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category theory
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Definition
The slice category or over category of a category over an object has
- objects that are all arrows such that , and
- morphisms from to such that .
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 which maps an object to its domain and a morphism (from to such that ) to the morphism .
The dual notion is an under category.
Examples
-
If is a poset and , then the slice category is the down set of elements with .
-
If is a terminal object in , then is isomorphic to .
Properties
Relation to codomain fibration
The assignment of overcategories to objects extends to a functor
C/(-) : C \to Cat
\,,
Under the Grothendieck construction this functor corresponds the the codomain fibration
cod : [I,C] \to C
from the arrow category of . (Note that unless has pullbacks, this functor is not actually a fibration, though it is always an opfibration.)
Adjunctions on overcategories
Proposition
Let
(L \dashv R) : D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}}
C
be a pair of adjoint functors, where the category has all pullbacks.
Then for every object there is induced a pair of adjoint functors between the slice categories
(L/X \dashv R/X)
: D/(L X) \stackrel{\overset{L/X}{\leftarrow}}{\underset{R/X}{\to}}
C/X
where
-
is the evident induced functor;
-
is the composite
R/X : D/{L X} \stackrel{R}{\to} C/{(R L X)} \stackrel{i_{X}^*}{\to}
C/X
of the evident functor induced by with the pullback along the -unit at .
Presheaves on over-categories and over-categories of presheaves
Let be a category, an object of and let be the over category of over . Write for the category of presheaves on and write for the over category of presheaves on over the presheaf , where 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 takes to the presheaf which is equipped with the natural transformation with component map .
A weak inverse of is given by the functor
\bar e : PSh(C)/Y(c) \to PSh(C/c)
which sends to given by
F : (f : d \to c) \mapsto F'(d)|_c
\,,
where 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 does not actually depend on the morphsims to , i.e. suppose that it factors through the forgetful functor from the over category to :
F : (C/c)^{op} \to C^{op} \to Set
\,.
Then and hence with respect to the closed monoidal structure on presheaves.
See also functors and comma categories.
For the analog statement in (∞,1)-category theory see (∞,1)-category of (∞,1)-presheaves -- Interaction with overcategories
at (∞,1)-category of (∞,1)-presheaves.
Limits and colimits
Proposition
A limit in an under category is computed as a limit in the underlying category.
Precisely: let be a category, an object, and the corresponding under category, and the obvious projection.
Let be any functor. Then, if it exists, the limit over in is the image under of the limit over :
p(\lim F) \simeq \lim (p F)
and is uniquely characterized by .
Proof
Over a morphism in the limiting cone over (which exists by assumption) looks like
\array{
&& \lim p F
\\
& \swarrow && \searrow
\\
p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d')
}
By the universal property of the limit this has a unique lift to a cone in the under category over :
\array{
&& t
\\
& \swarrow &\downarrow & \searrow
\\
&& \lim p F
\\
\downarrow & \swarrow && \searrow & \downarrow
\\
p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d')
}
It therefore remains to show that this is indeed a limiting cone over . Again, this is immediate from the universal property of the limit in . For let be another cone over in , then is another cone over in and we get in a universal morphism
\array{
&& t
\\
& \swarrow & \downarrow & \searrow
\\
&& Q
\\
\downarrow & \swarrow &\downarrow & \searrow & \downarrow
\\
&& \lim p F
\\
\downarrow & \swarrow && \searrow & \downarrow
\\
p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d')
}
A glance at the diagram above shows that the composite constitutes a morphism of cones in into the limiting cone over . Hence it must equal our morphism , by the universal property of , and hence the above diagram does commute as indicated.
This shows that the morphism which was the unique one giving a cone morphism on does lift to a cone morphism in , which is then necessarily unique, too. This demonstrates the required universal property of and thus identifies it with .
- Remark: One often says ” reflects limits” to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if is monadic (i.e., has a left adjoint such that the canonical comparison functor is an equivalence), then both reflects and preserves limits. In the present case, the projection is monadic, is essentially the category of algebras for the monad , at least if admits binary coproducts. (Added later: the proof is even simpler: if is the underlying functor for the category of algebras of an endofunctor on (as opposed to algebras of a monad), then reflects and preserves limits; then apply this to the endofunctor above.)
Proposition
For a category, a diagram, the comma category (over-category if is the point) and a diagram in the comma category, then the limit in coincides with the limit in .
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 we obtain the following statement, which of course is also immediately checked explicitly.