The language of categories

Transcription

1 The language of categories Mariusz Wodzicki March 15, Universal constructions 1.1 Initial and inal objects Initial objects An object i of a category C is said to be initial if for any object c Ob C, there exists a unique morphism i c. Any two initial objects are isomorphic and the isomorphism is unique Final objects An object f of a category C is said to be inal if for any object c Ob C, there exists a unique morphism c f. Any two inal objects are isomorphic and the isomorphism is unique Example: the category of sets In the category of sets, denoted hereafter Set, the empty set is the unique initial object. A set is a inal object in Set precisely when it has one element. The same holds in the category of topological spaces Top Example: a partially ordered set Given a partially ordered set (S, ), consider the category S whose objects are elements of S and, for any pair s, t S, there is exactly one morphism s t if s t, and Hom S (s, t) = otherwise. In what follows we shall not distinguish a partially ordered set and the corresponding category. 1

2 The empty set, 0 =, corresponds to the empty category: Ob 0 =. (1) An element s S is an initial object of S, if s = min S; it is inal, if s = max S. In particular, the initial object is unique when it exists. The same applies also to the inal object Example: the category of all (small) categories A category Γ is said to be small if its objects and arrows form sets. Small categories form a category denoted Cat and functors Γ Γ are morphisms from Γ to Γ : Hom Cat (Γ, Γ ) = Funct(Γ, Γ ) = Γ. The empty category, 0, is the unique initial object in Cat. Category 1 is a inal object Skeletal categories The category S associated with an arbitrary partially ordered set (S, ) is an example of a skeletal category: a category is said to be skeletal if there is at most one morphism between any pair of objects. The deinition of S only requires that the relation on set S be relexive and transitive. Any small skeletal category arises this way and small skeletal categories form a subcategory of Cat which is isomorphic to the category of sets equipped with a relation that is both relexive and transitive Discrete categories A category C is said to be discrete if the only morphisms in C are the identity morphisms id (c Ob C). Small discrete categories form a subcategory in Cat which is isomorphic to the category of sets, Set. Note that the initial and inal objects of Set are also initial and, respectively, inal objects in Cat. 2

3 1.1.8 Example: the category of R-modules An R-module is an initial as well as a inal object in the category of R-modules precisely when it has only one element. We speak of this module as the zero module even though it is not unique: any one-element set can be made into an R-module. This applies both to the category of left R-modules, which is denoted R-mod, and to the category of right R-modules which is denoted mod-r. A special case is provided by the category of vector spaces, Vect, over a ield K. 1.2 Direct and inverse its Diagrams Let Γ be a small category. For any category C, functors Γ C are often called Γ-diagrams (in C). This usage is particularly frequent when Γ has very few objects and morphisms Example: the empty diagram For any category C, there is just one 0-diagram in C, where 0 denotes the empty category, cf. (1). We shall refer to it as the empty diagram Example: an object An object in a category C is the same as a Γ-diagram in C where Γ is the partially ordered set 2 (the set of all subsets of 0 = ) Example: an arrow An arrow in a category C is the same as a Γ-diagram in C where Γ is the linearly ordered set 2 (the set of all subsets of 1 = {0}). 3

4 1.2.5 Example: a commutative square A commutative square in a category C c c (2) c c is the same as a Γ-diagram in C where Γ is the partially ordered set 2 (the set of all subsets of 2 = {0, 1}) Example: a composable pair of arrows A composable pair of arrows in a category C c c c (3) is the same as a Γ-diagram in C where Γ is the linearly ordered set 3 = {0, 1, 2} Example: a parallel pair Consider the category with just two objects, s and t, and two morphisms γ, γ s t. We shall denote this category by. A -diagram in a category C is the same as a pair of morphisms in C with the common source and target c c. (4) Example: a G-object The category of monoids Mon is naturally isomorphic with the full subcategory of Cat consisting of categories with a single object: G Γ, End ( ) = Hom (, ) = G, where denotes the only object of Γ. From now on we shall identify monoids with categories having a single object. Given a monoid G and a category C, a G-diagram in C is the same as an 4

5 object c Ob C equipped with the action of G on c, i.e., with a homomorphism of monoids G End C (c). We shall refer to it as a G-object. In the case of C = Set, Top, or Vect, we talk of G-sets, G-spaces and, respectively, (K-linear) representations of G The category of Γ-diagrams Since Γ-diagrams in C are just functors, Γ-diagrams in C form a category Γ C, C = Funct(Γ, C) with morphisms being natural transformations of functors The diagonal embedding C C There is a canonical embedding of C onto a full subcategory of C which sends c C to the constant diagram Δ : and Δ (g) = c for any g Ob Γ (5) Δ (γ) = id for any γ Arr Γ (6) In what follows we shall identify C with its image in C under the diagonal embedding Direct its For a given Γ-diagram D Γ C in a category C, consider the category C whose objects are families of morphisms φ = {φ D(g) c} Ob 5

6 with common target c Ob C which are compatible with the diagram, i.e., such that φ D(γ) = φ (γ Hom (g, g )). Morphisms φ φ in category C are deined as morphisms α c c between the targets such that α φ = φ (g Ob Γ). An initial object in C is called the direct it of diagram D. 1 The direct it of D is denoted D. In use there are also terms: the inductive it of D and the push-out of D. The latter usage is generally conined to diagrams with few vertices. Exercise 1 Denote by D the partially ordered set of positive integers ordered by the divisibility relation: m n if m n. Consider the following D-diagram in the category of abelian groups Ab D = Z, D(m n) = multiplication by n m, (m, n Z ). Show that D = Q with the morphisms D Q being the homomorphisms of abelian groups Z Q, i i n (i Z) Inverse its For a given Γ-diagram D Γ C in a category C, consider the category D C whose objects are families of morphisms ψ = {ψ c D(g)} Ob 1 Note that we are using the deinite article even though direct it is usually not unique; it is unique only up to a unique isomorphism. 6

7 with common source c Ob C which are compatible with the diagram, i.e., such that i.e., such that D(γ) ψ = ψ (γ Hom (g, g )). Morphisms ψ ψ in category D C are deined as morphisms α c c between the sources such that α φ = φ (g Ob Γ). A inal object in C is called the inverse it of diagram D. The inverse it of D is denoted D. In use there are also terms: the projective it of D and the pull-back of D. The latter usage is generally conined to diagrams with few vertices whereas the former one is more often applied to diagrams with ininitely many vertices Limits and coits Another usage gaining popularity is: its for inverse its, and coits for direct its. Exercise 2 Show that is a inal object of C, while is an initial object of C. Exercise 3 Let N be the set of natural numbers with the reverse linear order and let p be a prime. Consider the following N-diagram in the category of rings D = Z/p Z, D(m n) = the canonical epimorphism Z/p Z Z/p Z, where m, n Z. Show that the inverse it of D is the ring of p-adic numbers, D = Z, with the morphisms Z D being the quotient maps Z Z /p Z. 7

8 Exercise 4 Denote by D the partially ordered set introduced in Exercise 1. Consider the following D op -diagram in the category of rings D = Z/nZ, D (m n) op = the canonical epimorphism Z/mZ Z/nZ, where m, n Z. Show that D = Z (7) prime with the morphisms D D being the products over all primes of the canonical quotient maps Z Z/p () Z, where ω (n) denotes the p-order of n Z. Hint: we use here the canonical decomposition of a cyclic group into the product of the corresponding p-groups of order p () : Z/nZ prime The formal completion of Z Z/p () Z. The inverse it D in Exercise 4 is called the formal completion of the ring of integers. It is denoted Z. The abelianization 2 of the Galois group Gal(Q) = GalQ alg /Q is canonically isomorphic to the additive group of Z. This is a deep result of Algebraic Number Theory. Exercise 5 Show that Z is canonically isomorphic to the ring of endomorphisms End Ab (μ (C)) of the group of complex roots of identity μ (C) = {ζ C ζ = 1 for some n N}. 2 The abelianization G ab of a group G is the maximal abelian quotient of G; it coincides with the quotient G/[G, G] of G by its commuattor subgroup. 8

9 Equalizers and coequalizers The inverse it of a parallel pair (4) is called the equalizer of f and g. The direct it is called the coequalizer of f and g Cartesian squares A commutative square a b (8) in a category C is said to be Cartesian if a is the inverse it of c Cocartesian squares c c d b b d (9) (10) A commutative square (8) is said to be Cocartesian if, vice-versa, (10) is the direct it of (9) For a given Γ and a general category C, certain Γ-diagrams in C possess either the direct or the inverse it while others do not. A category C is said to be complete if D exists for any small category Γ and any Γ-diagram in C. If D exists for any Γ-diagram, we say that category C is cocomplete. Exercise 6 Show that the category of sets is complete. 9

10 Hint: consider the subset of the product of all D, consisting of the tuples Ob D, which satisfy the following relations x Ob x () = D(γ) x () (γ Arr Γ). Exercise 7 Show that the category of sets is cocomplete. Hint: consider the quotient of the disjoint union of all D, Ob D, by the equivalence relation generated by the following relation: if for some γ Arr Γ. x x x = D(γ) (x ), x D () and x D (), Exercise 8 Show that the category S associated with a partially ordered set (S, ) is complete if and only if (S, ) is inf-complete, i.e., every subset of S has in- imum. Similarly, show that S is cocomplete if and only if (S, ) is sup-complete, i.e., every subset of S has supremum Functorial direct its It is not infrequent that every Γ-diagram in a given category may have the corresponding its, and that those its depend functorially on the diagram. We say that a category C possesses functorial direct its (for Γ-diagrams), if there exists a functor C C (11) such that, for any object c Ob C and Γ-diagram D Ob C, there exists a natural in c and D bijection Hom C D, c Hom C (D, Δ ). (12) where Δ denotes the diagonal embedding functor introduced in

11 Functorial inverse its We say that a category C possesses functorial inverse its (for Γ-diagrams), if there exists a functor C C (13) such that, for any object c Ob C and Γ-diagram D Ob C, there exists a natural in c and D bijection Hom C c, D Hom C (Δ, D). (14) where Δ denotes the diagonal embedding functor introduced in Adjoint functors In (12) and (14) we encounter a fundamentally important concept: a pair of adjoint functors. Suppose that, for a pair of functors, C D, (15) be given. If, for any c Ob C and d Ob D, there exists a bijective correspondence Hom C (Fd, c) Hom D (d, Gc) (16) which is natural both in c and d, then we say that F is left adjoint to G, and G is right adjoint to F If two functors, F and F, are left adjoint to G, then they are isomorphic Example: functorial direct and inverse its A category C possesses functorial direct its if the diagonal embedding Δ C C admits a left adjoint. Similarly, if Δ admits a right adjoint, then C possesses functorial inverse its. 11

12 Exercise 9 Let G be a monoid. Show that the functor, ( ) Set Set, X X = Fix (X) = {x X gx = x for any g G}, which associates with a G-set X the ixed-point set, X = Fix (X), is right adjoint to the diagonal embedding functor Δ Set Set. In particular, X = X = Fix (X) for any G-set X. Exercise 10 Let G be a monoid. Show that the functor, ( ) Set Set, X X = X /G, which associates with a G-set X the set of orbits of G on X, i.e., the set of equivalence classes of the equivalence relation x x if x = g x and x = g x for some x X and g, g G. is left adjoint to the diagonal embedding functor Δ Set Set. In particular, X = X = X /G for any G-set X The sets of G-invariants and G-coinvariants Notation X and X is standard in representation theory where X is called the set of G-invariants, and X is called the set of G-coinvariants of G-representation X. Notation Fix (X) and X /G, and the corresponding terminology: the set of ixed-point set and the set of orbits, are primarily used for non-linear actions, i.e., for G-actions on sets, topological spaces, algebraic varieties. 12