Schreiber
universal constructions -- adjunction, limit and Kan extension

previous: presheaf categories with values in small sets

home: sheaves and stacks

next: accessing big categories – filtered colimits and ind-objects


  • 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 CC, it is of interest to ask if it is actually an ordinary object of CC, i.e. if it is representable;

    • an important class of examples we considered were presheaves induced by a functor L:CDL : C \to D and an object dDd \in D of the form D(L(),d):C opSetD(L(-),d) : C^{op} \to Set: when these are representable for all dd, then the assignment dR(d)d \mapsto R(d) of representing objects R(d)CR(d) \in C is functorial – the functor R:DCR : D \to C is the right adjoint of the functor FF.

  • 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.

Limits

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 DD be a small category and CC any category. A functor F:D opCF : D^{op} \to C may be thought of as a diagram of shape DD in CC.

Definition

Every object cCc \in C defines the constant diagram of shape DD in CC:

const c:D opC, const_c : D^{op} \to C \,,
const c:(d 1fd 2)(cId cc) const_c : (d_1 \stackrel{f}{\leftarrow} d_2) \mapsto (c \stackrel{Id_c}{\to} c)

The assignment of constant diagrams to objects in CC constitutes a functor

const:C[D op,C] const : C \to [D^{op}, C]

from CC to the functor category from D opD^{op} to CC.

Definition

Given categories CC and DD as above, the limit operation

lim:[D op,C]C lim : [D^{op}, C] \to C

is, if it exists, the right adjoint to const:C[D op,C]const : C \to [D^{op}, C];

the colimit operation

colim:[D op,C]C colim : [D^{op}, C] \to C

is, if it exists, the left adjoint.

Terminology

  • If the limit lim:[D op,C]Clim : [D^{op}, C] \to C exists, we say that “CC admits/has limits of shape DD”.

  • if all limits of shape DD for DD a small category exist in CC, we say that that CC admits small limits.

  • if all limits of shape DD for DD a finite category exist in CC, we say that that CC 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 opCF : D^{op} \to C may exists and be of interest, even if the limit construction on all of [D op,C][D^{op}, C] does not exist.

Limits as universal cones

We unwrap what the above definition of limit means more concretely in terms of components.

By the very definition of right adjoint, the limit limFlim F of a functor F[C op,D]F \in [C^{op}, D] has (and is defined by) the property that there is a bijection of sets

[C op,D](const c,F)C(c,limF) [C^{op}, D](const_c, F) \simeq C(c, lim F)

naturally for all FF. The set on the left is that of natural transformations η:const cF\eta : const_c \to F. Recall that one such transformation consists of a set of morphisms η d:cF(d)\eta_d : c \to F(d) for each dDd \in D, such that for all morphisms f:ddf : d \leftarrow d' in DD the diagram

c = c η d η d F(d) F(f) F(d)= c η d η d F(d) F(f) F(d) \array{ c &\stackrel{=}{\to}& c \\ \downarrow^{\eta_d} && \downarrow^{\eta_{d'}} \\ F(d) &\stackrel{F(f)}{\to}& F(d') } = \array{ && c \\ & {}^{\eta_{d}}\swarrow && \searrow^{\eta_{d'}} \\ F(d) && \stackrel{F(f)}{\to} && F(d') }

commutes. Such a diagram is called a cone over FF with tip cc, for the obvious reason.

Remark: limit as classifying space

By definition, morphisms from any cc into limFlim F are in bijection with cones over FF with tip cc. This means that limFlim F behaves like a classifying space for cones over FF: the object limFlim F subsumes in a way the entire diagram FF in one object, as far as probes into the diagram are concerned.

But more is true: the object limFlim F is not just an arbitrary object, but itself canonically the tip of a cone:

by setting c=limFc = lim F the defining bijection becomes

C(lim F,lim F)[C op,D](const limF,F). C(lim_F, lim_F) \simeq [C^{op},D](const_{lim F}, F) \,.

For this case there is a singled-out element on the left: the identity morphism Id lim FC(limF,limF)Id_{lim_F} \in C(lim F , lim F). Its image under the bijection hence defines a singled out cone η u:const limFF\eta^u : const_{lim F} \to F.

Moreover, for any other cone η u:const cF\eta^u : const_c \to F as above, with classifying morphism f:climFf : c \to lim F, the naturality of the defining bijection

C(limF,limF) [C op,D](const limF,F) C(f,limF) [C op,D](const f,F) C(c,limF) [C op,D](const c,F) \array{ C(lim F, lim F) &\stackrel{\simeq}{\to}& [C^{op}, D](const_{lim F}, F) \\ \downarrow^{C(f, lim F)} && \downarrow^{[C^{op}, D](const_{f}, F) } \\ C(c, lim F) &\stackrel{\simeq}{\to}& [C^{op}, D](const_{c}, F) }

is seen to imply that η\eta factors uniquely through η u\eta^u as

η=η uconst f \eta = \eta^u \circ const_f
c η d η d F(d) F(f) F(d)= c f limF η d u η d u F(d) F(f) F(d) \array{ && c \\ & {}^{\eta_{d}}\swarrow && \searrow^{\eta_{d'}} \\ F(d) && \stackrel{F(f)}{\to} && F(d') } = \array{ && c \\ && \downarrow^{f} \\ && lim F \\ & {}^{\eta^u_{d}}\swarrow && \searrow^{\eta^u_{d'}} \\ F(d) && \stackrel{F(f)}{\to} && F(d') }

special cases

  • the limit of the empty diagram D=D = \emptyset in CC is, if it exists the terminal object;

  • if DD is a discrete category, i.e. a category with only identity morphisms, then a diagram F:DCF : D \to C is just a collection c ic_i of objects of CC. Its limit is the product ic i\prod_i c_i of these.

  • if D={ab}D = \{a \stackrel{\to}{\to} b\} then limFlim F is the equalizer of the two morphisms F(b)F(a)F(b) \to F(a).

  • the limit over a diagram C A B \array{&& C \\ && \downarrow \\ A &\to& B} is a pullback or fiber product.

Properties

Limits in Set are hom-sets

For F:D opSetF : D^{op} \to Set any functor and const *:D opSetconst_{*} : D^{op} \to Set the functor constant on the point, the limit of FF is the hom-set

limF[D op,Set](const *,F) lim F \simeq [D^{op}, Set](const_{*}, F)

in the functor category, i.e. the set of natural transformations from the constant functor into FF.

Covariant Hom commutes with limits

For CC a locally small category, for F:D opCF : D^{op} \to C a functor and writing C(c,F()):D opSetC(c, F(-)) : D^{op} \to Set, we have

C(c,limF)limC(c,F()). C(c, lim F) \simeq lim C(c, F(-)) \,.

Proposition – limits in functor categories are computed pointwise

Let DD be a small category and let DD' be any category. Let CC be a category which admits limits of shape DD. Write [D,C][D',C] for the functor category. Then * [D,C][D',C] admits DD-shaped limits; * these limits are computed objectwise (“pointwise”) in CC: for F:D op[D,C]F : D^{op} \to [D',C] a functor we have for all dDd' \in D' that (limF)(d)lim(F()(d))(lim F)(d') \simeq lim (F(-)(d')). Here the limit on the right is in CC.

Proposition – small limits commute with small limits

Let DD and DD' be small catgeories and let CC be a category which admits limits of shape DD as well as limits of shape DD'. Then these limits commute with each other, in that

for F:D op×D opCF : D^{op} \times {D'}^{op} \to C a functor , with corresponding induced functors F D:D op[D op,C]F_D : {D'}^{op} \to [D^{op},C] and F D:D op[D op,C]F_{D'} : {D}^{op} \to [{D'}^{op},C], then

limFlim D(lim DF D)lim D(lim DF D). lim F \simeq lim_{D} (lim_{D'} F_D ) \simeq lim_{D'} (lim_{D} F_{D'} ) \,.

Proposition – right adjoints commute with limits

Let R:CCR : C \to C' be a functor that is right adjoint to some functor L:CCL : C' \to C. Let DD be a small category such that CC admits limits of shape DD. Then RR commutes with DD-shaped limits in CC in that

for F:D opCF : D^{op} \to C some diagram, we have

R(limF)lim(RF). R(lim F) \simeq lim (R \circ F) \,.

Proof

Using the adjunction isomorphism and the above fact that Hom commutes with limits, one obtains for every cCc' \in C'

C(c,R(limF)) C(L(c),limF) limC(L(c),F) limC(c,LF) C(c,lim(LF)).. \begin{aligned} C'(c', R (lim F)) & \simeq C(L(c'), lim F) \\ & \simeq lim C(L(c'), F) \\ & \simeq lim C'(c', L\circ F) \\ & \simeq C'(c', lim (L \circ F)) \,. \end{aligned} \,.

Since this holds naturally for every cc', the Yoneda lemma, corollary II on uniquenes of representing objects implies that R(limF)lim(GF)R (lim F) \simeq lim (G \circ F).

limits are equalizers of products

The limit of F:D opCF : D^{op} \to C is always a subobject of the product of the F(d)F(d), namely the equalizer of

dObj(D)F(d) fMor(d)(F(f)p t(f)) fMor(D)F(s(f)) \prod_{d \in Obj(D)} F(d) \stackrel{\prod_{f \in Mor(d)} (F(f) \circ p_{t(f)}) }{\to} \prod_{f \in Mor(D)} F(s(f))

and

dObj(D)F(d) fMor(d)(p s(f)) fMor(D)F(s(f)). \prod_{d \in Obj(D)} F(d) \stackrel{\prod_{f \in Mor(d)} (p_{s(f)}) }{\to} \prod_{f \in Mor(D)} F(s(f)) \,.

In particular therefore, a category has all limits already if it has all products and equalizers.

Examples

limits in Set

The limit over a Set-valued functor F:D opSetF : D^{op} \to Set is a subset of the product Π dObj(d)F(d)\Pi_{d \in Obj(d)} F(d) of all objects: limF={(s d) d dF(d)|forall(dfd):F(f)(s d)=s d}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\}.

limits in presheaf categories

Here consider limits of functors F:D opPSh(C)F : D^{op} \to PSh(C) with values in the category of presheaves on a small category CC.

Proposition

Limits of presheaves are computed objectwise:

limF:climF()(c) lim F : c \mapsto lim F(-)(c)

Here on the right the limit is over the functor F()(c):D opSetF(-)(c) : D^{op} \to Set.

Proposition

The Yoneda embedding Y:CPSh(C)Y : C \to PSh(C) commutes with small limits:

Let F:D opCF : D^{op} \to C be a functor, then

Y(limF)limYF Y(lim F) \simeq lim Y\circ F

(if either limit exists).

warning: the Yoneda embedding does not commute with colimits. See below.

Colimits

Colimits behave exactly dual to limits: a colimit in CC is a limit in C opC^{op}. Still, since they are important, we dicuss colimits in parallel to the discussion of limits above.

More precisely:

for F:DCF : D \to C a functor and F op:D opC opF^{op} : D^{op} \to C^{op} its opposite, we have

colimFlimF op. colim F \simeq lim F^{op} \,.

This becomes clear after the following.

Colimits as universal cocones

We unwrap what the above definition of colimit means more concretely in terms of components.

By the very definition of left adjoint, the colimit colimFcolim F of a functor F[C,D]F \in [C, D] has (and is defined by) the property that there is a bijection of sets

[C,D](F,const c,)C(colimF,c) [C, D](F, const_c, ) \simeq C(colim F, c)

naturally for all FF. The set on the left is that of natural transformations η:Fconst c\eta : F \to const_c. Recall that one such transformation consists of a set of morphisms η d:F(d)c\eta_d : F(d) \to c for each dDd \in D, such that for all morphisms f:ddf : d \to d' in DD the diagram

F(d) F(f) F(d) η d η d c = c=F(d) F(f) F(d) η d η d c \array{ F(d) &\stackrel{F(f)}{\to}& F(d') \\ \downarrow^{\eta_d} && \downarrow^{\eta_{d'}} \\ c &\stackrel{=}{\to}& c } = \array{ F(d) && \stackrel{F(f)}{\to} && F(d') \\ & {}^{\eta_{d}}\searrow && \swarrow^{\eta_{d'}} \\ && c }

commutes. Such a diagram is called a cocone under FF with tip cc.

But more is true: the object colimFcolim F is not just an arbitrary object, but itself canonically the tip of a cocone:

by setting c=colimFc = colim F the defining bijection becomes

C(colim F,colim F)[C,D](F,const colimF). C(colim_F, colim_F) \simeq [C,D](F, const_{colim F}) \,.

For this case there is a singled-out element on the left: the identity morphism Id colim FC(colimF,colimF)Id_{colim_F} \in C(colim F , colim F). Its image under the bijection defines a singled out cocone η u:Fconst colimF\eta^u : F \to const_{colim F} .

Moreover, for any other cocone η u:Fconst c\eta^u : F \to const_c as above, with co-classifying morphism f:colimFcf : colim F \to c, the naturality of the defining bijection implies that η\eta factors uniquely through η u\eta^u as

η=const fη u. \eta = const_f \circ \eta^u \,.

In components:

F(d) F(f) F(d) η d η d c=F(d) F(f) F(d) η d u η d u colimF f c \array{ F(d) && \stackrel{F(f)}{\to} && F(d') \\ & {}^{\eta_{d}}\searrow && \swarrow^{\eta_{d'}} \\ && c } = \array{ F(d) && \stackrel{F(f)}{\to} && F(d') \\ & {}^{\eta^u_{d}}\searrow && \swarrow^{\eta^u_{d'}} \\ && colim F \\ && \downarrow^{f} \\ && c }

special cases

  • the colimit of the empty diagram D=D = \emptyset in CC is, if it exists the initial object;

  • if DD is a discrete category, i.e. a category with only identity morphisms, then a diagram F:DCF : D \to C is just a collection c ic_i of objects of CC. Its colimit is the coproduct ic i\coprod_i c_i of these.

  • if D={ab}D = \{a \stackrel{\to}{\to} b\} then limFlim F is the coequalizer of the two morphisms F(b)F(a)F(b) \to F(a).

  • the limit over a diagram B C A \array{B &\to& C \\ \downarrow && \\ A } is a pushout.

Properties

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 CC a locally small category, for F:DCF : D \to C a functor, for cCc \in C and object and writing C(F(),c):CSetC(F(-), c) : C \to Set, we have

C(colimF,c)limC(F(),c). C(colim F, c) \simeq lim C(F(-), c) \,.

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 opC^{op} in terms of CC:

C(colimF,c) C op(c,colimF) C op(c,limF op) limC op(c,F op()) limC(F(),c) \begin{aligned} C(colim F, c) & \simeq C^{op}(c, colim F) \\ & \simeq C^{op}(c, lim F^{op}) \\ & \simeq lim C^{op}(c, F^{op}(-)) \\ & \simeq lim C(F(-), c) \end{aligned}

Notice that this actually says that C(,):C op×CSetC(-,-) : C^{op} \times C \to Set is continuous in both variables: in the first it sends limits in C opC^{op} and hence equivalently colimits in CC to limits in SetSet.

Proposition – left adjoints commute with colimits

Let L:CCL : C \to C' be a functor that is left adjoint to some functor R:CCR : C' \to C. Let DD be a small category such that CC admits limits of shape DD. Then LL commutes with DD-shaped colimits in CC in that

for F:DCF : D \to C some diagram, we have

L(colimF)colim(LF). L(colim F) \simeq colim (L \circ F) \,.

Proof

Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains for every cCc' \in C'

C(L(colimF),c) C(colimF,R(c)) limC(F(),R(c)) limC(LF(),c) C(colim(LF),c).. \begin{aligned} C'(L (colim F), c) & \simeq C(colim F, R(c')) \\ & \simeq lim C(F(-), R(c')) \\ & \simeq lim C'(L \circ F(-), c') \\ & \simeq C'(colim (L \circ F), c') \,. \end{aligned} \,.

Since this holds naturally for every cc', the Yoneda lemma, corollary II on uniquenes of representing objects implies that R(limF)lim(RF)R (lim F) \simeq lim (R \circ F).

colimits in Set

Here C=C = Set.

The colimit over a Set-valued functor F:DSetF : D \to Set is a quotient of the disjoint union

dObj(d)F(d)\coprod_{d \in Obj(d)} F(d) of all objects:

colimF=( dDF(d))/colim F = (\coprod_{d \in D} F(d))/\sim

where the equivalence relation is that generated by

(aF(d))(bF(d))if(dfd)D:F(f)(a)=b. (a \in F(d)) \sim (b \in F(d')) \;\; if \;\; \exists \; (d \stackrel{f}{\to} d') \in D : F(f)(a) = b \,.

filtered colimits

  • if DD is a poset, then the colimit over DD is the supremum over the F(d)F(d) with respect to (F(d)F(d))(F(d)F()F(d))(F(d) \leq F(d')) \Leftrightarrow (F(d) \stackrel{F(\leq)}{\to} F(d'));

The generalization of this is where the term “imit” for categorical (co)limit (probably) originates from: where DD a filtered category.

We discuss filtered colimits when discussing ind-objects in the next part accessing big categories – filtered colimits and ind-objects.

colimits in presheaf categories

Of particular relevance are colimits of functors F:DPSh(C)F : D \to PSh(C) with values in the category of presheaves on a small category CC.

Proposition

Colimits of presheaves are computed objectwise:

colimF:ccolimF()(c) colim F : c \mapsto colim F(-)(c)

Here on the right the colimit is over the functor F()(c):DSetF(-)(c) : D \to Set.

In particular, they always exist, because the colimits in Set always exist.

colimits and Yoneda embedding

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 XX be a topological space and U 1U_1, U 2U_2 two open subsets covering XX, i.e. such that U 1U 2=XU_1 \cup U_2 = X. Then

U 1U 2 U 2 U 1 U 1U 2 \array{ U_1 \cap U_2 &\to& U_2 \\ \downarrow && \downarrow \\ U_1 &\to& U_1 \cup U_2 }

is a pushout diagram. But the corresponding pushout of representable presheaves

Y(U 1U 2) Y(U 2) Y(U 1) sieve(U 1,U 2) \array{ Y(U_1 \cap U_2) &\to& Y(U_2) \\ \downarrow && \downarrow \\ Y(U_1) &\to& sieve(U_1, U_2) }

is not Y(X)=Y(U 1U 2)Y(X) = Y(U_1 \cup U_2) but is a presheaf called a sieve: it sends each VV to the set of maps VXV \to X that factor either through U 1U_1 or through U 2U_2.

For recall that a colimit of presheaves is computed objectwise, hence for each VCV \in C the set sieve(U 1,U 2)(V)sieve(U_1,U_2)(V) is the set given by the pushout diagram

C(V,U 1U 2) C(U 2) C(V,U 1) sieve(U 1,U 2)(V). \array{ C(V,U_1 \cap U_2) &\to& C(U_2) \\ \downarrow && \downarrow \\ C(V,U_1) &\to& sieve(U_1, U_2)(V) } \,.

On the other hand, we have

Proposition – co-Yoneda lemma Every presheaf is a colimit of representable presheaves:

For F:C opSetF : C^{op} \to Set we have

Fcolim (Y(V)F)Y(V). F \simeq colim_{(Y(V) \to F)} Y(V) \,.

Proof. By inspection one finds that the map

PSh(C)(F,G)lim Y(V)FPSh(Y(V),G)PSh(colim Y(V)FY(V),G) PSh(C)(F, G) \stackrel{}{\to} \lim_{Y(V) \to F} PSh(Y(V), G) \simeq PSh( colim_{Y(V) \to F} Y(V), G)

induced by all the precomposition operations with all functors Y(V)FY(V) \to F is a bijection. The result follows hence again from corollary II of the Yoneda lemma.

universality of Yoneda embedding

Moreover, a colimit of a diagram need not exist in CC at all, but will exist in PSh(C)PSh(C).

The Yoneda embedding is universal among functors from CC into (small) cocomplete categories, in the sense that given a functor F:CCF: C \to C' where CC' has all colimits, there exists a unique (up to isomorphism) extension

F^:PSh(C)C,\hat{F}: PSh(C) \to C',

so that F^YF\hat{F} \circ Y \cong F and F^\hat{F} preserves small colimits. Indeed, the desired F^\hat{F} is left adjoint to the functor

CPSh(C):chom C(F(),c)C' \to PSh(C): c' \mapsto \hom_{C'}(F(-), c')

This extension F^\hat F - called the Yoneda extension – is a Kan extension. To which we now turn.

Kan extension

We introduced limits and colimits as adjoints to the functor

const:C[D op,C]. const : C \to [D^{op}, C] \,.

Notice that with pt={}pt = \{\bullet\} the point, the category with a single object and a single morphism, we may identify CC with the functor category of functors from the point into it

C[pt,C]. C \simeq [pt, C] \,.

From this perspective the functor constconst may be understood as beeing precomposition with the terminal functor t:D opptt : D^{op} \to pt:

const=()t:[pt,C][D op,C]. const = (-) \circ t : [pt,C] \to [D^{op},C] \,.

This construction of course has an immediate generalization:

Definition

Let p:DBp : D \to B be a functor and

p *=()p:[B op,C][D op,C] p^* = (-)\circ p : [B^{op}, C] \to [D^{op}, C]

the functor of precomposition with pp. Then

left Kan extension along pp is, if it exists, the functor

Lan p:[D op,C][B op,C] Lan_p : [D^{op}, C] \to [B^{op}, C]

which is left adjoint to p *p^* and

right Kan extension is, if it exists, the functor

Ran p:[D op,C][B op,C] Ran_p : [D^{op}, C] \to [B^{op}, C]

which is right adjoint to p *p^*.

So right Kan extension along DptD \to pt is the same as limit. Left Kan extions along DptD \to pt the same as colimit.

Kan extension in terms of universal morphisms

Again the adjunction isomorphism yields universal morphisms when evaluated on identity morphisms.

Under the bijection

[B op,C](LanF,LanF)[D op,C](F,p *LanF) [B^{op}, C](Lan F, Lan F) \simeq [D^{op}, C](F, p^* Lan F)

the identity natural transformation Id LanFId_{Lan F} ís sent to a natural transformation η:Fp *LanF\eta : F \to p^* Lan F.

So the left Kan extension LanF=Lan pFLan F = Lan_p F of F:D opCF : D^{op} \to C along p:D opB opp:D^{op}\to B^{op} is a functor LanF:B opCLan F : B^{op} \to C equipped with a natural transformation η F:Fp *LanF\eta_F : F \Rightarrow p_* Lan F.

D op F C p η u Lan pF B op \array{ D^{op} &&\stackrel{F}{\to}&& C \\ & {}_{p}\searrow &\Downarrow^{\eta^u}& \nearrow_{Lan_p F} \\ && B^{op} }

with the property that every other natural transformation Fp *GF \Rightarrow p_* G factors uniquely through η u\eta^u by a transformation Lan pFGLan_p F \Rightarrow G.

Kan extension in terms of (co)limits

Whenever the limit on the right hand side of the following equivalence exists for all cc', the right Kan extension on the left exists and is specified by this expression:

(Ran pF)(c)lim cp(c)(c,p)F(c). (Ran_p F)(c') \simeq \lim_{c' \to p(c) \in (c',p)} F(c) \,.

Here

(c,p):=(const c,p)={ c p(c) p(f) p(c)} (c',p) := (const_{c'}, p) = \left\{ \array{ && c' \\ & \swarrow && \searrow \\ p(c) &&\stackrel{p(f)}{\to} && p(c') } \right\}

is the comma category.

Similarly if the following colimit exists, it computes a left Kan extension

(Lan pF)(c)colim p(c)c(p,c)F(c). (Lan_p F)(c') \simeq co\lim_{p(c) \to c' \in (p,c')} F(c) \,.

Here

(p,c):=(p,const c)={p(c) p(f) p(c) c} (p,c') := (p, const_{c'}) = \left\{ \array{ p(c) &&\stackrel{p(f)}{\to} && p(c') \\ & \searrow && \swarrow \\ && c' } \right\}

is the other comma category.

Examples

The two major examples of Kan extensions that will play a role are

Inverse image and extension

Given a functor f t:CCf^t : C \to C' the corresponding functor

()f t:PSh(C)PSh(C) (-) \circ f^t : PSh(C') \to PSh(C)

is called the direct image operation on sheaves.

The central example of this example is the following:

Let C=Op(X)C = Op(X) and C=Op(Y)C' = Op(Y) be categories of open subsets of topological spaces XX and YY. A continuous map f:XYf : X \to Y induces the obvious functor f t:=f 1:Op(Y)Op(X)f^t := f^{-1} : Op(Y) \to Op(X), since preimages of open subsets under continuous maps are open, hence presheaves push-forward along ff

f * 1:PSh(X)PSh(Y) f^{-1}_* : PSh(X) \to PSh(Y)

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:

NotationDefinition
(f 1) *:PSh(X,A)PSh(Y,A)(f^{-1})_* : PSh(X,A) \to PSh(Y,A)direct image
(f 1) :PSh(Y,A)PSh(X,A)(f^{-1})^\dagger : PSh(Y,A) \to PSh(X,A)left adjoint to direct image
(f 1) :PSh(Y,A)PSh(X,A)(f^{-1})^\ddagger : PSh(Y,A) \to PSh(X,A)right adjoint to direct image

By the above the inverse image (f 1) :PSh(Y)PSh(X)(f^{-1})^\dagger : PSh(Y) \to PSh(X) sends FPSh(Y)F \in PSh(Y) to

f F:Ucolim (Uf 1(V))(const U,f 1)F(V). f^\dagger F : U \mapsto colim_{(U \to f^{-1}(V)) \in (const_U, f^{-1})} F(V) \,.

So this approximates the possibly non-open subset f 1(V)f^{-1}(V) by all open subsets UU inside it.

On the other hand, the extension

(f 1):PSh(Y)PSh(X)(f^{-1})\ddagger : PSh(Y) \to PSh(X) sends FPSh(Y)F \in PSh(Y) to

f F:Ulim (f 1(V)U)(f 1,const U)F(V). f^\dagger F : U \mapsto lim_{(f^{-1}(V) \to U) \in (f^{-1},const_U)} F(V) \,.

So this approximates the possibly non-open subset f 1(V)f^{-1}(V) by all open subsets UU containing it.

Yoneda extension

Now we can come back to the question about how to extend a functor F:CDF : C \to D along the Yoneda embedding.

Definition

For CC a small category and F:CDF : C \to D a functor, its Yoneda extension

F˜:[C op,Set]D \tilde F : [C^{op},Set] \to D

is the left Kan extension Lan YF:[C op,Set]DLan_Y F : [C^{op}, Set] \to D of FF along the Yoneda embedding YY:

F˜:=Lan YF. \tilde F := Lan_Y F \,.

Remark

Often it is of interest to Yoneda extend not F:CDF : C \to D itself, but the composition YF:C[D op,Set]Y \circ F : C \to [D^{op}, Set] to get a functor entirely between presheaf categories

F^:=YF˜:[C op,Set][D op,Set]. \hat F := \tilde{Y \circ F} : [C^{op},Set] \to [D^{op}, Set] \,.

Proposition

Recalling the general formula for the left Kan extension of a functor F:CDF : C \to D through a functor p:CCp : C \to C'

(LanF)(c)colim (p(c)c)(p,c)F(c) (Lan F)(c') \simeq \colim_{(p(c) \to c') \in (p,c')} F(c)

one finds for the Yoneda extension the formula

F˜(A) :=(LanF)(A) colim (Y(U)A)(Y,A)F(U). \begin{aligned} \tilde F (A) & := (Lan F)(A) \\ & \simeq \colim_{(Y(U) \to A) \in (Y,A) } F(U) \end{aligned} \,.

(Recall the notation for the comma category (Y,A):=(Y,const A)(Y,A) := (Y, const_A) whose objects are pairs (UC,(Y(U)A)[C op,Set])(U \in C, (Y(U) \to A) \in [C^{op}, Set] ).

For the full extension F^:[D op,Set][C op.Set]\hat F : [D^{op}, Set] \to [C^{op}. Set] this yields

F^(A)(V) =(colim (Y(U)A)(Y,A)F(U))(V) colim (Y(U)A)(Y,A)F(U)(V) colim (Y(U)A)(Y,A)Hom D(V,F(U)). \begin{aligned} \hat F(A)(V) &= (\colim_{(Y(U) \to A) \in (Y,A) } F(U))(V) \\ &\simeq \colim_{(Y(U) \to A) \in (Y,A) } F(U)(V) \\ &\simeq \colim_{(Y(U) \to A) \in (Y,A) } Hom_{D}(V,F(U)) \end{aligned} \,.

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 CC coincides with the original functor: F˜YF \tilde F \circ Y \simeq F .

  • The Yoneda extension commutes with small colimits in CC in that for α:AC\alpha : A \to C a diagram, we have F˜(colim(Yα))colimFα\tilde F (colim (Y\circ \alpha)) \simeq colim F \circ \alpha .

  • Moreover, F˜\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.