nLab
Kan extension

Context

Category theory

2-Category theory

Enriched category theory

Limits and colimits

Contents

Idea

The Kan extension of a functor F:CDF : C \to D with respect to a functor

C p C \array{ C \\ \downarrow^p \\ C' }

is, if it exists, a kind of best approximation to the problem of finding a functor CDC' \to D such that

C F D p C, \array{ C &\stackrel{F}{\to}& D \\ \downarrow^p & \nearrow \\ C' } \,,

hence to extending the domain of FF through pp from CC to CC'.

More generally, this makes sense not only in Cat but in any 2-category.

Similarly, a Kan lift is the best approximation to lifting a morphism F:CDF : C \to D through a morphism

D D \array{ D' \\ \downarrow \\ D }

to a morphism F^\hat F

D F^ C F D. \array{ && D' \\ & {}^{\hat F}\nearrow & \downarrow \\ C &\stackrel{F}{\to}& D } \,.

Kan extensions are ubiquitous. See the discussion at Examples below.

Definitions

There are various slight variants of the definition of Kan extension . In good cases they all exist and all coincide, but in some cases only some of these will actually exist.

We (have to) distinguish the following cases:

  1. “ordinary” or “weak” Kan extensions

    These define the extension of an entire functor, by an adjointness relation.

    Here we (have to) distinguish further between

    1. global Kan extensions,

      which define extensions of all possible functors of given domain and codomain (if all of them indeed exist);

    2. local Kan extensions,

      which define extensions of single functors only, which may exist even if not every functor has an extension.

  2. “pointwise” or “strong” Kan extensions

    These define the value of an extended functor on each object (each “point”) by a weighted (co)limit.

    Furthermore, a pointwise Kan extension can be “absolute”.

If the pointwise version exists, then it coincides with the “ordinary” or “weak” version, but the former may exist without the pointwise version existing. See below for more.

Some authors (such as Kelly) assert that only pointwise Kan extensions deserve the name “Kan extension,” and use the term as “weak Kan extension” for a functor equipped with a universal natural transformation. It is certainly true that most Kan extensions which arise in practice are pointwise. This distinction is even more important in enriched category theory.

Ordinary or weak Kan extensions

Global Kan extensions

Let

p:CC p : C \to C'

be a functor. For DD any other category, write

p *:[C,D][C,D] p^* : [C',D] \to [C,D]

for the induced functor on the functor categories: this sends a functor h:CDh : C' \to D to the composite functor p *h:CpChDp^* h : C \stackrel{p}{\to} C' \stackrel{h}{\to} D.

Definition

If p *p^* has a left adjoint, typically denoted

p !:[C,D][C,D] p_! : [C,D] \to [C',D]

or

Lan p:[C,D][C,D] Lan_p : [C,D] \to [C',D]

then this left adjoint is called the ( ordinary or weak ) left Kan extension operation along pp. For h[C,D]h \in [C,D] we call p !hp_! h the left Kan extension of hh along pp.

Similarly, if p *p^* has a right adjoint, this right adjoint is called the right Kan extension operation along pp. It is typically denoted

p *:[C,D][C,D] p_* : [C,D] \to [C',D]

or

Ran=Ran p:[C,D][C,D]. Ran = Ran_p: [C,D] \to [C',D] \,.

The analogous definition clearly makes sense as stated in other contexts, such as in enriched category theory.

Observation

If C=*C' = * is the terminal category, then

  • the left Kan extension operation forms the colimit of a functor;

  • the right Kan extension operation forms the limit of a functor.

Proof

The functor p *p^* in this case sends objects dd of DD to the constant functor Δ d\Delta_d on dd. Notice that for F[C,D]F \in [C,D] any functor,

Therefore the natural hom-isomorphisms of the adjoint functors (p !p *)(p_! \dashv p^*) and (p *p *)(p^* \dashv p_*)

D(d,p *F)Func(Δ d,F) D(d, p_* F) \simeq Func(\Delta_d, F)

and

D(p !F,d)Func(F,Δ d) D(p_! F, d) \simeq Func(F, \Delta_d)

assert that

  • p *Fp_* F corepresents the cones over FF: this means by definition that p *F=lim Fp_* F = \lim_\leftarrow F is the limit over FF;

  • p !Fp_! F represents the cocones under FF: this means by definition that p !F=lim Fp_! F = \lim_\to F is the colimit of FF.

Local Kan extension

There is also a local definition of “the Kan extension of a given functor FF along pp” which can exist even if the entire functor defined above does not. This is a generalization of the fact that a particular diagram of shape CC can have a limit even if not every such diagram does. It is also a special case of the fact discussed at adjoint functor that an adjoint functor can fail to exist completely, but may still be partially defined. If the local Kan extension of every single functor exists for some given p:CCp\colon C\to C' and DD, then these local Kan extensions fit together to define a functor which is the global Kan extension.

Thus, by the general notion of “partial adjoints”; we say

Definition

The local left Kan extension of a functor F[C,D]F\in [C,D] along p:CCp : C \to C'* is, if it exists, a functor

Lan pF:CD Lan_p\,F : C'\to D

equipped with a natural isomorphism

Hom [C,D](F,p *())Hom [C,D](Lan pF,), Hom_{[C,D]}(F,p^*(-))\cong Hom_{[C',D]}(Lan_p\,F,-) \,,

hence a (co)representation of the functor Hom [C,D](F,p *())Hom_{[C,D]}(F,p^*(-)).

The local definition of right Kan extensions along pp is dual.

As for adjoints and limits, by the usual logic of representable functors this can equivalently be rephrased in terms of universal morphisms:

Definition

The left Kan extension LanF=Lan pFLan F = Lan_p F of F:CDF : C \to D along p:CCp :C\to C' is a functor LanF:CDLan F : C' \to D equipped with a natural transformation η F:Fp *LanF\eta_F : F \Rightarrow p^* Lan F.

with the property that every other natural transformation Fp *GF \Rightarrow p^* G factors uniquely through η F\eta_F as

Similarly for the right Kan extension, with the direction of the natural transformations reversed:

By the usual reasoning (see e.g. Categories Work, chapter IV, theorem 2), if these representations exist for every FF then they can be organised into a left (right) adjoint Lan pLan_p (Ran pRan_p) to p *p^*.

Remark

The definition in this form makes sense not just in Cat but in every 2-category. In slightly different terminology, the left Kan extension of a 1-cell F:CDF:C\to D along a 1-cell pK(C,C)p\in K(C,C') in a 2-category KK is a pair (Lan pF,α)(Lan_p F,\alpha) where α:FLan pFp\alpha : F\to Lan_p F\circ p is a 2-cell which reflects the object FK(C,D)F\in K(C,D) along the functor p *=K(p,D):K(C,D)K(C,D)p^* = K(p,D):K(C',D)\to K(C,D). Equivalently, it is such a pair such that for every G:CDG\colon C' \to D, the function

K(C,D)(Lan pF,G)αK(C,D)(F,Gp) K(C',D)(Lan_p F, G) \xrightarrow{- \cdot \alpha} K(C,D)(F, G \circ p)

is a bijection.

In this form, the definition generalizes easily to any n-category for any n2n\ge 2. If KK is an nn-category, we say that the left Kan extension of a 1-morphism F:CDF:C\to D along a 1-morphism pK(C,C)p\in K(C,C') is a pair (Lan pF,α)(Lan_p F,\alpha), where Lan pF:CDLan_p F \colon C' \to D is a 1-morphism and α:FLan pFp\alpha : F\to Lan_p F\circ p is a 2-morphism, with the property that for any 1-morphism G:CDG\colon C'\to D, the induced functor

K(C,D)(Lan pF,G)αK(C,D)(F,Gp) K(C',D)(Lan_p F, G) \xrightarrow{- \cdot \alpha} K(C,D)(F, G \circ p)

is an equivalence of (n2)(n-2)-categories.

Preservation of Kan extensions

We say that a Kan extension Lan pFLan_p F is preserved by a functor GG if the composite GLan pFG \circ Lan_p F is a Kan extension of GFG F along pp, and moreover the universal natural transformation GFG(Lan pF)pG F \to G(Lan_p F)p is the composite of GG with the universal transformation F(Lan pF)fF\to (Lan_p F)f.

Pointwise or strong Kan extensions

If the codomain category DD admits certain (co)limits, then left and right Kan extensions can be constructed, over each object (“point”) of the domain category CC' out of these: Kan extensions that admit this form are called pointwise. (Reviews include (Riehl, I 1.3)).

The notion of pointwise Kan extensions deserves to be discussed in the general context of enriched category theory, which we do below. The reader may want to skip ahead to the section

which discusses the situation in ordinary (Set-enriched) category theory in terms of ordinary limits (“conical” limits, defined in terms of cones, to be distinguished from the more general weighted limits). While the formulas in that case are classical and fundamentally useful in practice, they do rely heavily on special properties of the enriching category Set.

The general formulation of pointwise Kan extensions in general enriched contexts is

In the case that the codomain category is (co)tensored these may be expressed equivalently

First, here is a characterization that doesn’t rely on any computational framework:

Definition

A Kan extension, def. 2, is called pointwise if and only if it is preserved by all representable functors.

(Categories Work, theorem X.5.3)

in terms of weighted (co)limits

Suppose given F:CDF : C \to D and p:CCp : C \to C' such that for every cCc' \in C', the weighted limit

(Ran pF)(c)lim C(c,p())F. (Ran_p F)(c') \coloneqq lim^{C'(c',p(-))} F \,.

exists. Then these objects fit together into a functor Ran pFRan_p F which is a right Kan extension of FF along pp. Dually, if the weighted colimit

(Lan pF)(c)colim C(p(),c)F. (Lan_p F)(c') \coloneqq colim^{C'(p(-),c')} F \,.

exists for all cc', then they fit together into a left Kan extension Lan pFLan_p F. These definitions evidently make sense in the generality of VV-enriched category theory for VV a closed symmetric monoidal category. (In fact, they can be modified slightly to make sense in the full generality of a 2-category equipped with proarrows.)

In particular, this means that if CC is small and DD is complete (resp. cocomplete), then all right (resp. left) Kan extensions of functors F:CDF\colon C\to D exist along any functor p:CCp\colon C\to C'.

One can prove that any Kan extension constructed in this way must be pointwise, in the sense of being preserved by all representables as above. Moreover, conversely, if a Kan extension Lan pFLan_p F is pointwise, then one can prove that (Lan pF)(c)(Lan_p F)(c') must be in fact a C(p(),c)C'(p(-),c')-weighted colimit of FF, and dually; thus the two notions are equivalent.

in terms of (co)ends

If the VV-enriched category DD is powered over VV, then the above weighted limit may be re-expressed in terms of an end as

(Ran pF)(c) cCC(c,p(c))F(c). (Ran_p F)(c') \simeq \int_{c \in C} C'(c',p(c))\pitchfork F(c) \,.

So in particular when D=VD = V this is

(Ran pF)(c) cC[C(c,p(c)),F(c)]. (Ran_p F)(c') \simeq \int_{c \in C} [C'(c',p(c)),F(c)] \,.

Similarly, if DD is tensored over VV, then the left Kan extension is given by a coend.

(Lan pF)(c) cCC(p(c),c)F(c). (Lan_p F)(c') \simeq \int^{c \in C} C'(p(c),c')\otimes F(c) \,.

in terms of conical (co)limits

In the case of functors between ordinary locally small categories, hence in the special case of VV-enriched category theory for V=V = Set, there is an expression of a weighted (co)limit and hence a pointwise Kan extension as an ordinary (“conical”, meaning: in terms of cones) (co)limit over a comma category:

Proposition

Let

Then the right Kan extension of a functor F:CDF : C \to D of locally small categories along a functor p:CCp : C \to C' exists and its value on an object cCc' \in C' is given by the limit

(Ran pF)(c)lim ((Δ c/p)CFD), (Ran_p F)(c') \simeq \lim_\leftarrow \left((\Delta_{c'}/p) \to C \stackrel{F}{\to} D\right) \,,

where

Likewise, if DD admits small colimits, the left Kan extension of a functor exists and is pointwise given by the colimit

(Lan pF)(c)lim ((p/Δ c)CFD). (Lan_p F)(c') \simeq \lim_\to \left((p/\Delta_{c'}) \to C \stackrel{F}{\to} D\right) \,.

This appears for instance as (Borceux, I, thm 3.7.2). Discussion in the context of enriched category theory is in (Kelly, section 3.4).

A cartoon picture of the forgetful functor out of the comma category p/Δ cCp/\Delta_{c'} \to C, useful to keep in mind, is

(p(c 1) p(ϕ) p(c 2) c)(c 1ϕc 2). \left( \array{ p(c_1) &&\stackrel{p(\phi)}{\to}&& p(c_2) \\ & \searrow && \swarrow \\ && c' } \right) \;\; \mapsto \;\; \left( c_1 \stackrel{\phi}{\to} c_2 \right) \,.

The comma category here is equivalently the category of elements of the functor C(p(),c):C opSetC'(p(-), c') : C^{op} \to Set

(p/Δ c)el(C(p(),c)). (p/\Delta_{c'}) \simeq el( C'(p(-), c') ) \,.
Proof

Consider the case of the left Kan extension, the other case works analogously, but dually.

First notice that the above pointwise definition of values of a functor canonically extends to an actual functor:

for ϕ:c 1c 2\phi : c'_1 \to c'_2 any morphism in CC' we get a functor

ϕ *:p/Δ c 1p/Δ c 2 \phi_* : p/\Delta_{c'_1} \to p/\Delta_{c'_2}

of comma categories, by postcomposition. This morphism of diagrams induces canonically a corresponding morphism of colimits

(Lan pF)(c 1)(Lan pF)(c 2). (Lan_p F)(c'_1) \to (Lan_p F)(c'_2) \,.

Now for the universal property of the functor Lan pFLan_p F defined this way. For cCc' \in C' denote the components of the colimiting cocone (Lan pF)(c):=lim (p/Δ cCFD)(Lan_p F)(c') := \lim_{\to}( p/\Delta_{c'} \to C \stackrel{F}{\to} D) by s ()s_{(-)}, as in

(p(c 1)ϕc) p(h) (p(c 2)λc) F(c 1) F(h) F(c 2) s ϕ s λ (Lan pF)(c). \array{ (p(c_1)\stackrel{\phi}{\to} c') &&\stackrel{p(h)}{\to}&& (p(c_2)\stackrel{\lambda}{\to} c') \\ \\ \\ F(c_1) &&\stackrel{F(h)}{\to}&& F(c_2) \\ & {}_{\mathllap{s_\phi}}\searrow && \swarrow_{\mathrlap{s_{\lambda}}} \\ && (Lan_p F)(c') } \,.

We now construct in components a natural transformation

η F:Fp *Lan pF \eta_F : F \to p^* Lan_p F

for Lan pFLan_p F defined as above, and show that it satisfies the required universal property. The components of η F\eta_F over cCc \in C are morphisms

η F(c):F(c)(Lan pF)(p(c)). \eta_F(c) : F(c) \to (Lan_p F)(p (c)) \,.

Take these to be given by

η F(c):=s Id p(c) \eta_F(c) := s_{Id_{p(c)}}

(this is similar to what happens in the proof of the Yoneda lemma, all of these arguments are variants of the argument for the Yoneda lemma, and vice versa). It is straightforward, if somewhat tedious, to check that these are natural, and that the natural transformation defined this way has the required universal property.

Comparing the definitions

We have seen that if DD has enough limits or colimits, then a pointwise Kan extension can be defined in terms of these limits, and will necessarily satisfy the universal property described first. However, not all Kan extensions are pointwise: that is, having a universal transformation F(Lan pF)pF \to (Lan_p F)p does not necessarily imply that the individual values of Lan pFLan_p F are limits or colimits in its codomain. Non-pointwise Kan extensions can exist even when DD does not admit very many limits.

It should be noted, though, that pointwise Kan extensions can still exist, and hence the particular requisite limits/colimits exist, even if DD is not (co)complete. For instance, the Kan extensions that arise in the study of derived functors are pointwise, and in fact absolute (preserved by all functors), even though their codomains are homotopy categories which generally do not admit all limits and colimits.

Non-pointwise Kan extensions seem to be very rare in practice. However, the abstract notion of Kan extension (sometimes called simply “extension”) in a 2-category, and its dual notion of lifting, can be useful in 2-category theory. For instance, bicategories such as Prof admit all right extensions and right liftings; a bicategory with this property may be considered a horizontal categorification of a closed monoidal category.

Absolute Kan extensions

Am absolute Kan extension Lan pFLan_p F is one which is preserved by all functors GG out of the codomain of FF:

(1)G(Lan pF)Lan p(GF) G (Lan_p F) \simeq Lan_p(G F)

(same for right Kan extensions).

The most prominent example of absolute Kan extensions is given by adjoint functors; in fact they can be defined as certain absolute Kan extensions. See there for the precise statement.

absolute vs pointwise

Absolute Kan extensions are always pointwise, as the later can be defined as those preserved by representables; there are (lots of) examples of pointwise Kan extensions which are not absolute.

Note that in a general 2-category, absolute Kan extensions make perfect sense, while for defining pointwise ones more structure is needed: comma objects and/or some structure which would let us work with (co)limits inside that 2-category (such as a (co)Yoneda structure? or a proarrow equipment).

Of (,1)(\infty,1)-functors

The global definition of Kan extensions for functors in terms of left/right adjoints to pullbacks may be interpreted essentially verbatim in the context of (∞,1)-categories

See at (∞,1)-Kan extension.

In a general 2-category

The Kan extension of a functor may be regarded more abstractly as an extension-problem in the 2-category Cat of categories. The same extension problem can be stated verbatim in any 2-category and hence there is a corresponding more general notion of Kan extensions of 1-morphisms in 2-categories.

This is discussed in (Lack 09, section 2.2).

Properties

Left Kan extension on representables / fully faithfulness

Let 𝒱\mathcal{V} be a suitable enriching category (a cosmos). Notably 𝒱\mathcal{V} may be Set.

Proposition

For F:CDF : C \to D a 𝒱\mathcal{V}-enriched functor between small 𝒱\mathcal{V}-enriched categories we have

  1. the left Kan extension along FF takes representable presheaves C(c,):C𝒱C(c,-) : C \to \mathcal{V} to their image under FF:

    Lan FC(c,)D(F(c),) Lan_F C(c, -) \simeq D(F(c), -)

    for all cCc \in C.

  2. if FF is a full and faithful functor then F *(Lan FH)HF^* (Lan_F H) \simeq H and in fact the (Lan FF *)(Lan_F \dashv F^*)-unit of an adjunction is a natural isomorphism

    IdF *Lan F Id \stackrel{\simeq}{\to} F^* \circ Lan_{F}

    whence it follows (see the basic properties of adjoint functors) that Lan F:[C,𝒱][D,𝒱]Lan_F : [C,\mathcal{V}] \to [D,\mathcal{V}] is itself a full and faithful functor.

The second statement appears for instance as (Kelly, prop. 4.23).

Proof

For the first statement, using the coend formula for the left Kan extension we have naturally in dDd' \in D the expression

Lan FC(c,):d cCD(F(c),d)C(c,)(c) cCD(F(c),d)C(c,c) D(F(c),d). \begin{aligned} Lan_F C(c,-) : d' \mapsto & \int^{c' \in C} D(F(c'), d') \cdot C(c,-)(c') \\ & \simeq \int^{c' \in C} D(F(c'), d') \cdot C(c,c') \\ & \simeq D(F(c), d') \end{aligned} \,.

Here the last step is called sometimes the co-Yoneda lemma. It follows for instance by observing that cCD(F(c),d)C(c,c)\int^{c' \in C} D(F(c'), d') \cdot C(c,c') is equivalently dually the expression for the left Kan extension of the non-representable D(F(),d):C op𝒱D(F(-),d') : C^{op} \to \mathcal{V} along the identity functor.

Similarly for the second, if H:DEH : D \to E is any 𝒱\mathcal{V}-enriched functor with EE tensored over 𝒱\mathcal{V}, then its left Kan extension evaluated on the image of FF is

Lan FH:F(d) cCD(F(c),F(d))H(c) cCC(c,d)H(c) H(d). \begin{aligned} Lan_F H : F(d) \mapsto & \int^{c \in C} D(F(c), F(d)) \cdot H(c) \\ & \simeq \int^{c \in C} C(c, d) \cdot H(c) \\ & \simeq H(d) \end{aligned} \,.

Left Kan extensions preserving certain limits

The following statement says that left exact functors into toposes have left exact left Kan extension along the Yoneda embedding (Yoneda extension) and that this is the inverse image of a geometric morphism of sheaf toposes of the original functor preserves covers.

(We state this in (∞,1)-category theory, the same statement holds true in plain category theory by just disregarding all occurences of”\infty”.)

Proposition

Let H\mathbf{H} be an (∞,1)-topos and let 𝒞\mathcal{C} be an (∞,1)-site with (∞,1)-sheaf (∞,1)-category Sh(𝒞)Sh(\mathcal{C}). Then the (∞,1)-functor

Top(𝒳,Sh(𝒞))Func lex,leftadj(Sh(𝒞),𝒳)()LFunc(PSh(𝒞),𝒳)()YFunc(𝒞,𝒳) Top(\mathcal{X}, Sh(\mathcal{C})) \simeq Func^{lex, leftadj}(Sh(\mathcal{C}), \mathcal{X}) \stackrel{(-)\circ L }{\longrightarrow} Func(PSh(\mathcal{C}), \mathcal{X}) \stackrel{(-)\circ Y }{\longrightarrow} Func(\mathcal{C}, \mathcal{X})

given by precomposition with ∞-stackification/sheafification LL and with the (∞,1)-Yoneda embedding YY is a full and faithful (∞,1)-functor. Moreover, its essential image consisist of those (∞,1)-functors f:𝒞𝒳f \colon \mathcal{C} \longrightarrow \mathcal{X} which are left exact and which preserve covers in that for {U iX} i\{U_i \to X\}_i a covering in 𝒞\mathcal{C}, then if(U i)f(X)\coprod_i f(U_i) \to f(X) is an effective epimorphism in 𝒳\mathcal{X}.

This appears as Lurie, HTT, prop. 6.2.3.20.

Remark

Prop. 4 is a central statement in the theory of classifying toposes. See there for more.

For more discussion of left exactness properties preserved by left Kan extension see also (Borceux-Day, Karazeris-Protsonis).

Kan extension along (op)fibration

Proposition

Let f:CDf : C \to D be a small opfibration of categories, and let 𝒞\mathcal{C} be a category with all small colimits. Then for each dDd \in D the inclusion

f 1(d)f/d f^{-1}(d) \to f/d

of the fiber over dd into the comma category given by

c(c,Id d=Id f(c)) c \mapsto (c, Id_{d} = Id_{f(c)})

has a left adjoint, given by

(c,f(c)d)c. (c, f(c) \to d) \mapsto c \,.

Therefore (by the discussion here) it is a cofinal functor. Accordingly, the local formula for the left Kan extension

f !:[C,𝒟][D,𝒟] f_! : [C, \mathcal{D}] \to [D, \mathcal{D}]

is equivalently given by taking the colimit over the fiber:

f !X:dlim f 1(d)X. f_! X : d \mapsto \lim_{\underset{f^{-1}(d)}{\to}} X \,.

Examples

The central point about examples of Kan extensions is:

Kan extensions are ubiquitous .

To a fair extent, category theory is all about Kan extensions and the other universal constructions: limits, adjoint functors, representable functors, which are all special cases of Kan extensions – and Kan extensions are special cases of these.

Listing examples of Kan extensions in category theory is much like listing examples of integrals in analysis: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see coend for more).

Keeping that in mind, we do list some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.

General

  • For C=C' = the point, the right Kan extension of FF is the limit of FF, RanFlimFRan F \simeq \lim F and the left Kan extension is the colimit LanFcolimFLan F \simeq colim F.

  • For f:XYf : X \to Y a morphism of sites coming from a functor f t:S YS Xf^t : S_Y \to S_X of the underlying categories, the left Kan extension of functors along f tf^t is the inverse image operation f 1:PSh(Y)PSh(X)f^{-1} : PSh(Y) \to PSh(X).

  • see also examples of Kan extensions

Non-pointwise Kan extensions

We discuss examples of Kan extensions that are not point-wise

Restriction and extension of sheaves

For more on the following see also

The basic example for left Kan extensions using the above pointwise formula, is in the construction of the pullback of sheaves along a morphism of topological spaces. Let f:XYf:X\to Y be a continuous map and FF a presheaf over XX. Then the formula (f *F)(U)=F(f 1(U))(f_* F)(U) = F(f^{-1}(U)) clearly defines a presheaf f *Ff_* F on YY, which is in fact a sheaf if FF is. On the other hand, given a presheaf GG over YY we can not define pullback presheaf (f 1G)(V)=G(f(V))(f^{-1} G)(V)=G(f(V)) because f(V)f(V) might not be open in general (unless ff is an open map). For Grothendieck sites such f(V)f(V) would not make even sense. But one can consider approximating from above by G(W)G(W) for all Wf(V)W\supset f(V) which are open and take a colimit of this diagram of inclusions (all WW are bigger, so getting down to the lower bound means going reverse to the direction of inclusions). But inclusion f(V)Wf(V)\subset W implies Vf 1(f(V))f 1(W)V\subset f^{-1}(f(V))\subset f^{-1}(W). The latter identity Vf 1(W)V\subset f^{-1}(W) involves only open sets. Thus we take a colimit over the comma category (Vf 1)(V\downarrow f^{-1}) of GG. If GG is a sheaf, the colimit G(V)G(V) understood as a rule VG(V)V\mapsto G(V) is still not a sheaf, we need to sheafify. The result is sheaf-theoretic pullback

f 1G=sheafify(Vcolim Vf 1WG(W))=sheafify(Vcolim (Vf 1)G) f^{-1}G = \mathrm{sheafify}(V\mapsto\mathrm{colim}_{V\hookrightarrow f^{-1}W} G(W)) = \mathrm{sheafify}(V\mapsto\mathrm{colim}_{(V\downarrow f^{-1})} G)

which is a sheaf, and one can analyze this construction to show that f 1f^{-1} is a left adjoint to f *f_*. This usage of left Kan extension persists in the more general case of Grothendieck topologies.

Kan extension in physics

We list here some occurences of Kan extensions in physics.

Notice that since, by the above discussion, Kan extensions are ubiquitous in category theory and are essentially equivalent to other standard universal constructions such as notably co/limits, to the extenent that there is a relation between category theory and physics at all, it necessarily also involves Kan extensions, in some guise. But here is a list of some example where they appear rather explicitly.

Remark on terminology: pushforward vs. pullback

Generally, for p:CCp : C \to C' a functor, the induced “precomposition” functor on functor categories

[C,D]p[C,D] [C', D] \stackrel{- \circ p}{\to} [C,D]

is spoken of as pulling back a functor on CC' to a functor on CC, as this operation goes in the direction opposite to that of pp itself. For this reason, we have above denoted this functor by p *p^*. Likewise, one might call the (left or right) Kan extensions along pp a push forward of functors from CC to functors on CC'.

This notation also coincides with that for geometric morphisms in one case: any functor p:CCp\colon C\to C' between small categories induces a geometric morphism [C,Set][C,Set][C,Set] \to [C',Set] of presheaf toposes, whose inverse image is the above p *p^* and whose direct image p *p_* is the right Kan extension functor. Note that p *p^* preserves (finite) limits, as required of an inverse image functor, since it has a left adjoint, namely left Kan extension.

On the other hand, if pp is additionally a flat functor, then the above precomposition functor is also the direct image of a geometric morphism, whose inverse image is given by left Kan extension (which preserves finite limits when pp is flat). More generally, if C opC^{op} and (C) op(C')^{op} are sites and p op:C op(C) opp^{op}\colon C^{op}\to (C')^{op} is flat and preserves covering families (i.e. it is a morphism of sites), then precomposition is the direct image of a geometric morphism Sh(C op)Sh((C) op)Sh(C^{op})\to Sh((C')^{op}) between sheaf toposes.

For example, C opC^{op} and (C) op(C')^{op} might be the posets Open(X)Open(X) and Open(X)Open(X') of open subsets of topological spaces (or locales) XX and XX' and inclusions, in which case

Open(X) opOpen(X) op Open(X)^{op} \to Open(X')^{op}

come from continuous maps of topological spaces going the other way

XX:f, X \leftarrow X' : f \,,

via the usual inverse image f 1:O(X) opO(X) opf^{-1} : O(X)^{op} \to O(X')^{op} of open subsets.

Thus, in such cases, the functor p *p^*, which looks like a pullback of functors along p=f 1p = f^{-1}, corresponds geometrically to a push-forward of (pre)sheaves along ff. Therefore, in presheaf literature (such as Categories and Sheaves) the precomposition functor induced by pp is usually denoted p *p_* and not p *p^*.

It is however noteworthy that also the opposite perspective does occur in geometrically motivated examples. For instance

  • if CC is the discrete category on smooth space and D=U(1)D = U(1) is the discrete category on the smooth space XX underlying the Lie group U(1)U(1), then smooth functors (i.e. functors internal to smooth spaces) F:CDF : C \to D can be identified with smooth U(1)U(1)-valued functions on XX, and the functor on these functor categories induced by a smooth functor p:CCp : C \to C' does correspond to the familiar notion of pullback of functions;

  • and similar in higher degrees: if C=P 1(X)C = P_1(X) is the smooth path groupoid of a smooth space and D=BU(1)D = \mathbf{B} U(1) the smooth group U(1)U(1) regarded as a one-object Lie groupoid, then smooth functors CDC \to D correspond to smooth 1-forms Ω 1(X)\in \Omega^1(X) on XX, and precomposition with a smooth functor p:P 1(X)P 1(X)p : P_1(X) \to P_1(X') corresponds to the familiar notion of pullback of 1-forms.

This means that whether or not Kan extension corresponds geometrically to pushforward or to pullback depends on the way (covariant or contravariant) in which the domain categories CC, CC' are identified with geometric entities.

References

Textbook sources include

The book

has a famous treatment of Kan extensions with a statement: “The notion of Kan extensions subsumes all the other fundamental concepts in category theory”. Of course, many other fundamental concepts of category theory can also be regarded as subsuming all the others.

Lecture notes with an eye towards applications in homotopy theory include

For Kan extensions in the context of enriched category theory see

  • Eduardo Dubuc, Kan extensions in enriched category theory, Lecture Notes in Mathematics, Vol. 145 Springer-Verlag, Berlin-New York 1970 xvi+173 pp.

and chapter 4 of

  • Max Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64, 1982, Republished in: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 (pdf)

The (∞,1)-category theory notion is discussed in section 4.3 of

For uses of Kan extension in the study of algebras over an algebraic theory see

Preservation of certain limits by left Kan extended functors is discussed in

  • Francis Borceux, and Brian Day, On product-preserving Kan extension, Bulletin of the Australian Mathematical Society, Vol 17 (1977), 247-255 (pdf)

  • Panagis Karazeris, and Grigoris Protsonis, Left Kan extensions preserving finite products (pdf)

The general notion of extensions of 1-morphisms in 2-categories is discussed in

Revised on February 6, 2014 01:46:31 by Noam Zeilberger (195.83.213.132)