Chapter 6. All Concepts are Kan Extensions
FC03 — Riehl, Category Theory in Context.
This chapter contains 11 definitions, 3 theorems, 9 propositions, 3 lemmas, 2 corollaries, 16 examples, 1 construction and 4 remarks.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition/terminology
Kan extensions solve the categorical extension problem along K:C→D and come in left and right dual forms; they are partial adjoints to restriction K*:ED→EC.
Definition 6.1.1
A left Kan extension of F:C→E along K:C→D is a functor Lan_K F:D→E with η:F⇒(Lan_K F)K universal among such extensions; right Kan extensions are defined dually.
Remark 6.1.2
When \(K \colon \mathsf { C } \mathsf { D }\) is fully faithful, then assuming certain (co)limits exist, the left and right Kan extensions do in fact extend the functor \(F\) along K, at least up to natural isomorphism; see Corollary 6.3.9. But even a genuine on-the-nose extension of F along K might not necessarily define a Kan extension; see Exercises 6.1.i and 6.1.ii.
Example 6.1.3
For any object \(c \in \mathsf { C }\) and any \(F \colon \mathsf { C } \to\) Set, there is a bijection between elements \(x \in F c\) and natural transformations with boundary as displayed below, from the functor picking out the singleton set to \(F\) along the representable at \(c\); this is the Yoneda lemma.
Example 6.1.4
For any group G, there is a unique functor 1 BG picking out the single object in this category. For any category \(\complement ,\) object \(c \in \mathsf { C }\) , and G-object X in \(\complement ,\) natural transformations as displayed below-left and below-right correspond to maps in \(\mathsf{C}\) between \(c\) and the underlying object of the \(G\)-object \(X\), with the group action forgotten.
Proposition 6.1.5
If, for fixed \(K \colon \mathsf { C } \mathsf { D }\) and E, the left and right Kan extensions of any functor \(F \colon { \mathsf { C } } \mathsf { E }\) along K exist, then these define left and right adjoints to the pre-composition functor \(K ^ { * } \colon \mathsf { E } ^ { \mathsf { D } } \to \mathsf { E } ^ { \mathsf { C } }\)
Example 6.1.7
The objects of the functor category \(\mathsf { V e c t } _ { \Bbbk } ^ { \mathsf { B } G }\) are G-representations over the field k, and arrows are G-equivariant linear maps. If H is a subgroup of \(G ,\) restriction $ { } _ { H } ^ { G } _ { } ^ { } _ { } ^ { H }$ of a G-representation to an H-representation is simply pre-composition by the inclusion functor \(\mathsf { B } H \hookrightarrow \mathsf { B } G\) . This functor has a left adjoint, called induction, and a right adjoint, called coinduction.
Example 6.1.8
The category ∆ of finite non-empty ordinals and order-preserving maps is a full subcategory containing all but the initial object of the category \(\mathbb { \Delta } _ { + }\) , of finite ordinals and order-preserving maps. Presheaves on ∆ are simplicial sets, while presheaves on \(\mathbb { \Delta } _ { + }\) + are called augmented simplicial sets. Left Kan extension defines a left adjoint to restriction that augments a simplicial set \(X\) with its set \(\pi_{0}X\) of path components, while right Kan extension augments it with the singleton set.
Construction: For d∈D, K↓d parametrizes approximations Kc→d from C and projects to C; the…
For d∈D, K↓d parametrizes approximations Kc→d from C and projects to C; the colimit of F over K↓d supplies the pointwise left Kan-extension value when it exists.
Theorem 6.2.1
For F:C→E and K:C→D, if for each d the colimit of K↓d→C→E exists, then it computes Lan_K F(d); dually Ran_K F(d) is the limit over d↓K when those limits exist.
Corollary 6.2.6
\(I f K \colon { \mathsf { C } } \to { \mathsf { D } }\) is a functor so that C is small and D is locally small then: (i) \(I f \mathsf { E }\) is cocomplete, then left Kan extensions \(\mathsf { E } ^ { \mathsf { D } } \xrightarrow [ { K ^ { * } } ] { \mathsf { L a n } _ { K } } \mathsf { E } ^ { \mathsf { C } }\) exist and are given by the colimit formula (6.2.2). (ii) If E is complete, then right Kan extensions \(\mathsf { E } ^ { \mathsf { D } } \overset { K ^ { * } } { \underset { \mathrm { R a n } _ { K } } { \longleftrightarrow } } \mathsf { E } ^ { \mathsf { C } }\) exist and are given by the limit formula (6.2.3).
Example 6.2.7
Consider the posets \(( \mathbb { Q } , \leq )\) of rationals and \(( \mathbb { R } _ { > 0 } , \leq )\) of non-negative reals. We have the functor \(2 ^ { - } \colon \mathbb { Q } \to \mathbb { R } _ { > 0 }\) defined at the introduction to this chapter. The poset \(\mathbb { R } _ { > 0 }\) has all limits (infima) and colimits (suprema) of bounded subsets. In particular, we may use Theorem 6.2.1 to define the left Kan extension along \(\mathbb { Q } \hookrightarrow \mathbb { R }\) . The value of Lan \(2 ^ { x }\) is the supremum (the colimit) of all \(2 ^ { q }\) with \(q \in \mathbb { Q }\) and \(q \leq x ,\) . This is the usual definition of \(2 ^ { x }\) , so we conclude that the exponential function \(2 ^ { - } \colon { \mathbb { R } } \to { \mathbb { R } } _ { > 0 }\) is the left Kan extension of \(2 ^ { - } \colon \mathbb { Q } \to \mathbb { R } _ { > 0 }\) along the inclusion \(\mathbb { Q } \hookrightarrow\) R.
Example 6.2.8
Returning to Example 6.1.7, we can use Theorem 6.2.1 to give explicit formulae for the induction and coinduction functors associated to a subgroup inclusion \(H \hookrightarrow G\): when \(\mathsf{C}\) is complete and cocomplete, Corollary 6.2.6 supplies the induction, restriction and coinduction adjunctions (6.2.9).
Example 6.2.11
Recall from Example 1.3.15 that any group G has an orbit category \(O _ { G }\) whose objects are subgroups \(H \subset G .\) , which we identify with the left G-sets \(G / H\) of left cosets of H. Morphisms \(G / H \to G / K\) are G-equivariant maps, each of which is represented by a mapping \(g H \mapsto g \gamma K\) , where \(\gamma \in G\) is an element so that \(\gamma ^ { - 1 } H \gamma \subset K\) From this description, we see that the monoid of endomorphisms of the object \(G / e \in O _ { G }\) is the group Gop. So there is an embedding \({ \mathsf { B } } G ^ { \mathrm { o p } } \hookrightarrow O _ { G }\) sending the single object to \(G / e\)
Example 6.2.12
Write \(\mathbb { \Delta } _ { \leq n }\) for the full subcategory spanned by the first \(n + 1\) ordinals \([ 0 ] , \ldots , [ n ]\) in ∆. Restriction along the inclusion functor \(i _ { n } \colon \Delta _ { \leq n } \hookrightarrow \Delta\) is called ntruncation. As Set is complete and cocomplete, Corollary 6.2.6 implies that n-truncation admits both left and right adjoints: The composite comonad on \({ \mathsf { S e t } } ^ { \mathbb { \Delta } ^ { \mathrm { { o p } } } }\) is \(\mathbf { s k } _ { n } ,\) the functor that maps a simplicial set to its n-skeleton. The composite monad on \({ \mathsf { S e t } } ^ { \Delta ^ { \mathrm { o p } } }\) is \(\mathrm { c o s k } _ { n } ,\) , the functor that maps a simplicial set to its n-coskeleton. Furthermore, \(\mathbf { s k } _ { n }\) is left adjoint to cos \(\zeta _ { n } ,\) as is the case for any comonad and monad arising in this way (see Exercise 4.1.iii). Again, Theorem 6.2.1 provides an explicit de scription of the skeleton and coskeleton functors.
Definition 6.3.1
A functor \(L \colon \mathsf { E } \to \mathsf { F }\) preserves a left Kan extension \(( \mathrm { L a n } _ { K } F , \eta )\) if the whiskered composite \(( L \mathrm { L a n } _ { K } F , L \eta )\) is the left Kan extension of \(L F\) along \(K .\)
Lemma 6.3.2
Left adjoints preserve left Kan extensions. Proof 1. Suppose given a left Kan extension (LanK \(\cdot F , \eta )\) with codomain E and suppose further that \(L \colon \mathsf { E } \to \mathsf { F }\) has a right adjoint R with unit \(\iota \colon 1 _ { \mathsf { E } } \Rightarrow R L\) and counit ν : \(L R \Rightarrow 1 _ { \mathsf { F } }\) Then given \(H \colon \mathsf { D } \mathsf { F }\) there are natural isomorphisms \[ \mathsf {F} ^ {\mathsf {D}} (L \operatorname{Lan} _ {K} F, H) \cong \mathsf {E} ^ {\mathsf {D}} (\operatorname{Lan} _ {K} F, R H) \cong \mathsf {E} ^ {\mathsf {C}} (F, R H K) \cong \mathsf {F} ^ {\mathsf {C}} (L F, H K). \] Taking \(H = L \mathrm { L a n } _ { K } F\) , these isomorphisms act on the identity natural transformation as follows:
Example 6.3.3
The forgetful functor U : Top Set has both left and right adjoints. It follows from Lemma 6.3.2 that U preserves the left and right Kan extensions of Example 6.1.7, as constructed in Example 6.2.8.
Example 6.3.4
The forgetful functor $U _ { } $ Set preserves limits, because it is a right adjoint, but not colimits because the underlying set of a direct sum is not simply the coproduct of the underlying sets of vectors. Indeed, it is clear from the construction given in Example 6.2.8 that the underlying set of a G-representation induced from an Hrepresentation is not equal to the G-set induced from the underlying H-set. Lemma 6.3.2 implies, however, that the underlying set of the coinduced representation does agree with the coinduced G-set from the underlying H-set.
Definition 6.3.5
When E is locally small, a right Kan extension, as displayed below-right, is a pointwise right Kan extension if it is preserved by all representable functors \(\mathsf E ( e , - )\)
Definition 6.3.6
When E is locally small, a left Kan extension, as displayed below-left, is a pointwise left Kan extension if for all \(e \in \mathsf { E }\) the data displayed below-right is a right Kan extension: Because covariant representables preserve all limits (Proposition 3.4.5), it is clear that if a right Kan extension is given by the formula of Theorem 6.2.1, then that Kan extension is pointwise; dually, the left Kan extensions defined by Theorem 6.2.1 are pointwise. The surprise is that the converse also holds. This characterization justifies the terminology: a pointwise Kan extension can be computed pointwise as a limit in E.
Theorem 6.3.7
For locally small D,E, a right Kan extension is pointwise iff Ran_K F(d) ≅ lim(d↓K→C→E) for every d; dually a left Kan extension is pointwise iff Lan_K F(d) ≅ colim(K↓d→C→E).
Lemma 6.3.8
Given functors \(F \colon { \mathsf { C } } \to { \mathsf { E } } a n d K \colon { \mathsf { C } } \to { \mathsf { D } }\) with D and E locally small and an object d D, from which we define \(\Pi _ { d } \colon d \downarrow K \to \mathsf { C } ,\) , there is a natural isomorphism \[ \operatorname{Cone}(e, F\Pi_{d}) \cong \mathsf{Set}^{\mathsf{C}}\big(\mathsf{D}(d,K-),\ \mathsf{E}(e,F-)\big). \]
Corollary 6.3.9
If K is fully faithful, then the counit of any pointwise right Kan extension defines a natural isomorphism RanK \(F \cdot K \cong F ,\) , and dually the unit of any pointwise left Kan extension defines a natural isomorphism \(F \cong \operatorname { L a n } _ { K } F \cdot K .\)
Proposition 6.3.10
A category \(\mathsf{C}\) admits all colimits of diagrams indexed by a small category \(\mathsf{J}\) if and only if the restriction functor \(\mathsf{C}^{\mathsf{J}^{\triangleright}} \to \mathsf{C}^{\mathsf{J}}\) admits a left adjoint, defined by pointwise left Kan extension: \[ \operatorname{colim} \dashv \operatorname{res} \colon \mathsf{C}^{\mathsf{J}^{\triangleright}} \rightleftarrows \mathsf{C}^{\mathsf{J}}. \] Dually, \(\mathsf{C}\) admits all \(\mathsf{J}\)-indexed limits if and only if the restriction functor \(\mathsf{C}^{\mathsf{J}^{\triangleleft}} \to \mathsf{C}^{\mathsf{J}}\) admits a right adjoint, defined by pointwise right Kan extension: \[ \operatorname{res} \dashv \lim \colon \mathsf{C}^{\mathsf{J}} \rightleftarrows \mathsf{C}^{\mathsf{J}^{\triangleleft}}. \]
Definition 6.4.1
A class of weak equivalences is a class of morphisms in a category that contains the identities and satisfies the 2-of-6 property: if for any composable triple of arrows \(f\), \(g\), \(h\), if \(hg\) and \(gf\) are in \(\mathcal{W}\), then so are \(f\), \(g\), \(h\) and \(hgf\).
Lemma 6.4.2
Any class of morphisms in a category that contains the identities and satisfies the 2-of-6 property necessarily also contains the isomorphisms.
Definition 6.4.3
A homotopical category is a category equipped with a class of weak equivalences. A functor between such categories is homotopical if it preserves the weak equivalences.
Remark 6.4.4
Any homotopical category C has an associated homotopy category HoC characterized by the following universal property: the homotopy category HoC is initial among categories E equipped with a homotopical functor C E, that is, a functor that sends weak equivalences in C to isomorphisms.6 In analogy with ring theory, the universal functor C HoC is referred to as a localization functor.
Example 6.4.5
When C has no specified subcategory of weak equivalences, as a default we assign it the minimal homotopical structure, where the weak equivalences are precisely the isomorphisms and the localization functor is the identity.
Example 6.4.6
There are two homotopical structures of interest on the category \(\mathsf { C h } _ { R }\) of chain complexes of R-modules. In the first, the class of weak equivalences is taken to be the chain homotopy equivalences. In the second, the class of weak equivalences is enlarged to contain all quasi-isomorphisms, that is, chain maps \(f \colon A _ { \bullet } \to B _ { \bullet }\) that induce an isomorphism on graded abelian groups after applying the homology functor introduced in Example 1.3.2(viii).
Example 6.4.7
The category of topological spaces also has two homotopical structures of interest. In the first, the class of weak equivalences is taken to be the class of homotopy equivalences. In the second, this class is enlarged to contain all weak homotopy equivalences, which are maps inducing isomorphisms on all homotopy groups. The homotopy category of spaces is the homotopy category associated to the second homotopical structure. It is equivalent to the category \(\mathsf { H t p y } _ { \mathrm { C W } }\) of CW complexes and homotopy classes of maps that appears in Examples 2.1.6(vi) and (vii).
Definition 6.4.8
For F:C→D between homotopical categories, the total left derived functor LF is the right Kan extension of δF along γ:C→HoC; the total right derived functor RF is the left Kan extension along γ, whenever these Kan extensions exist.
Definition 6.4.9
A point-set left derived functor is a homotopical LF:C→D with LF⇒F inducing the total left derived functor after localization; dually a point-set right derived functor is homotopical RF with F⇒RF inducing the total right derived functor.
Definition 6.4.10
• A left deformation on a homotopical category C consists of an endofunctor Q together with a natural weak equivalence \(q \colon Q { \stackrel { \sim } { \Rightarrow } } 1 { \mathsf { c } }\) A left deformation for a functor \(F \colon { \mathsf { C } } \to { \mathsf { D } }\) between homotopical categories consists of a left deformation \(( Q , q )\) for \(\complement\) such that \(F\) preserves all weak equivalences between objects in a full subcategory containing the image of \(Q .\) There is an obvious dual notion of right deformation \(r \colon 1 \mathsf { c } \ { \stackrel { \sim } { \Rightarrow } } R\) Interestingly, left deformations can be used to construct point-set left derived functors, a homotopical universal property that is independent of the deformation used.
Proposition 6.4.11
If F:C→D has a left deformation q:Q⇒~1_C, then (LF=FQ, Fq:LF⇒F) is a point-set left derived functor; dually for right deformations.
Proposition 6.4.12
The total left derived functor of a left deformable functor is an absolute right Kan extension, and in particular pointwise.
Proposition 6.4.13
Suppose \(F \to G\) is an adjunction between homotopical categories and suppose also that F has a total left derived functor LF, G has a total right derived functor RG, and both derived functors are absolute Kan extensions. Then the total derived functors form an adjunction \(\mathbf { L } F \lnot \ \mathbf { R } G\) between the homotopy categories.
Remark 6.4.14
If \(F \to G\) is an adjunction satisfying the hypotheses of Proposition 6.4.13, then the adjunction descends to an adjunction \(\mathbf{L}F \dashv \mathbf{R}G\) between the homotopy categories \(\operatorname{Ho}\mathsf{C}\) and \(\operatorname{Ho}\mathsf{D}\), compatible with the localization functors.
Proposition 6.5.1
The left Kan extension of F:C→D along the unique functor C→1 is exactly colim F, whenever either exists; dually the right Kan extension along C→1 is lim F.
Proposition 6.5.2
- \(I f F \vdash G\) is an adjunction with unit η : \(1 \Rightarrow G F\) and counit ǫ : \(F G \Rightarrow 1\) , then \(( G , \eta )\) is a left Kan extension of the identity functor along F and \(( F , \epsilon )\) is a right Kan extension of the identity functor along G. Moreover, both Kan extensions are absolute, and in particular pointwise. (ii) Conversely, \(i f ( G , \eta \colon 1 _ { \mathsf { C } } \Rightarrow G F )\) is a left Kan extension of the identity along F and if F preserves this Kan extension, then \(F \to G\) with unit \(\eta .\)
Proposition 6.5.4
For small F:C→D with D admitting products and equalizers, Fc is the equalizer of the two canonical maps ∏{c→x}Fx ⇉ ∏{c→x→y}Fy (Yoneda equalizer formula).
Proposition 6.5.6
For small F:C→D with D admitting coproducts and coequalizers, Fc is the coequalizer of the two canonical maps ∐{y→x→c}Fy ⇉ ∐{x→c}Fx (coYoneda coequalizer formula).
Theorem 6.5.7
Density I: for locally small C, every F:C→Set is the colimit of representables indexed by (∫F)^op; dually every presheaf F:C^op→Set is the colimit of representables indexed by ∫F.
Remark 6.5.9
By Corollary 6.2.6, any functor \(F \colon { \mathsf { C } } \to { \mathsf { E } }\) whose domain is small and whose codomain is locally small and cocomplete admits a left Kan extension along the Yoneda embedding \(y \colon { \mathsf { C } } \to { \mathsf { S e t } } ^ { \mathsf { C } ^ { \mathrm { o p } } }\)
Definition 6.5.10
The codensity9 monad of \(G \colon \mathsf { D } \to \mathsf { C }\) is given by the right Kan extension of G along itself, whenever this exists.10
Example 6.5.11
Consider the inclusion \(\mathsf{Field} \hookrightarrow \mathsf{CRing}\). It has no left adjoint, because such a functor would carry \(\mathbb{Z}\) to an initial field, which does not exist. But it does have a density monad \(T\) defined by \[ TA := \prod_{\mathfrak{p} \in \operatorname{Spec}(A)} \operatorname{Frac}(A/\mathfrak{p}), \] where \(\operatorname{Spec}(A)\) is the set of prime ideals \(\mathfrak{p} \subset A\) and \(\operatorname{Frac}(A/\mathfrak{p})\) is the field of fractions of the integral domain \(A/\mathfrak{p}\).
Example 6.5.12
Familiar monads also arise as codensity monads. For instance, the codensity monad of the inclusion Fin ֒ Set is the ultrafilter monad β of Example 5.1.4(v). Theorem 6.2.1 characterizes the set of ultrafilters on a set X as a limit \[ \beta(X) \cong \lim\left(X \downarrow \operatorname{Fin} \xrightarrow{\ \Pi\ } \operatorname{Fin} \hookrightarrow \operatorname{Set}\right). \]
Example 6.5.13
The Giry monad of Example 5.1.5(iv) is also a codensity monad for the inclusion of a subcategory of \({ \mathsf { C } } \hookrightarrow\) Meas. The objects of C are the finite powers of the unit interval \(I = [ 0 , 1 ] \subset \mathbb { R } .\) equipped with the Borel σ-algebra, plus the measurable space of sequences in I that converge to 0. The objects can be regarded as convex subspaces of Euclidean space, from which perspective the morphisms are defined to be all afine maps between these spaces. Omitting the space of sequences, the subcategory of finite powers of the interval and afine maps generates a variant of the Giry monad for finitely additive probability measures as its codensity monad. See [Ave16].