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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 , it is of interest to ask if it is actually an ordinary object of , i.e. if it is representable;
an important class of examples we considered were presheaves induced by a functor and an object of the form : when these are representable for all , then the assignment of representing objects is functorial – the functor is the right adjoint of the functor .
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 be a small category and any category. A functor may be thought of as a diagram of shape in .
Definition
Every object defines the constant diagram of shape in :
The assignment of constant diagrams to objects in constitutes a functor
from to the functor category from to .
Definition
Given categories and as above, the limit operation
is, if it exists, the right adjoint to ;
the colimit operation
is, if it exists, the left adjoint.
Terminology
If the limit exists, we say that “ admits/has limits of shape ”.
if all limits of shape for a small category exist in , we say that that admits small limits.
if all limits of shape for a finite category exist in , we say that that 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 may exists and be of interest, even if the limit construction on all of 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 of a functor has (and is defined by) the property that there is a bijection of sets
naturally for all . The set on the left is that of natural transformations . Recall that one such transformation consists of a set of morphisms for each , such that for all morphisms in the diagram
commutes. Such a diagram is called a cone over with tip , for the obvious reason.
Remark: limit as classifying space
By definition, morphisms from any into are in bijection with cones over with tip . This means that behaves like a classifying space for cones over : the object subsumes in a way the entire diagram in one object, as far as probes into the diagram are concerned.
But more is true: the object is not just an arbitrary object, but itself canonically the tip of a cone:
by setting the defining bijection becomes
For this case there is a singled-out element on the left: the identity morphism . Its image under the bijection hence defines a singled out cone .
Moreover, for any other cone as above, with classifying morphism , the naturality of the defining bijection
is seen to imply that factors uniquely through as
the limit of the empty diagram in is, if it exists the terminal object;
if is a discrete category, i.e. a category with only identity morphisms, then a diagram is just a collection of objects of . Its limit is the product of these.
if then is the equalizer of the two morphisms .
the limit over a diagram is a pullback or fiber product.
Limits in Set are hom-sets
For any functor and the functor constant on the point, the limit of is the hom-set
in the functor category, i.e. the set of natural transformations from the constant functor into .
Covariant Hom commutes with limits
For a locally small category, for a functor and writing , we have
Proposition – limits in functor categories are computed pointwise
Let be a small category and let be any category. Let be a category which admits limits of shape . Write for the functor category. Then * admits -shaped limits; * these limits are computed objectwise (“pointwise”) in : for a functor we have for all that . Here the limit on the right is in .
Proposition – small limits commute with small limits
Let and be small catgeories and let be a category which admits limits of shape as well as limits of shape . Then these limits commute with each other, in that
for a functor , with corresponding induced functors and , then
Proposition – right adjoints commute with limits
Let be a functor that is right adjoint to some functor . Let be a small category such that admits limits of shape . Then commutes with -shaped limits in in that
for some diagram, we have
Proof
Using the adjunction isomorphism and the above fact that Hom commutes with limits, one obtains for every
Since this holds naturally for every , the Yoneda lemma, corollary II on uniquenes of representing objects implies that .
limits are equalizers of products
The limit of is always a subobject of the product of the , 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 is a subset of the product of all objects: .
Here consider limits of functors with values in the category of presheaves on a small category .
Proposition
Limits of presheaves are computed objectwise:
Here on the right the limit is over the functor .
Proposition
The Yoneda embedding commutes with small limits:
Let 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 is a limit in . Still, since they are important, we dicuss colimits in parallel to the discussion of limits above.
More precisely:
for a functor and 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 of a functor has (and is defined by) the property that there is a bijection of sets
naturally for all . The set on the left is that of natural transformations . Recall that one such transformation consists of a set of morphisms for each , such that for all morphisms in the diagram
commutes. Such a diagram is called a cocone under with tip .
But more is true: the object is not just an arbitrary object, but itself canonically the tip of a cocone:
by setting the defining bijection becomes
For this case there is a singled-out element on the left: the identity morphism . Its image under the bijection defines a singled out cocone .
Moreover, for any other cocone as above, with co-classifying morphism , the naturality of the defining bijection implies that factors uniquely through as
In components:
the colimit of the empty diagram in is, if it exists the initial object;
if is a discrete category, i.e. a category with only identity morphisms, then a diagram is just a collection of objects of . Its colimit is the coproduct of these.
if then is the coequalizer of the two morphisms .
the limit over a diagram 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 a locally small category, for a functor, for and object and writing , 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 in terms of :
Notice that this actually says that is continuous in both variables: in the first it sends limits in and hence equivalently colimits in to limits in .
Proposition – left adjoints commute with colimits
Let be a functor that is left adjoint to some functor . Let be a small category such that admits limits of shape . Then commutes with -shaped colimits in in that
for some diagram, we have
Proof
Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains for every
Since this holds naturally for every , the Yoneda lemma, corollary II on uniquenes of representing objects implies that .
Here Set.
The colimit over a Set-valued functor is a quotient of the disjoint union
of all objects:
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 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 with values in the category of presheaves on a small category .
Proposition
Colimits of presheaves are computed objectwise:
Here on the right the colimit is over the functor .
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 be a topological space and , two open subsets covering , i.e. such that . Then
is a pushout diagram. But the corresponding pushout of representable presheaves
is not but is a presheaf called a sieve: it sends each to the set of maps that factor either through or through .
For recall that a colimit of presheaves is computed objectwise, hence for each the set 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 we have
Proof. By inspection one finds that the map
induced by all the precomposition operations with all functors 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 at all, but will exist in .
The Yoneda embedding is universal among functors from into (small) cocomplete categories, in the sense that given a functor where has all colimits, there exists a unique (up to isomorphism) extension
so that and preserves small colimits. Indeed, the desired is left adjoint to the functor
This extension - 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 the point, the category with a single object and a single morphism, we may identify with the functor category of functors from the point into it
From this perspective the functor may be understood as beeing precomposition with the terminal functor :
This construction of course has an immediate generalization:
Definition
Let be a functor and
the functor of precomposition with . Then
left Kan extension along is, if it exists, the functor
which is left adjoint to and
right Kan extension is, if it exists, the functor
which is right adjoint to .
So right Kan extension along is the same as limit. Left Kan extions along the same as colimit.
Again the adjunction isomorphism yields universal morphisms when evaluated on identity morphisms.
Under the bijection
the identity natural transformation ís sent to a natural transformation .
So the left Kan extension of along is a functor equipped with a natural transformation .
with the property that every other natural transformation factors uniquely through by a transformation .
Whenever the limit on the right hand side of the following equivalence exists for all , 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 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 and be categories of open subsets of topological spaces and . A continuous map induces the obvious functor , since preimages of open subsets under continuous maps are open, hence presheaves push-forward along
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 |
---|---|
direct image | |
left adjoint to direct image | |
right adjoint to direct image |
By the above the inverse image sends to
So this approximates the possibly non-open subset by all open subsets inside it.
On the other hand, the extension
sends to
So this approximates the possibly non-open subset by all open subsets containing it.
Now we can come back to the question about how to extend a functor along the Yoneda embedding.
Definition
For a small category and a functor, its Yoneda extension
is the left Kan extension of along the Yoneda embedding :
Remark
Often it is of interest to Yoneda extend not itself, but the composition to get a functor entirely between presheaf categories
Proposition
Recalling the general formula for the left Kan extension of a functor through a functor
one finds for the Yoneda extension the formula
(Recall the notation for the comma category whose objects are pairs .
For the full extension 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 coincides with the original functor: .
The Yoneda extension commutes with small colimits in in that for a diagram, we have .
Moreover, 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.