induced functor between over-categories. If $\mathbf{C}/c$ has finite products, then a morphism $f: d \to c$ in $\mathbf{C}$ induces the base change functor $f^*: \mathbf{C}/c \to \mathbf{C}/d$, which takes an object $\alpha: z \to c$ in $\mathbf{C}/c$ to the projection from the fibered product $y \times_{f,\alpha} d \to_{\pi_1} d$ as an object of $\mathbf{C}/d$. The right adjoint to this functor is (by definition) the dependent product.

Slicing of adjoint functors

\begin{proposition}\label{SliceAdjoints} (sliced adjoints) \linebreak Let

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 $\epsilon_b$).

\end{proposition} 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. \begin{proof} (in 1-category theory; see the proof of Proposition \ref{SliceAdjointsFurtherGeneralized} in the case of (∞,1)-categories.)

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:

\begin{tikzcd} L(c) \ar[rr, f] && d &{\phantom{AAA}}\leftrightarrow{\phantom{AAA}}& c \ar[rr, \eta_c] \ar[rrrr, rounded corners, to path={ (yshift=+12pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8.5pt\tikztotarget.north) (\tikztotarget.north)}] && R \circ L(c) \ar[rr, R(f)] && R(d) \end{tikzcd}

\begin{tikzcd} c \ar[rr, \widetilde f] && R(d) &{\phantom{AAA}}\leftrightarrow{\phantom{AAA}}& L(c) \ar[rr, L(\widetilde{f})] \ar[rrrr, rounded corners, to path={ (yshift=+8pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8.5pt\tikztotarget.north) (\tikztotarget.north)}] && L \circ R(d) \ar[rr, \epsilon_d] && d \end{tikzcd}

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

(1a)

\begin{tikzcd} L(c) \ar[rr, f, dashed] \ar[dr] && d \ar[dl, p] \ & L(b) \end{tikzcd}

(2a)

\begin{tikzcd} c \ar[rr, {\eta_c}] \ar[dr] \ar[rrrr, rounded corners, to path={ (yshift=+30pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+27pt\tikztotarget.north) (\tikztotarget.north)}] && R \circ L(c) \ar[rr, {R(f)}, dashed] \ar[dr] && R(d) \ar[dl, R(p)] \ & b \ar[rr, \eta_b{below}] & & R \circ L(b) \end{tikzcd}

(2b)

\begin{tikzcd} c \ar[rr, dashed] \ar[dr] \ar[rrrr, rounded corners, to path={ (yshift=+12pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8pt\tikztotarget.north) (\tikztotarget.north)}] && \eta_b^\ast\big(R(d)\big) \ar[rr] \ar[dl] \ar[dr, phantom, \mbox{\tiny\rmfamily(pb)}] && R(d) \ar[dl, R(p)] \ & b \ar[rr, \eta_b{below}] & & R \circ L(b) \end{tikzcd}

(1b)

\begin{tikzcd} L(c) \ar[dr] \ar[rrrr, L(\widetilde{f})] \ar[rrrrrr, rounded corners, to path={ (yshift=+8pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8pt\tikztotarget.north) (\tikztotarget.north)}] && && L \circ R(d) \ar[rr, {\epsilon_d}] \ar[dl] && d \ar[dl, p] \ & L(b) \ar[rr, L(\eta_b){below}] \ar[rrrr, rounded corners, to path={ (yshift=-8pt\tikztostart.south) nodebelow{ \scalebox{} } (yshift=-8pt\tikztotarget.south) (\tikztotarget.south)}] & & L \circ R \circ L(b) \ar[rr, \epsilon_{L(b)}{below}] && L(b) \end{tikzcd}

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).

\linebreak

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

(1a)

\begin{tikzcd} c \ar[rr, dashed, f] \ar[dr] && R(d) \ar[dl] \ & R(b) \end{tikzcd}

(2)

\begin{tikzcd} L(c) \ar[rr, dashed, L(f)] \ar[dr] \ar[rrrr, rounded corners, to path={ (yshift=+8pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8pt\tikztotarget.north) (\tikztotarget.north)}] && L \circ R(d) \ar[rr, \epsilon_d{above}] \ar[dl] && d \ar[dl] \ & L \circ R(b) \ar[rr, \epsilon_b{below}] && b \end{tikzcd}

(1b)

\begin{tikzcd} c \ar[rr, \eta_c{above}] \ar[dr] \ar[rrrrrr, rounded corners, to path={ (yshift=+12pt\tikztostart.north) nodeabove{ \scalebox{} } (yshift=+8pt\tikztotarget.north) (\tikztotarget.north)}] && R \circ L(c) \ar[rrrr,R(\widetilde{f})] \ar[dr] && %R \circ L \circ R(d) %\ar[rr, R(\epsilon_d){above}] %\ar[dl] && R(d) \ar[dl] \ & R(b) \ar[rr, \eta_{R(b)}{below}] \ar[rrrr, rounded corners, to path={ (yshift=-8pt\tikztostart.south) nodebelow{ \scalebox{} } (yshift=-8pt\tikztotarget.south) (\tikztotarget.south)}] && R \circ L \circ R(b) \ar[rr, R(\epsilon_b){below}] && R(b) \end{tikzcd}

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

\begin{remark} \label{LeftAdjointOfSlicedAdjunctionFormsAdjuncts} (left adjoint of sliced adjunction forms adjuncts) \linebreak The sliced adjunction (Prop. \ref{SliceAdjoints}) 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.):

\end{remark}

The two adjunctions in \ref{SliceAdjoints} 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 $K = \Delta^0$.)

\begin{proposition}\label{SliceAdjointsGeneralized} (sliced adjoints) \linebreak Let

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

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 $\alpha$).

\end{proposition}

Proposition \ref{SliceAdjointsGeneralized} (as well as HTT, lem. 5.2.5.2) can be deduced from the following general result (applied to the case where $\mathcal{X}$ and $\mathcal{Y}$ are the arrow categories of $\mathcal{C}$ and $\mathcal{D}$):

\begin{proposition}\label{SliceAdjointsFurtherGeneralized} Let \begin{tikzcd} {\mathcal{X}} & {\mathcal{Y}} \ {\mathcal{C}} & {\mathcal{D}} \arrow[L, from=1-1, to=1-2] \arrow[U, from=1-1, to=2-1] \arrow[V, from=1-2, to=2-2] \arrow[F, from=2-1, to=2-2] \end{tikzcd} be a (strictly) commutative diagram of quasicategories (a model of (∞,1)-categories), where $U$ is a cartesian fibration and $V$ is a cocartesian fibration. Suppose that $L$ and $F$ have right adjoints $R$ and $G$, and that the canonical map $U\circ R\to G\circ V$ is the identity natural transformation. Let $\alpha \colon c\to G d$ be a morphism with transpose $\beta \colon F c \to d$. Then the composite

admits a right adjoint, given by the composite

Here the subscripts indicate taking fibers, and $\alpha^*$ and $\beta_!$ are co/cartesian transports for $U$ and $V$.

\end{proposition}

\begin{proof} We show that $\beta_!\circ L_c$ has a right adjoint; the description for the right adjoint follows by inspecting the proof. By HTT, Proposition 5.2.4.1, it suffices to show that for each $y\in \mathcal{Y}_d$, the $\infty$-category $\mathcal{X}_c \times_{\mathcal{Y}_d} \times (\mathcal{Y}_d)_/y$ has a terminal object. For this, set $\mathcal{Y}_\beta =\Delta^1 \times _{\mathcal{D}} \mathcal{Y}$. The retraction $\mathcal{Y}_\beta \to \mathcal{Y}_d$ induces an equivalence $\mathcal{X}_c \times_{\mathcal{Y}_\beta} (\mathcal{Y}_\beta)_{/y} \overset{\simeq}{\to} \mathcal{X}_c \times_{\mathcal{Y}_d} (\mathcal{Y}_d)_{/y}$, so it suffices to show that $\mathcal{X}_c \times_{\mathcal{Y}_\beta} (\mathcal{Y}_\beta)_{/y}$ has a terminal object. This $\infty$-category can be written as $(\mathcal{X}\times_{\mathcal{Y}}\mathcal{Y}_{/y})\times _{\mathcal{C}\times_{\mathcal{D}}\mathcal{D}_{/V y}}\{(c,\beta)\}$. Using the fact that $U$ is a cartesian fibration and $\mathcal{Y}_{/y} \to \mathcal{Y}$ is a right fibration, we find that the projection $\mathcal{X}\times _{\mathcal{Y}}\mathcal{Y}_{/y} \to \mathcal{C}\times _{\mathcal{D}}\mathcal{D}_{/V y}$ is a cartesian fibration (with cartesian edges given by those whose images in $\mathcal{X}$ are cartesian). Moreover, $p$ preserves terminal objects by hypothesis. It follows that the fiber $p^{-1}(c,\alpha)$ has a terminal object, namely, the cartesian transport of the terminal object of $\mathcal{X}\times_{\mathcal{Y}}\mathcal{Y}_{/y}$. The claim follows. \end{proof}

Presheaves on over-categories and over-categories of presheaves

See slice of presheaves is presheaves on slice.

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.

See also functors and comma categories.

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

Limits and colimits

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

For proof

• with formalization in Cubical Agda: see 1lab, arbitrary-limits-in-slices,

• in the generality of (∞,1)-category theory: 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.

Related concepts

• over-category

  • enriched over category

  • under category

  • coslice category / over topos

• slice 2-category

• over (∞,1)-category,

  • model structure on an over category

  • over-(∞,1)-topos

References

Textbook accounts:

• Francis Borceux. Ex. 1.2.7 in: Handbook of Categorical Algebra, Vol 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

• Emily Riehl, Ex. 1.1.3 in: Category Theory in Context, Dover Publications (2017) [[ISBN:9780486809038](https://store.doverpublications.com/products/9780486809038), pdf, webpage]

Formalization in cubical Agda:

• 1lab: Slice categories