SOME WORDS ON KAN EXTENSIONS. MARCO VERGURA We start with what is, in many respects, a fundamental example for Kan extensions. Let C be a small category and let PSh(C) be the category of presheaves (of sets) over C, i.e. the functor op category SetC of contravariant functors from C to Set. Such a category comes equipped with a fully faithful functor y: C −! PSh(C);A 7! C(−;A); the Yoneda embedding. Given a presheaf P : Cop −! Set, recall that the category of elements of P is the category Z P C with objects given by pairs (A; x), for A 2 Ob(C) and x 2 P (A), and morphisms (A; x) ! (B; y) between two such pairs given by morphisms A ! B in C such that (P (A ! B))(y) = x. Yoneda's lemma implies that the category of elements of P is isomorphic to the category (y # P ) having as objects pairs (A; τ), where A 2 Ob(C) and τ : yA ! P is a natural transformation, and as morphisms (A; τ) ! (B; σ) those arrows A ! B in C such that σ ◦ y(A ! B) = τ. Note that there is a projection (or forgetful) functor πP :(y # P ) −! C; (A; τ) 7! A: We can now state the universal property of the category PSh(C) or, better, of the pair (PSh(C); y). Theorem 1. Let C be a small category and let F : C −! D be a functor into a cocomplete category D. Then there exist a unique (up to isomorphisms) cocontinuous functor LF : PSh(C) −! D such ∼ that LF ◦ y = F , i.e. such that the following diagram commute y C / PSh(C) L F F D =∼ up to an invertible natural transformation LF ◦ y −! F . In fact, LF can be defined as LF (P ) := colim(F ◦ πP ); for each P 2 PSh(C). Furthermore, LF has a right adjoint RF given by D(F (−); ?): D −! PSh(C);X 7! (D(F (−);X): Cop −! Set): A proof of this classical result can be found (for example) in [MLM94, x1.5]. As a byproduct, we get another well-known fact about presheaves, usually referred to by saying that every presheaf is a colimit of representables Corollary 2. Let C be a small category and let P 2 PSh(C). Then there is a canonical isomorphism =∼ colim(y ◦ πP ) ! P: Date: January 21, 2016. 1 Proof. With respect to the situation in Theorem1, take D := PSh(C) and F := y. Then the ∼ identity functor IdPSh(C) : PSh(C) −! PSh(C) certainly verifies IdPSh(C) ◦y = y and is cocontinuous, so there must be a canonical isomorphism Ly ! IdPSh(C). Here are two fundamental examples of Theorem1, which are both instances of what is sometimes called the nerve-realization paradigm. Example 3. Let ∆ be the simplex category whose objects are the finite non-empty ordinals [n] = f0; 1; : : : ; ng, for n 2 N and whose morphisms are monotone maps between them. Presheaves ∆op −! Set are commonly known as simplicial sets and the category PSh(∆) is denoted by sSet. There are embeddings ( n ) n n+1 X ∆ −! Top; [n] 7! ∆ := (t0; : : : ; tn) 2 R : ti = 1 and 8i 2 f0; : : : ; ng (ti ≥ 0) i=0 and ∆ −! Cat; [n] 7! [n]; where [n] 2 Cat is seen as a category having as sets of object f0; : : : ; ng and with a morphism i ! j (for i; j 2 f0; : : : ; ng) precisely when i ≤ j. We have then diagrams of functors y y ∆ / sSet ∆ / sSet Top Cat Thus, applying Theorem1, we get pairs of adjoint functors |·| τ1 & & sSet? Top sSet? Cat d d Sing N The functor j·j is usually called the geometric realization functor, whereas Sing is called the singular complex functor. Furthermore, τ1 is known as the fundamental category functor, while N is the nerve functor. Note, that, for each X 2 Top, C 2 Cat and [n] 2 ∆, we have, by definition n Sing(X)n = Top(∆ ;X);NCn = Cat([n]; C); where, for a simplicial set S, Sn denotes S([n]), the set of n−simplices of S. These adjoint pairs play a fundamental role in the homotopy theory of spaces. Kan extensions allow one to generalize (to some extent) the situation of Theorem1 by replacing the Yoneda embedding C −! PSh(C) with an arbitrary functor C −! B. Before seeing how this generalization works, we first need to introduce the core concept of these notes. Definition 4. Let F : A −! B and G: A −! D be functors. (1) A pair (K; α), where K : B −! D is a functor and α: G −! KF is a natural transfor- mation, is a left Kan extension of G along F if it satisfies the following universal property. Given any pair (H; β) where H : B −! D is a functor and β : G −! HF is a natural trans- formation, there exists a unique natural transformation γ : K −! H such that γF ◦ α = β. (2) Dually, a pair (K; τ), where K : B −! D and τ : KF −! G, is a right Kan extension of G along F if it satisfies the following universal property. Given any pair (H; σ) where H : B −! D and σ : HF −! G is a natural transformation, there exists a unique natural transformation δ : H −! K such that τ ◦ δF = σ. 2 As usual when dealing with universal properties, a left and a right Kan extension of G along F , if they exist, are unique up to within a unique isomorphism. One then commonly talks about the left and right Kan extension of G along F and denote them by LanF (G) and RanF (G) respectively. Also, one commonly uses the term "left Kan extension\ and the notation LanF (G) both to mean the pair (K; α) as in Definition4 and to just indicate the underlying functor K. The same holds for right Kan extensions. The promised (partial) generalization of Theorem1 comes in terms of the existence of left Kan extensions along a functor into a cocomplete category. Theorem 5. Let F : A −! B and G: A −! D be functors, where A is small. (1) Suppose D is a cocomplete category. Then LanF (G) exists and is given, for B 2 B, by F (LanF (G)) (B) := colim(G ◦ πB ); F where πB :(F # B) −! A is the projection functor (A; F A ! B) 7! A. Here (F # B) is the category having as objects pairs (A; F A ! B), where A 2 A and FA ! B is a morphism in B, and as morphism (A; t: FA ! B) ! (A0; t0 : FA0 ! B) those morphisms a: A ! A0 in A such that t0 ◦ F (a) = t. (2) Dually, suppose D is a complete category. Then RanF (G) exists and is given, for B 2 B, by 0F (RanF (G)) (B) := lim(G ◦ πB ); 0F where πB :(B # F ) −! A is the projection functor (A; B ! FA) 7! A and (B # F ) is the opposite of (F # B) above. Proof. The proof is quite formal. We only prove (1), as (2) follows dually. First of all we show how F 0 the assignment B 7! colim(G ◦ πB ) extends to a functor B −! D. Fix then g : B ! B in B. To F F get a map colim(G ◦ πB ) ! colim(G ◦ πB0 ) in D, by the universal property of the colimit, we need F F to find a cocone from G ◦ πB over colim(G ◦ πB0 ). If we let 0 0 F λ = λ(A;f 0) : GA ! colim(G ◦ πB0 ) (A;f 0)2(F #B0) be the colimiting cocone, we can take our needed cocone to be 0 F λ(A;gf) : GA ! colim(G ◦ πB0 ) : (A;f)2(F #B) F F We then get a uniquely induced map colim(G ◦ πB ) ! colim(G ◦ πB0 ) which we take to define 0 (LanF (G))(g) and which is such that, for each (A; f) 2 (F # B), (LanF (G))(g) ◦ λ(A;f) = λ(A;gf). Here λ : GA ! colim(G ◦ πF ) (A;f) B (A;f)2(F #B) is the colimiting cocone. The uniqueness property of (LanF (G))(g) ensures that we get a functor LanF (G): B −! D. We can find a natural transformation α: G −! LanF (G) ◦ F by simply taking the family α = α := λ : GA ! colim(G ◦ πF ) ; A (A;1FA) FA A2A whose naturality follows from the definition of LanF (G) on arrows of B. Finally, we need to show the universal property of the pair (LanF (G); α). Let then H : B −! D be a functor and β : G −! HF be a natural transformation. We are going to define the required F factorization γ : LanF (G) −! H as follows. Given B 2 B, we have a cocone from the functor G◦πB given by β Hf GA −!A HFA −! HB (A;f)2(F #B) 3 F and thus, we get a unique map γB : colim(G ◦ πB ) ! HB such that γB ◦ λ(A;f) = Hf ◦ βA, for each (A; f) 2 (F # B). To see that the familiy of maps (γB)B2B gives rise indeed to a 0 natural transformation γ : LanF (G) −! H, let g : B ! B be a map in B and consider, for any (A; f) 2 (F # B), the following diagram F γB colim(G ◦ πB ) / HB O O (LanF (G))(g) Hg " γB0 | λ F 0 Hf (A;f) colim(G ◦ πB0 ) / HB < b 0 H(gf) λ(A;gf) GA / HFHFA A βA Here the outer square and the lower trapezoid commute by definition of γB and of γB0 respectively, whereas the left triangle is commutative by definition of (LanF (G))(g) and the right triangle com- mutes because H is a functor.