over category (changes) in nLab

Showing changes from revision #51 to #52: Added | Removed | Changed

Contents

Definition

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

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

• morphisms g:X→X′∈Cg: X \to X' \in \mathbf{C} from f:X→cf:X \to c to f′:X′→cf': X' \to c such that f′∘g=ff' \circ g = f.

C/c={X →g X′ f↘ ↙ f′ c} 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:C/c→CU_c: \mathbf{C}/c \to \mathbf{C} which maps an object f:X→cf:X \to c to its domain XX and a morphism g:X→X′∈C/cg: X \to X' \in \mathbf{C}/c (from f:X→cf:X \to c to f′:X′→cf': X' \to c such that f′∘g=ff' \circ g = f) to the morphism g:X→X′g: X \to X'.

The dual notion is an under category.

Examples

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

• If 11 is a terminal object in C\mathbf{C}, then C/1\mathbf{C}/1 is isomorphic to C\mathbf{C}.

• For XX a topological space then the category of covering spaces over XX is a full subcategory of the slice category Top /XTop_{/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 CC admits binary coproducts with the fixed object cc, then the forgetful functor C/c→CC/c \to C is comonadic. See coreader comonad for more details.

Relation to codomain fibration

The assignment of overcategories C/cC/c to objects c∈Cc \in C extends to a functor

C/(−):C→Cat C/(-) : C \to Cat

Under the Grothendieck construction this functor corresponds to the codomain fibration

cod:[I,C]→C cod : [I,C] \to C

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

Slicing of adjoint functors

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

𝒟⊥⟶R⟵L𝒞 \mathcal{D} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}

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

Then:

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

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

   where:

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

   • R /bR_{/b} is the composite

     R /b:𝒟 /L(b)⟶R𝒞 /(R∘L(b))⟶(η b) *𝒞 /b 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 RR;

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

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

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

   where:

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

   • L /bL_{/b} is the composite

     L /b:𝒟 /R(b)⟶L𝒞 /(L∘R(b))⟶(ϵ b) !𝒞 /b 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 LL;

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

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 η c:c→R∘L(c)\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)

• the adjunction counit ϵ d:L∘R(d)→d\;\;\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 ff in 𝒟 /L(b)\mathcal{D}_{/L(b)}, by (at the top of the diagrams) the above expression of adjuncts between 𝒞\mathcal{C} and 𝒟\mathcal{D} and (at the bottom) by the triangle identity.

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

Hence:

• starting with a morphism as in (1a) and transforming it to (2)(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 τ\tau to their adjuncts τ˜\widetilde{\tau}, in that (again by this Prop.):

L /d(c ↓ τ R(b))=(L(c) ↓ τ˜ b)∈𝒟 /b 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}

\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=Δ 0K = \Delta^0.)

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

𝒞⊥⟵R⟶L𝒟 \mathcal{C} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D}

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

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

(3)𝒞 /c⊥⟵R /b⟶L /b𝒟 /d, \mathcal{C}_{/c} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longleftarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longrightarrow}} {\bot} \mathcal{D}_{/d} \mathrlap{\,,}

where:

• L /cL_{/c} is the composite

  L /c:𝒞 /c⟶L𝒟 /L(c)⟶β !𝒟 /d 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 LL;

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

• R /dR_{/d} is the composite

  R /d:𝒟 /d⟶R𝒞 /R(d)⟶(α *𝒞 /c 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 RR;

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

\end{proposition}

Presheaves on over-categories and over-categories of presheaves

See slice of presheaves is presheaves on slice.

Let CC be a category, cc an object of CC and let C/cC/c be the over category of CC over cc. Write PSh(C/c)=[(C/c) op,Set]PSh(C/c) = [(C/c)^{op}, Set] for the category of presheaves on C/cC/c and write PSh(C)/Y(c)PSh(C)/Y(c) for the over category of presheaves on CC over the presheaf Y(c)Y(c), where Y:C→PSh(c)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 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 𝒞 /X\mathcal{C}_{/X} we obtain the following statement, which of course is also immediately checked explicitly.

• over-category

  • enriched over category

  • under category

  • over topos

• slice 2-category

• over (∞,1)-category,

  • model structure on an over category

  • over-(∞,1)-topos

References

Formalization in cubical Agda:

• 1lab: Slice categories