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.
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.
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 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.)
(sliced adjoints)
Let
be a pair of adjoint functors (adjoint ∞-functors), where the category (∞-category) $\mathcal{C}$ has all pullbacks (homotopy pullbacks).
Then:
For every object $b \in \mathcal{C}$ there is induced a pair of adjoint functors between the slice categories (slice ∞-categories) of the form
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
of
the evident functor induced by $R$;
the (homotopy) pullback along the $(L \dashv R)$-unit at $b$ (i.e. the base change along $\eta_b$).
For every object $b \in \mathcal{D}$ there is induced a pair of adjoint functors between the slice categories of the form
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
of
the evident functor induced by $L$;
the composition with the $(L \dashv R)$-counit at $b$ (i.e. the left base change along $\epsilon_b$).
(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 $\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)$
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).
(left adjoint of sliced adjunction forms adjuncts)
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.):
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.
There is an equivalence of categories
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
which sends $\eta : F' \to Y(C))$ to $F \in PSh(C/c)$ given by
where $F'(d)|_c$ is the pullback
Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphsims to $C$, i.e. suppose that it factors through the forgetful functor from the over category to $C$:
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.
See also functors and comma categories.
For the analogous statement in (∞,1)-category theory see at (∞,1)-category of (∞,1)-presheaves – Interaction with overcategories?.
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 $F$:
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.
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.
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.
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 $\mathrm{id}_X$.
Last revised on October 9, 2021 at 04:12:51. See the history of this page for a list of all contributions to it.