GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS

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1 GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS CHRIS HENDERSON Abstract. This paper will move through the basics o category theory, eventually deining natural transormations and adjunctions and showing the equivalence o two dissimilar deinitions o adjoint unctors, in order to state the Adjoint Functor Theorem. This theorem provides a nice result allowing us to classiy exactly which unctors have let adjoints. We borrow heavily rom Awodey s Category Theory but almost all proos are independent solutions. Almost no background in math is necessary or understanding. Contents 1. The Basics 1 2. Isomorphisms and Natural Transormations 2 3. Adjoint Functors: Two Deinitions and Their Equivalence 4 4. Limits 8 5. Preservation o Limits 9 6. Completeness and the Adjoint Functor Theorem Acknowledgements 14 Reerences The Basics Just as a group is made o a set paired with an operation and a topological space is a set with a topology, a category is made up o objects, arrows, and a rule o composition. One should note that the objects need not be a set, but I will illustrate this point later. For a ormal deinition: Deinition 1.1. A category, C, consists o: Objects: A, B, C,... Arrows:, g, h,... such that every arrow,, has a domain and codomain which are objects in C. We can represent this as: : A B where A = dom(), B = cod(). Composition: I, g C and : A B, g : B C, then g : A C. Identity: For any object, A C, there is an arrow: 1 A : A A. These must satisy our usual concepts o: Date: August 15,

2 2 CHRIS HENDERSON Associativity: h (g ) = (h g) i cod() = dom(g) and cod(g) = dom(h). Unit: i : A B. 1 A = = 1 B These concepts are not unamiliar, so let s give two quick examples o categories and move on. Example 1.2. Set: The category o sets and unctions. Example 1.3. Top: The category o topological spaces and continuous maps. Deinition 1.4. Hom C (A, B), where A, B are objects in a category C, is the set o morphisms, or arrows, rom A to B. Deinition 1.5. I C is a category, then its dual category, C op, is the category with the same objects as C but with arrows reversed. For instance, i : A B is in the category C, then op : B A would be an arrow in C op. The next concept to cover is the idea o unctors, which intuitively are like homomorphisms which we can use to map categories to categories, implying that they act on both objects and arrows and preserve composition. For a ormal deinition: Deinition 1.6. A unctor, F : C D where C and D are categories, is a mapping o objects to objects and arrows to arrows with: F (A) D or A C F () : F (A) F (B) with F () D i, A, B C and : A B F (g ) = F (g) F () F (1 A ) = 1 F (A) Example 1.7. The orgetul unctor: Take Top and Set and deine a unctor, U : Top Set, by U(X, G) = X. In other words, U orgets the topology on the space X. 2. Isomorphisms and Natural Transormations Beore we can deine an adjoint unctor we must deine isomorphisms and natural transormations. Deinition 2.1. For a category C, an arrow : A B, is an isomorphism i there exists an arrow g : B A such that: g = 1 B g = 1 A. In this case we would denote this by A = B.

3 GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS 3 A B 1 A g B g A 1 B A B This concept o an isomorphism appears identical to the normal deinition o a bijective homomorphism but does not in act always coincide. For instance, consider the category Top and take the set X = {1, 2, 3} and denote the discrete and coarse topologies as D and C, respectively. Consider the unction : (X, D) (X, C) where (1) = 1, (2) = 2, (3) = 3. This is clearly continuous, so it is in our category, and clearly bijective, in that it is one-to-one and onto, but its inverse, g : (X, C) (X, D) sending g(1) = 1, g(2) = 2, g(3) = 3 is not continuous because or example, {2} is an open set in (X, D) but g 1 ({2}) = {2} is not an open set in (X, C). Thus, g is not in Top and so does not have an inverse and is thereore not an isomorphism. Proposition 2.2. Inverses are unique. Proo. Take an arrow : A B and suppose there are inverses h, g : B A such that h = g = 1 B and g = h = 1 A. Then: h ( g) = (h ) g h 1 B = 1 A g h = g Thus, we can write the inverse o as 1. Deinition 2.3. Given categories C, D and unctors F, G : C D, a natural transormation η : F G, is a amily o arrows η C : F C GC or C C such that or any : C C, with C, C C, the ollowing diagram commutes: C F C η C GC F () C F C η C GC G() Example 2.4. Let D : Set Top be the unctor which applies the discrete topology (all sets are open) and let C : Set Top be the unctor which takes a set and applies the coarse topology (only the entire set and the empty set are open), then we see that the natural transormation η : D C could be the amily o arrows mapping every x X Top to itsel, giving us, or any : X Y D() DX η X =1 X CX DY ηy =1 Y CY C() This commutes by our requirement on the identity in Deinition 1.1 (Since D, C do not change where maps elements o X, they simply make topologies such that

4 4 CHRIS HENDERSON is continuous). We can see this more explicitly by taking x X and: (C() η X )(x) = C(x) = (x) = D(x) = (η Y D)(x) 3. Adjoint Functors: Two Deinitions and Their Equivalence Deinition 3.1. Given categories C, D and unctors F : C D, U : D C, a amily o arrows, φ : Hom D (F C, D) Hom C (C, UD), is natural in C i given any h : C C the ollowing diagram commutes: Hom D (F C, D) φ C,D Hom C (C, UD) (F h) Hom D (F C, D) φc,d Hom C (C, UD) (Where h is simply h composed on the right.) In other words, or any : F C D, we have: φ C,D () h = φ C,D( F h) φ is said to be natural in D i given any g : D D the ollowing diagram commutes: h Hom D (F C, D) φ C,D Hom C (C, UD) g (Ug) Hom D (F C, D ) Hom C (C, UD ) φc,d (Where g is simply g composed on the let.) In other words, or any : F C D, we have: φ C,D (g ) = U(g) φ C,D () A more sophisticated deinition o what it means to be natural in C or D requires the use o covariant and contravariant representable unctors, a topic discussed in section ive, but the deinition above will suice as we only need it to deine adjoint unctors. We will now give two dierent deinitions or adjoint unctors, show in an example how they match up, and then prove their equivalence. Deinition 3.2. Hom-Set: Given categories C, D and unctors F : C D, U : D C, then F is let adjoint to U i or any C C, D D, there is an isomorphism, φ : Hom D (F C, D) Hom C (C, UD) that is natural in both C and D. Deinition 3.3. Unit: Given categories C, D and unctors F : C D, U : D C, then F is let adjoint to U i there is a natural transormation η : 1 C U F such that or any C C, D D and : C U(D) there exists a unique g : F C D so that the ollowing diagram commutes: F C g D U F (C) η C C U(g) U(D)

5 GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS 5 In other words: = U(g) η C. We call η the unit. As an aside, there is a dual deinition using a counit, or a natural transormation rom U F to 1 D, but as its equivalence is proved similarly we will do no more than mention it. We can write that F is let adjoint to U (or equivalently that U is right adjoint to F ) as F U. Example 3.4. Consider the unctors, U : Top Set and F : Set Top, where U is the orgetul unctor and F is the unctor which applies the discrete topology, denoted by D. (For instance, i we have a set X, then F X = (X, D)). We will show that F U. Let s start with the unit deinition: Take any objects X Set, Y Top, and : X UY, let η X = 1 X, and the ollowing commutes: F X U F (X) η X X U() Y U(Y ) By deinition o the identity and since U(x) = (x) or all x X, is unique. (Noting that since all subsets o F X = (X, D) are open, any unction with F X as the domain is continuous, so : F X Y is continuous and thus Top.) Now or the hom-set deinition: Take X Set and Y Top. Let φ X,Y : Hom Top (F X, Y ) Hom Set (X, UY ) be the unction φ X,Y = U(g), where g : F X Y (since U F (X) = X). We now need to show that the ollowing diagram commutes, or any : X X in Set: φ(x,y Hom Top (F X, Y ) ) Hom Set (X, UY ) (F ) Hom Top (F X, Y ) φx,y Hom Set (X, UY ) (φ X,Y (g)) = U(g) = U(g) UF () = φ X,Y ((F ) (g)) Thus, φ is natural in X. For naturality in Y we need to check that the ollowing commutes or h : Y Y : φ(x,y Hom Top (F X, Y ) ) Hom Set (X, UY ) h (Uh) Hom Top (F X, Y ) Hom Set (X, UY ) φx,y

6 6 CHRIS HENDERSON To check this: (Uh) (φ X,Y (g)) = Uh U(g) = U(h g) = φ X,Y (h (g)) Thus φ is natural in Y. Now we need only check that φ is in act an isomorphism. Let ψ X,Y : Hom Set (X, UY ) Hom Top (F X, Y ) take any unction in Set and change its domain to F X and its range to Y, without changing where it sends individual elements o the set. We can do this because any unction with F X as its domain is continuous. Then it is easy to check that ψ X,Y φ X,Y = 1 HomTop (F X,Y ) and φ X,Y ψ X,Y = 1 HomSet (X,UY ) and thus, φ is an isomorphism. Theorem 3.5. Deinitions 3.2 and 3.3 are equivalent. Proo. Deinition 3.3 implies Deinition 3.2: Deine φ C,D (g) = U(g) η C (where g : F C D and η C is as deined in De We can see immediately that φ C,D : Hom D (F C, D) Hom C (C, UD). That φ C,D is an isomorphism ollows rom the uniqueness o g, given : C UD, such that = U(g) η C (i.e. because or any : C UD there is a unique g : F C D such that = U(g) η C, the unction ψ C,D () = g is an inverse to φ C,D ). So we must show naturality in C and D. For naturality in C, take : C C and h : F C D: (φ C,D (g) = (U(h) η C ) = U(h) UF () η C (rom the commutative diagram or natural transormations) Thus, φ is natural in C. For naturality in D examine g : D D Thus φ is natural in D. = U(h F ()) η C = φ C,D((F ) (h)) U(g) (φ C,D (h)) = U(g) (U(h) η C ) = U(g h) η C = φ C,D (g h) = φ C,D (g (h)) Now: Deinition 3.2 implies Deinition 3.3 Consider any : C UD and note that φ 1 C,D () : F C D. Let η C : C UF C be deined by η C = φ C,F C (1 F C ). To show that η : 1 C U F is a natural

7 GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS 7 transormation, take γ : C C and consider the ollowing commutative diagrams: Hom D (F C, F C) φ C,F C Hom C (C, UF C) Hom D (F C, F C ) φ C,F C Hom C (F, UF C ) (F γ) (UF γ) (F γ) Hom D (F C, F C ) Hom C (C, UF C ) Hom D (F C, F C ) Hom C (C, UF C ) φc,f C φc,f C γ Naturality in D Naturality in C Naturality in D gives us that: UF (γ) φ C,F C (1 F C ) = φ C,F C (F γ 1 F C ) Naturality in C gives us that: φ C,F C (1 F C ) γ = φ C,F C (1 F C F γ) Working rom one to the other: UF (γ) φ C,F C (1 F C ) = φ C,F C (F γ 1 F C ) = φ C,F C (F γ) = φ C,F C (1 F C F γ) = φ C,F C (1 F C ) γ η C γ = UF (γ) η C Thus since the ollowing diagram commutes, η is a natural transormation. C η C UF C γ C η C UF C UF (γ) Now we notice rom the commutative diagram given by De. 3.2: φ C,F C Hom D (F C, F C) Hom C (C, UF C) (φ 1 C,D ()) Hom D (F C, D) φ C,D (Uφ 1 C,D ()) Hom C (C, UD) Plugging in 1 F C : Uφ 1 C,D () φ C,F C = φ C,D (φ 1 C,D ()) = φ C,D (φ 1 C,D () Uφ 1 C,D () φ C,F C(1 F C ) = φ C,D (φ 1 C,D ((1 F C))) = φ C,D (φ 1 C,D ()) =

8 8 CHRIS HENDERSON Thus, or : C UD there exists g : F C D (where g = φ 1 C,D ()) such that the ollowing commutes: F C g=φ 1 C,D () D U F (C) η C C U(g) U(D) Now we need show that such a g is unique. Since φ is an isomorphism, the uniqueness o g requires only that we show φ C,D (g) = U(g) η C. U(g) η C = U(g) φ C,F C (1 F C ) = φ C,D (g 1 F C ) = φ C,D (g) (The second line comes rom naturality in D.) 4. Limits Deinition 4.1. A terminal object in a category C is an object 1 C such that or any C C, there is a unique morphism C 1. Proposition 4.2. Terminal objects are unique up to unique isomorphism. Proo. Take two terminal objects, 1 and 1, and ind the unique maps 1 α 1 β Since 1 is a terminal object then the only arrow rom 1 to itsel is its identity, 1 1. Since β α is a map rom 1 to itsel then it must be that 1 1 = β α. Similarly, 1 1 = α β. Thus, 1 = 1. Since 1 and 1 are terminal objects, then α and β are the unique maps or which this holds. Deinition 4.3. Let J and C be categories. A diagram o type J in C is a unctor D : J C. We write objects in the index category, J as i, j,... and values o the unctor D as D i, D j,.... Deinition 4.4. A cone to a diagram D consists o an object C C and a amily o arrows c i : C D i in C such that or each α : i j in J the ollowing commutes: C c i D i c j D α With this we can deine a category Cone(D) o cones to the diagram D, where a morphism o cones ν : (C, c j ) (C, c j ) is an arrow ν C such that the ollowing commutes or all j J: C D j ν C 1 c j c j D j

9 GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS 9 Deinition 4.5. A limit or a diagram D : J C is a terminal object in Cone(D). A inite limit is a limit or a diagram on a inite index category. We denote a limit as p i : lim D j D i j noting that the deinition says that given any cone (C, c i ) to D, there is a unique u : C lim D j j such that or all i p i u = c i. Example 4.6. Take 0 to be the empty category. Then there is only one diagram D : 0 C. Then cones are simply objects in C and so the limit is just a terminal object in C: lim D j = 1 j 0 5. Preservation o Limits Deinition 5.1. A unctor F : C D preserves limits o type J i whenever p j : L D j is a limit or a diagram D : J C, the cone F p j : F L F D j is a limit or the diagram F D : J D. We can also write this as: F (lim D j ) = lim F (D j ) For an example o this we look at representable untors. A representable unctor, denoted Hom C (C, ) : C Set or some object C C, takes an object X C to the set Hom C (C, X) and takes an arrow, : X Y, to Hom C (C, ) = : Hom C (C, X) Hom C (C, Y ) deined by (g) = g where g : C X in C. It is easy to check that this satisies all the properties o unctors. (Note: we can only use categories C considered locally small, meaning that or any X, Y C, Hom C (X, Y ) is small enough to constitute a set. A small category would be one where the objects can be considered a set and the morphisms can be considered a set.) Similarly, we also have contravariant representable unctors, denoted Hom C (, C) : C op Set which takes any : X Y to : Hom C (Y, C) Hom C (X, C) such that or g : X C, (g) = g. Lemma 5.2. Representable unctors preserve limits. Proo. Take any diagram D : J C or some index category J and assume it has a limit L. Then or any cone (A, c i ) to D there is a unique arrow u : A L such that or all i, p i u = c i. Now take any cone (S, i ) to Hom C (C, ) : C Set. Given any element s S we have a map and a cone (since that L is a limit): C i(s) L pi D i u s

10 10 CHRIS HENDERSON Note that u s is unique by the deinition o limits. Hom C (C, L) deined by g(s) = u s as given beore. commutes: S Now deine a map g : S Then the ollowing diagram g i Hom C (C, L) (pi) Hom C (C, D i ) Uniqueness o g ollows rom the uniqueness o each element u s. Deinition 5.3. The Yoneda embedding is the unctor y : C Set Cop which takes C C to the contravariant representable unctor (5.4) and takes : C D to the natural transormation (5.5): (5.4) (5.5) yc = Hom C (, C) : C op Set y = Hom C (, ) : Hom C (, C) Hom C (, D) Beore we start the next proposition, we introduce the notation C D or the category with unctors taking D to C as objects and natural transormations between such unctors as arrows. Proposition 5.6. Given objects A and B in a locally small category, C, i ya = yb, then A = B. Proo. Let ν be the natural isomorphism rom ya to yb. We will now show that the unction g : Hom C (A, B) Hom Set C op (ya, yb), deined by g() = y or any : A B, is an isomorphism. First, take any ν : ya yb and deine a unction φ : A B as φ = ν A (1 A). From the commutative diagram ollowing we see that ν = Hom C (, ν A (1 A)): Hom C (A, A) ν A Hom C (A, B) c Hom C (C, A) ν C c Hom C (C, B) where c : C A. Applying these unctions to 1 A, we get that: ν C(c (1 A )) = c (ν A(1 A )) ν C(1 A c) = φ c ν C(c) = φ c Because o this, we can deine an arrow h : Hom Set C op (ya, yb) Hom C (A, B) by h(ν ) = ν A (1 A). Now, to check that h and g are mutually inverse we irst note that or any : A B, = Hom C (A, )(1 A ). Also, or any ν : ya yb, we have ν = Hom C (, ν A (1 A)), as shown above. Thereore: g(h(ν)) = y(ν A (1 A )) = Hom C (, ν A (1 A )) = ν

11 GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS 11 and also: h(y()) = h(hom C (, )) = Hom C (A, )(1 A ) = We also write g and h or the analogous isomorphisms between Hom C (B, A) and Hom Set C op (yb, ya). Now, to show that A = B we observe, by the commutative diagram or naturality o ν: and similarly, ν A (1 A ν 1 A (1 B)) = ν A (1 A ) ν 1 A (1 B) ν A (ν 1 A (1 B) = h(ν)) h(ν 1 ) 1 B = h(ν) h(ν 1 ) Thus, A = B. ν 1 A (1 B ν A (1 A )) = ν 1 A (1 B) ν A (1 A ) ν 1 A (ν A(1 A )) = h(ν 1 ) h(ν) 1 A = h(ν 1 ) h(ν) One consequence o Proposition 5.6 is that i we have some φ : Hom C (X, Y ) = Hom C (X, Z) that is natural in X, then Y = Z. Lemma 5.7. I we have a diagram D : J C and i: φ : Hom C (X, L) = lim Hom C (X, D j ) which is natural in X, then L is a limit o the diagram D. Proo. First note that rom our proo or Proposition 5.6, we can denote any natural transormation rom Hom C (, L) to Hom C (, D i ) as (g i ) since it corresponds to some g i : L D i. Notice that as we vary X, p Xi φ X orms a natural transormation Hom C (, L) to Hom C (, D i ) which we denote (k i ) or some k i : L D i. We need to show that (L, k i ) is, in act, a cone. Take a map α : i j, where i, j J. Since (lim Hom C (X, D j ), p Xi ) is a cone (Diagram 1), then we have that: p Xj = (Dα) p Xi Thus, (L, k i ) is a cone (Diagram 2). p Xj φ X = (Dα) p Xi φ X k j = Dα k j p Xj lim Hom C (X, D j ) Hom C (X, D j ) L p Xi k i (Dα) Hom C (X, D i ) D i k j Dα Diagram 1 Diagram 2 D j

12 12 CHRIS HENDERSON Now, take any cone (X, x i ) to D and we will show that there is a map u such that the ollowing commutes: X x i u L D i ki Diagram 3 Take any one element set, 1 = { }, (which particular set is irrelevant, since any one element set is isomorphic to any other one element set) and deine a map h i : 1 Hom C (X, D i ) such that h i ( ) = x i. To show that (1, h i ) is a cone, we again take the arrow (Dα), and since (X, x i ) is a cone (Diagram 4): x j = Dα x i since star is the only element o 1, then: h j ( ) = (Dα) h i ( ) h j = (Dα) h i X xj D j 1 Hom C (X, D j ) h i Dα (Dα) Hom C (X, D i ) x i D i Diagram 4 Diagram 5 Thus, (1, h i ) is a cone (Diagram 5), and there is some unique v : 1 lim Hom C (X, D j ) such that the ollowing commutes: 1 h i v lim Hom C (X, D j ) pxi Hom C (X, D i ) h j Thus: p Xi (v( )) = h i ( ) = x i Deine a map u : X L by u = φ 1 X (v( )). To check that Diagram 3 commutes: p Xi (v( )) = x i p Xi φ X φ 1 X (v( ))) = x i (k i ) (u) = x i k i u = x i Now, we need to show that such a map u is unique, and i it is, we are inished with the proo. Since (k i ) = p Xi φ X, to show that our arrow u is uniquely determined by its composites with all the k i, we need only show that the unctions p Xi φ X are jointly one-to-one. Essentially, since φ X is an isomorphism, what we

13 GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS 13 need to show is that p Xi are jointly one-to-one. Take the commutative diagram below: Hom C (X, L) p Xi φ X φ X lim Hom C (X, D j ) pxi Hom C (X, D i ) Diagram 6 Suppose p Xi is not jointly one-to-one, or in other words, suppose that or some a and b in Hom C (X, L), p Xi φ X (a) = p Xi φ X (b) or all i. Then deine: φ X (a) i x = b m(x) = φ X (b) i x = a φ X (x) otherwise Clearly, p Xi φ X = p Xi m even though φ X m, so φ X is not unique or Diagram 6. Since this contradicts our deinition o limit, p Xi must be one-to-one. Thus, our arrow u is unique. Thus, k i : L D i is a limit o the diagram D. We have already shown that representable unctors preserve limits, and now we will use Lemma 5.7 to show that right adjoints preserve limits as well. Proposition 5.8. Right adjoints preserve limits, or more ormally, i F : C D and G : D C are unctors and F G, then i there exists a diagram D : J D with a limit, we have that the diagram GD : J C has a limit and that: G(lim D j ) = lim GD j Proo. Since F G, then or any X C we have: Hom C (X, G(lim D j )) = Hom D (F X, lim D j ) Since representable unctors preserve limits: = lim Hom D (F X, D j ) = lim Hom C (X, GD j ) By Lemma 5.7, this implies that G(lim D j ) is a limit o the diagram G. In other words: G(lim D j ) = lim GD j In other words, G preserves limits. 6. Completeness and the Adjoint Functor Theorem Deinition 6.1. A category, C, is complete i or any small category J and diagram D : J C, there is a limit in C. We are now at the point where we can state the Adjoint Functor Theorem, an important tool because it tells us a necessary and suicient condition or a unctor to have a let adjoint.

14 14 CHRIS HENDERSON Theorem 6.2. Adjoint Functor Theorem (Freyd) Let C be locally small and complete. Given any category D and a unctor F : C D between them, then the ollowing are equivalent: F has a let adjoint. F preserves limits, and or each object D D the unctor F satisies the ollowing condition: There exists a set o objects (C i ) i I C such that or any C C and any arrow : D F C, there exists an i I and arrows φ : D F C i and : C i C such that the ollowing diagram commutes: D φ F C i F C F = F () φ C i C 7. Acknowledgements I would like to thank Peter May and the REU program or providing a motivation or this paper and mentors to guide it, Michael Shulman or providing guidance and putting up with numerous drats and last minute questions, Emily Riehl and Claire Tomesch or being stand-in mentors early in the process, and Steve Awodey or writing the clearest, most coherent text on category theory. Reerences [1] Steve Awodey. Category Theory. Oxord University Press [2] Colin McLarty. Elementary Categories, Elementary Toposes. Oxord University Press