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in the previous section we had seen how every category is embedded into its category of presheaves as the full subcategory of representable functors
conversely, given a presheaf on $C$, it is of interest to ask if it is actually an ordinary object of $C$, i.e. if it is representable;
an important class of examples we considered were presheaves induced by a functor $L : C \to D$ and an object $d \in D$ of the form $D(L(-),d) : C^{op} \to Set$: when these are representable for all $d$, then the assignment $d \mapsto R(d)$ of representing objects $R(d) \in C$ is functorial – the functor $R : D \to C$ is the right adjoint of the functor $F$.
of particular importance are adjoints to precomposition functors on functor categories – such adjoints are called
we look at some examples and discuss the interaction with the Yoneda embedding: the Yoneda extension of functors from categories to presheaf categories;
these concepts are at the heart of category and sheaf theory, they are the standard tools that will be used throughout.
Idea There is a natural notion of morphism from a single object into a diagram of objects. The limit of the diagram is the classifying object of such maps: it is a single object which subsumes the entire diagram as far as maps into it are concerned.
The dual notion, that of a colimit to be discussed below, is about objects with subsume an entire diagram as far as maps out of the diagram are concerned.
Let $D$ be a small category and $C$ any category. A functor $F : D^{op} \to C$ may be thought of as a diagram of shape $D$ in $C$.
Definition
Every object $c \in C$ defines the constant diagram of shape $D$ in $C$:
The assignment of constant diagrams to objects in $C$ constitutes a functor
from $C$ to the functor category from $D^{op}$ to $C$.
Definition
Given categories $C$ and $D$ as above, the limit operation
is, if it exists, the right adjoint to $const : C \to [D^{op}, C]$;
the colimit operation
is, if it exists, the left adjoint.
Terminology
If the limit $lim : [D^{op}, C] \to C$ exists, we say that “$C$ admits/has limits of shape $D$”.
if all limits of shape $D$ for $D$ a small category exist in $C$, we say that that $C$ admits small limits.
if all limits of shape $D$ for $D$ a finite category exist in $C$, we say that that $C$ admits finite limits.
Frequently neither adjoint to the constant functor exists, but one is still interested in the local adjoints: the (co)limit of a fixed diagram $F : D^{op} \to C$ may exists and be of interest, even if the limit construction on all of $[D^{op}, C]$ does not exist.
We unwrap what the above definition of limit means more concretely in terms of components.
By the very definition of right adjoint, the limit $lim F$ of a functor $F \in [C^{op}, D]$ has (and is defined by) the property that there is a bijection of sets
naturally for all $F$. The set on the left is that of natural transformations $\eta : const_c \to F$. Recall that one such transformation consists of a set of morphisms $\eta_d : c \to F(d)$ for each $d \in D$, such that for all morphisms $f : d \leftarrow d'$ in $D$ the diagram
commutes. Such a diagram is called a cone over $F$ with tip $c$, for the obvious reason.
Remark: limit as classifying space
By definition, morphisms from any $c$ into $lim F$ are in bijection with cones over $F$ with tip $c$. This means that $lim F$ behaves like a classifying space for cones over $F$: the object $lim F$ subsumes in a way the entire diagram $F$ in one object, as far as probes into the diagram are concerned.
But more is true: the object $lim F$ is not just an arbitrary object, but itself canonically the tip of a cone:
by setting $c = lim F$ the defining bijection becomes
For this case there is a singled-out element on the left: the identity morphism $Id_{lim_F} \in C(lim F , lim F)$. Its image under the bijection hence defines a singled out cone $\eta^u : const_{lim F} \to F$.
Moreover, for any other cone $\eta^u : const_c \to F$ as above, with classifying morphism $f : c \to lim F$, the naturality of the defining bijection
is seen to imply that $\eta$ factors uniquely through $\eta^u$ as
the limit of the empty diagram $D = \emptyset$ in $C$ is, if it exists the terminal object;
if $D$ is a discrete category, i.e. a category with only identity morphisms, then a diagram $F : D \to C$ is just a collection $c_i$ of objects of $C$. Its limit is the product $\prod_i c_i$ of these.
if $D = \{a \stackrel{\to}{\to} b\}$ then $lim F$ is the equalizer of the two morphisms $F(b) \to F(a)$.
the limit over a diagram $\array{&& C \\ && \downarrow \\ A &\to& B}$ is a pullback or fiber product.
Limits in Set are hom-sets
For $F : D^{op} \to Set$ any functor and $const_{*} : D^{op} \to Set$ the functor constant on the point, the limit of $F$ is the hom-set
in the functor category, i.e. the set of natural transformations from the constant functor into $F$.
Covariant Hom commutes with limits
For $C$ a locally small category, for $F : D^{op} \to C$ a functor and writing $C(c, F(-)) : D^{op} \to Set$, we have
Proposition – limits in functor categories are computed pointwise
Let $D$ be a small category and let $D'$ be any category. Let $C$ be a category which admits limits of shape $D$. Write $[D',C]$ for the functor category. Then * $[D',C]$ admits $D$-shaped limits; * these limits are computed objectwise (“pointwise”) in $C$: for $F : D^{op} \to [D',C]$ a functor we have for all $d' \in D'$ that $(lim F)(d') \simeq lim (F(-)(d'))$. Here the limit on the right is in $C$.
Proposition – small limits commute with small limits
Let $D$ and $D'$ be small catgeories and let $C$ be a category which admits limits of shape $D$ as well as limits of shape $D'$. Then these limits commute with each other, in that
for $F : D^{op} \times {D'}^{op} \to C$ a functor , with corresponding induced functors $F_D : {D'}^{op} \to [D^{op},C]$ and $F_{D'} : {D}^{op} \to [{D'}^{op},C]$, then
Proposition – right adjoints commute with limits
Let $R : C \to C'$ be a functor that is right adjoint to some functor $L : C' \to C$. Let $D$ be a small category such that $C$ admits limits of shape $D$. Then $R$ commutes with $D$-shaped limits in $C$ in that
for $F : D^{op} \to C$ some diagram, we have
Proof
Using the adjunction isomorphism and the above fact that Hom commutes with limits, one obtains for every $c' \in C'$
Since this holds naturally for every $c'$, the Yoneda lemma, corollary II on uniquenes of representing objects implies that $R (lim F) \simeq lim (G \circ F)$.
limits are equalizers of products
The limit of $F : D^{op} \to C$ is always a subobject of the product of the $F(d)$, namely the equalizer of
and
In particular therefore, a category has all limits already if it has all products and equalizers.
The limit over a Set-valued functor $F : D^{op} \to Set$ is a subset of the product $\Pi_{d \in Obj(d)} F(d)$ of all objects: $lim F = \left\{ (s_d)_d \in \prod_d F(d) | for all (d \stackrel{f}{\to} d') : F(f)(s_{d'}) = s_d \right\}$.
Here consider limits of functors $F : D^{op} \to PSh(C)$ with values in the category of presheaves on a small category $C$.
Proposition
Limits of presheaves are computed objectwise:
Here on the right the limit is over the functor $F(-)(c) : D^{op} \to Set$.
Proposition
The Yoneda embedding $Y : C \to PSh(C)$ commutes with small limits:
Let $F : D^{op} \to C$ be a functor, then
(if either limit exists).
warning: the Yoneda embedding does not commute with colimits. See below.
Colimits behave exactly dual to limits: a colimit in $C$ is a limit in $C^{op}$. Still, since they are important, we dicuss colimits in parallel to the discussion of limits above.
More precisely:
for $F : D \to C$ a functor and $F^{op} : D^{op} \to C^{op}$ its opposite, we have
This becomes clear after the following.
We unwrap what the above definition of colimit means more concretely in terms of components.
By the very definition of left adjoint, the colimit $colim F$ of a functor $F \in [C, D]$ has (and is defined by) the property that there is a bijection of sets
naturally for all $F$. The set on the left is that of natural transformations $\eta : F \to const_c$. Recall that one such transformation consists of a set of morphisms $\eta_d : F(d) \to c$ for each $d \in D$, such that for all morphisms $f : d \to d'$ in $D$ the diagram
commutes. Such a diagram is called a cocone under $F$ with tip $c$.
But more is true: the object $colim F$ is not just an arbitrary object, but itself canonically the tip of a cocone:
by setting $c = colim F$ the defining bijection becomes
For this case there is a singled-out element on the left: the identity morphism $Id_{colim_F} \in C(colim F , colim F)$. Its image under the bijection defines a singled out cocone $\eta^u : F \to const_{colim F}$.
Moreover, for any other cocone $\eta^u : F \to const_c$ as above, with co-classifying morphism $f : colim F \to c$, the naturality of the defining bijection implies that $\eta$ factors uniquely through $\eta^u$ as
In components:
the colimit of the empty diagram $D = \emptyset$ in $C$ is, if it exists the initial object;
if $D$ is a discrete category, i.e. a category with only identity morphisms, then a diagram $F : D \to C$ is just a collection $c_i$ of objects of $C$. Its colimit is the coproduct $\coprod_i c_i$ of these.
if $D = \{a \stackrel{\to}{\to} b\}$ then $lim F$ is the coequalizer of the two morphisms $F(b) \to F(a)$.
the limit over a diagram $\array{B &\to& C \\ \downarrow && \\ A }$ is a pushout.
The properties of colimits are of course dual to those of limits. It is still worthwhile to make some of them explicit.
Contravariant Hom sends colimits to limits
For $C$ a locally small category, for $F : D \to C$ a functor, for $c \in C$ and object and writing $C(F(-), c) : C \to Set$, we have
Depending on how one introduces limits this holds by definition or is an easy consequence. In fact, this is just rewriting the respect of the covariant Hom of limits (as described there) in $C^{op}$ in terms of $C$:
Notice that this actually says that $C(-,-) : C^{op} \times C \to Set$ is continuous in both variables: in the first it sends limits in $C^{op}$ and hence equivalently colimits in $C$ to limits in $Set$.
Proposition – left adjoints commute with colimits
Let $L : C \to C'$ be a functor that is left adjoint to some functor $R : C' \to C$. Let $D$ be a small category such that $C$ admits limits of shape $D$. Then $L$ commutes with $D$-shaped colimits in $C$ in that
for $F : D \to C$ some diagram, we have
Proof
Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains for every $c' \in C'$
Since this holds naturally for every $c'$, the Yoneda lemma, corollary II on uniquenes of representing objects implies that $R (lim F) \simeq lim (R \circ F)$.
Here $C =$ Set.
The colimit over a Set-valued functor $F : D \to Set$ is a quotient of the disjoint union
$\coprod_{d \in Obj(d)} F(d)$ of all objects:
$colim F = (\coprod_{d \in D} F(d))/\sim$
where the equivalence relation is that generated by
The generalization of this is where the term “imit” for categorical (co)limit (probably) originates from: where $D$ a filtered category.
We discuss filtered colimits when discussing ind-objects in the next part accessing big categories – filtered colimits and ind-objects.
Of particular relevance are colimits of functors $F : D \to PSh(C)$ with values in the category of presheaves on a small category $C$.
Proposition
Colimits of presheaves are computed objectwise:
Here on the right the colimit is over the functor $F(-)(c) : D \to Set$.
In particular, they always exist, because the colimits in Set always exist.
Warning The Yoneda embedding (which, recall, commutes with small limits) does not commute with small colimits.
The counterexample crucial for the discussion of sheaves is:
let $X$ be a topological space and $U_1$, $U_2$ two open subsets covering $X$, i.e. such that $U_1 \cup U_2 = X$. Then
is a pushout diagram. But the corresponding pushout of representable presheaves
is not $Y(X) = Y(U_1 \cup U_2)$ but is a presheaf called a sieve: it sends each $V$ to the set of maps $V \to X$ that factor either through $U_1$ or through $U_2$.
For recall that a colimit of presheaves is computed objectwise, hence for each $V \in C$ the set $sieve(U_1,U_2)(V)$ is the set given by the pushout diagram
On the other hand, we have
Proposition – co-Yoneda lemma Every presheaf is a colimit of representable presheaves:
For $F : C^{op} \to Set$ we have
Proof. By inspection one finds that the map
induced by all the precomposition operations with all functors $Y(V) \to F$ is a bijection. The result follows hence again from corollary II of the Yoneda lemma.
Moreover, a colimit of a diagram need not exist in $C$ at all, but will exist in $PSh(C)$.
The Yoneda embedding is universal among functors from $C$ into (small) cocomplete categories, in the sense that given a functor $F: C \to C'$ where $C'$ has all colimits, there exists a unique (up to isomorphism) extension
so that $\hat{F} \circ Y \cong F$ and $\hat{F}$ preserves small colimits. Indeed, the desired $\hat{F}$ is left adjoint to the functor
This extension $\hat F$ - called the Yoneda extension – is a Kan extension. To which we now turn.
We introduced limits and colimits as adjoints to the functor
Notice that with $pt = \{\bullet\}$ the point, the category with a single object and a single morphism, we may identify $C$ with the functor category of functors from the point into it
From this perspective the functor $const$ may be understood as beeing precomposition with the terminal functor $t : D^{op} \to pt$:
This construction of course has an immediate generalization:
Definition
Let $p : D \to B$ be a functor and
the functor of precomposition with $p$. Then
left Kan extension along $p$ is, if it exists, the functor
which is left adjoint to $p^*$ and
right Kan extension is, if it exists, the functor
which is right adjoint to $p^*$.
So right Kan extension along $D \to pt$ is the same as limit. Left Kan extions along $D \to pt$ the same as colimit.
Again the adjunction isomorphism yields universal morphisms when evaluated on identity morphisms.
Under the bijection
the identity natural transformation $Id_{Lan F}$ ís sent to a natural transformation $\eta : F \to p^* Lan F$.
So the left Kan extension $Lan F = Lan_p F$ of $F : D^{op} \to C$ along $p:D^{op}\to B^{op}$ is a functor $Lan F : B^{op} \to C$ equipped with a natural transformation $\eta_F : F \Rightarrow p_* Lan F$.
with the property that every other natural transformation $F \Rightarrow p_* G$ factors uniquely through $\eta^u$ by a transformation $Lan_p F \Rightarrow G$.
Whenever the limit on the right hand side of the following equivalence exists for all $c'$, the right Kan extension on the left exists and is specified by this expression:
Here
is the comma category.
Similarly if the following colimit exists, it computes a left Kan extension
Here
is the other comma category.
The two major examples of Kan extensions that will play a role are
Given a functor $f^t : C \to C'$ the corresponding functor
is called the direct image operation on sheaves.
The left adjoint to the direct image is the inverse image functor.
The right adjoint to the direct image functor, if it exists, is the extension operation on (pre)sheaves.
The central example of this example is the following:
Let $C = Op(X)$ and $C' = Op(Y)$ be categories of open subsets of topological spaces $X$ and $Y$. A continuous map $f : X \to Y$ induces the obvious functor $f^t := f^{-1} : Op(Y) \to Op(X)$, since preimages of open subsets under continuous maps are open, hence presheaves push-forward along $f$
One can’t in the same simple way pull them back, since images of open subsets need not be open. The Kan extension computes the best possible approximation:
Notation | Definition |
---|---|
$(f^{-1})_* : PSh(X,A) \to PSh(Y,A)$ | direct image |
$(f^{-1})^\dagger : PSh(Y,A) \to PSh(X,A)$ | left adjoint to direct image |
$(f^{-1})^\ddagger : PSh(Y,A) \to PSh(X,A)$ | right adjoint to direct image |
By the above the inverse image $(f^{-1})^\dagger : PSh(Y) \to PSh(X)$ sends $F \in PSh(Y)$ to
So this approximates the possibly non-open subset $f^{-1}(V)$ by all open subsets $U$ inside it.
On the other hand, the extension
$(f^{-1})\ddagger : PSh(Y) \to PSh(X)$ sends $F \in PSh(Y)$ to
So this approximates the possibly non-open subset $f^{-1}(V)$ by all open subsets $U$ containing it.
Now we can come back to the question about how to extend a functor $F : C \to D$ along the Yoneda embedding.
Definition
For $C$ a small category and $F : C \to D$ a functor, its Yoneda extension
is the left Kan extension $Lan_Y F : [C^{op}, Set] \to D$ of $F$ along the Yoneda embedding $Y$:
Remark
Often it is of interest to Yoneda extend not $F : C \to D$ itself, but the composition $Y \circ F : C \to [D^{op}, Set]$ to get a functor entirely between presheaf categories
Proposition
Recalling the general formula for the left Kan extension of a functor $F : C \to D$ through a functor $p : C \to C'$
one finds for the Yoneda extension the formula
(Recall the notation for the comma category $(Y,A) := (Y, const_A)$ whose objects are pairs $(U \in C, (Y(U) \to A) \in [C^{op}, Set] )$.
For the full extension $\hat F : [D^{op}, Set] \to [C^{op}. Set]$ this yields
Here the first step is from above, the second uses that colimits in presheaf categories are computed objectwise and the last one is again using the Yoneda lemma.
Proposition
The restriction of the Yoneda extension to $C$ coincides with the original functor: $\tilde F \circ Y \simeq F$.
The Yoneda extension commutes with small colimits in $C$ in that for $\alpha : A \to C$ a diagram, we have $\tilde F (colim (Y\circ \alpha)) \simeq colim F \circ \alpha$ .
Moreover, $\tilde F$ is defined up to isomorphism by these two properties.
Last revised on September 9, 2009 at 00:49:15. See the history of this page for a list of all contributions to it.