# Why is category theory useful

## Introduction to category theory

Transcript

1 Introduction to Category Theory Christian Dzierzon 1 June 7, University of Bremen,

2 Table of contents 1 Introduction Motivation Basics / Set theory Bibliography Categories Deinition and examples Constructions with categories Special morphisms and objects Functors and natural transformations Functors Operations with functors Natural transformations Equivalence of categories Functor categories Limits sources and sinks Products and coproducts Limits and colimits Functors and limits Limits in functor categories Properties of hom-functors & generators Adjunctions Universal bearings & characterizations Adjoints and (co-) limits Existence of adjuncts Factoring structures Factoring structures for sources Special factorizing structures Existence of factoring structures and the connection to limits. 83 1

3 A Exercise Exercises 85 A.1 Categories and Functors A.2 Natural Transormations A.3 Products and Coproducts A.4 Adjunctions A.5 Limits and Completeness A.6 Factorization Structures Bibliography 91 Index 92 2

4 Chapter 1 Introduction This script was written parallel to the lecture Algebra II, WiSe 03/04, University of Bremen, held by Pro. H.-E. Porst, Christian Dzierzon and Christoph Schubert. It is the elaboration of a two-hour (!) Introductory event to category theory. It is therefore only intended to complete and supplement the lecture notes and to follow up the content and does not claim to be complete. It cannot and should therefore not replace the numerous textbooks on category theory. The third section of the introduction contains a few literature recommendations, including Category Theory by H. Herrlich and G.E. Strecker, which served as the basis of the content of the categorical part of the lecture. Contents of the second part of the lecture Algebra II, which mainly beas with module theory, are not included in this script. The canonical content of a lecture on algebra is assumed to be known. The new German spelling is also used. At this point, the author would like to thank Christoph Schubert, who helped him with the preparation of the script and especially the exercises, as well as Frauke Günther, who made her notes available. 1.1 Motivation Why category theory at all? This question may be asked by many mathematicians who learn about category theory for the first time in the course of their studies or who are confronted with terms of this theory. In just a few sentences, category theory could be rewritten as follows: Category theory is a relatively new area of mathematics, at least in contrast to the classical areas such as analysis, differential geometry, etc. In 1942, S. Eilenberg and S. MacLane first introduced the term category, Functor and natural transformation an elementary term of today's category theory with the intention of formulating a uniform language for homology and cohomology within the algebraic topology. A short time later the first article 2 appeared, which dealt exclusively with categories and functors. In the course of time, an independent metamathematic theory developed which, by examining common structures in various mathematical theories, attempts to apply these theories to a higher 1 In: Natural Isomorphisms in Group Theory, Proc. Nat. Acad. Sci. USA 28,, General Theory o Natural Equivalences, Trans. AMS 58,,

Characterize 5 levels of abstraction. The attempt is made to identify and normalize existing analogies. Therefore, category theory can be viewed as a kind of (meta) language about mathematics on the one hand, and as an independent sub-area within mathematics on the other. There have even been some attempts to use category theory as the basis of mathematics 3, but we will not go into that here. Because of this claim of category theory to represent a kind of metatheory, it is closely linked to the problem of a sufficiently strong, set theoretical basis of mathematics. More details can be found in the second section of this introduction. Bearing diagrams are an important aid in category theory. They enable the simple representation of complex facts, serve to provide evidence and support a calculation that is as element-free as possible. In so-called concrete categories in particular, the quantities underlying the objects and their elements are deliberately not considered, if this is possible in a proof. Rather, the attempt is made to express properties abstractly using the morphisms between the objects. This makes it easier to recognize analogies, to generalize properties to other categories and to arrive at a better understanding. This relatively high level of abstraction is what makes this elegant theory so strong on the one hand, and it makes it difficult to access it on the other. So here are some motivating and hopefully illuminating examples. Example products in different variations. Let A 1, A 2, C be either (a) sets, or (b) groups, or (c) topological spaces; then applies in all three cases: There is a set (or group, top. space) A 1 A 2, two mappings (or group homomorphisms, continuous fig.) π i: A 1 A 2 A i, i = 1, 2, so that for each pair of mappings (or group hom., Continuous fig.) 1: CA 1, 2: CA 2 there is exactly one map (or group hom., Continuous fig.) With: CA 1 A 2 π 1 = 1 and π 2 = 2. (Of course, π 1 and π 2 are the usual projections with π i (x 1, x 2) = xi and are deified by (x) = (1 (x), 2 (x)).) In category theory this is indicated by a commutative diagram 4 clarifies: 3 Cf. MacLane / Moerdijk A detailed description of the term is given in. First of all, we understand a diagram to be a bearing scheme such as in (), where the corners stand for quantities, groups, etc. and the arrows stand for the corresponding figures in between. A diagram commutes when the (only possible) compound of two figures in each of its sub-triangles results in the third figure. In this introductory chapter we only use commutative diagrams. 4th

6 C 1 2 A 1! A 2 π 1 A 1 A 2 π2 () It is said that there is exactly one, so that the diagram commutes and this is called a universal property of the triple (A 1 A 2, π 1, π 2). Similar known universal properties are discussed below. They make it possible to clearly characterize constructions up to isomorphism. If in one of the cases (a), (b) or (c) there is a corresponding triple (P, ρ 1: PA 1, ρ 2: PA 2) which has the same universal property, then there is a bijection (or . Isomorphism, homeomorphism) g: PA 1 A 2 with π ig = ρ i, i = 1, 2. The following analogies can be recognized immediately: Analogies: categorical: set group top. Space object illustration homomorph. steady fig. morphism bijection isomorphic. Homeomorphic. Isomorphism kart. Prod. Direct. Prod. Top. Prod. Product An abstract, categorical proof of the above-mentioned uniqueness up to isomorphism then looks something like this: Univ. Property of (A 1 A 2, π 1, π 2) :! : π i = ρ i id P P ρ 1 ρ 2 A 1 g A 2 π 1 π2 A 1 A 2 id Univ. Property of (P, ρ 1, ρ 2): From this it follows! g: ρ i g = π i π i (g) = π i and ρ i (g) = ρ i. For id and id P we also have π i id = π i and ρ i id P = ρ i. Because of the uniqueness based on the universal properties, then g = id P and g = id, i.e. = g 1 and P = A 1 A 2.5

7 (For the sake of clarity, id A1 A 2 was briefly written id.) Note that this proof is completely independent of the special properties such as being a group, possessing a topological structure, etc. One proves a theorem for different ones in one go Areas of mathematics! Example (duality) The universal property expressed in the diagram () abstractly defines a product which, in the three cases described, provides the respectively known construction. The idea of duality or the dual term for a categorical term such as that of the product can be illustrated very clearly using this example. Roughly speaking, the dual term emerges from the original by reversing all bearings involved. In this case, this provides the following universal property: A 1 C 1 2! A 2 µ 1 µ 2 A 1 + A 2 (i: A i C) i = 1.2! : µ i = i What does that result in the three cases under consideration? (a): Disjoint union of two sets. (b): Free product of two groups. (c): Topological sum of two topological spaces. This is categorically referred to as a co-product (µ 1, µ 2, A 1 + A 2). The prefix Co- makes it clear that this term emerged from the term product by dualizing. Examples (universal bearings) We consider some well-known examples from (linear) algebra and topology, which are intended to clarify the categorical concept of universal bearings. (1) Free objects (algebra): Let X be an (Abelian) group or a ring and B X a subset. Then, as is well known, X is called rei over B, all of which applies: B ι X! Hom. Y from. Grp. Or ring In the case of vector spaces or modules, this is formulated accordingly. (2) Tensor product of vector spaces (lin. Alg.): Let U, V, W be (finitely dimensional) vector spaces, then the tensor product U V of U with V is a vector space, the following 6

8 universal properties fulfilled: There is a homomorphism g: U Hom (V, U V) with U g Hom (V, U V) Hom. () Hom (V, W) U V! Hom. W (Note that Hom (,) is measured as a vector space.) Here g is deined by (g (u)) (v): = uv for u U, v V, and the following is deined from the basic elements: ( uv): = ((u)) (v). (3) Čech-Stone compactification (topology): Let X be a completely regular topological space (which is therefore also T 2), then there is a compact T 2 space βx and a continuous mapping β: X βx with the following universal Property: X β βx! steady Y comp. T2 continuous (4) Abelian relation (algebra): Let G be a group, then there is an Abelian group A G and a group homomorphism r G: G A G with: G r G A G! Hom. Hom. A from. Grp. Here, A G is the quotient G / [G, G], where [G, G] denotes the commutator of G. The horizontal bearings appearing in all examples are examples of so-called universal bearings that are closely linked to the important categorical concept of adjoint situations, as we shall see later. The pairs (ι, X), (g, UV), (β, βx) and (r G, AG) are then called universal bearings for the corresponding objects B, U, X, and G. It seems that example (2 ) by the fact that the target area of the universal bearing is not UV, but Hom (V, UV). As we will see later, this is due to the fact that a universal bearing refers to a so-called functor, i.e. the functor Hom (V,) is also applied to the object U V. In example (1) or (3), (4), however, this functor U or E is a so-called forgetting functor (U) or embedding functor (E), which is notationally suppressed for reasons of clarity. For example, in (4) it should read E (A G) instead of A G, E () instead and E (A) instead of A. Thus, the four examples do not differ in their categorical structure. 7th

9 1.2 Fundamentals / set theory As already mentioned, one needs a sufficiently strong set theoretical basis in order to be able to operate category theory meaningfully. There are some soothing approaches to this, some of which meet different requirements. The set-theoretical details of these different approaches should not concern us further here. Rather, in the following we briefly outline the ontological standpoint that we have chosen, which represents a solid basis and is advocated by many authors. From now on it should be used tacitly and be available to us as a carefully chosen foundation. Our basic assumption is that there is a model for the Zermelo-Fraenkel axiom system with axiom of choice (ZFC). In addition, that there exists a sufficiently large set U, the so-called (set) universe, which is closed under the usual set-theoretic constructions. For example, U should contain the set of natural numbers ω, for every two sets u, v U, {u, v}, (u, v) and uv should be elements of U, xu U should hold x U, etc. Then Let us distinguish and consider only the following two types of entities: Classes: subsets A U. Sets: elements possibly. This is a very intuitive approach: Everything that is outside of our universe does not exist for us! Since ZFC can of course only talk about sets, some authors may describe the elements as small sets in contrast to the other sets. Obviously, according to the regularity axiom U / U, the universe U is not a set in our way of speaking! Because of the ordered closing properties, of course, the following also applies: x u U = x U u U = u U. Every set is a class, but not the other way around. A U is therefore also called a proper class if A is not a set. For example, the class of all sets is a real class, namely U. The distinction between sets and classes helps in axiomatic set theory to overcome the well-known antinomies and problems of naive set theory, such as those arising from the assumption of a set of all sets or similar result. Finally, the following two well-known slogans should help to clarify the difference between sets and classes: Classes are parts of the universe, sets are its elements. Sets are classes that appear as an element of another class. (Because B A U = B U.) 8

10 1.3 Bibliography Only a few works are mentioned here that are relatively well suited as introductory literature for beginners. The first chapters of 1. in a) served as content guidelines for the lecture. a) For an introduction to category theory: 1. H. Herrlich, G. E. Strecker: Category Theory, Heldermann Verlag Berlin, [Very suitable for self-study, especially for beginners. Is available in the textbook collection of the SuUB.] 2. J. Adámek, H. Herrlich, G. E. Strecker: Abstract and Concrete Categories. The Joy o Cats, John Wiley and Sons, [Soon also available as an online version.] 3. D. Pumplün: Elements of category theory, Spektrum-Verlag, [Positive: Easily understandable, easy-to-solve exercises integrated in the running text. Negative: Object-free deinition of the category.] Further reading: 4. F. Borceaux: Handbook o Categorical Algebra, Cambridge University Press, S. Mac Lane: Categories or the Working Mathematician, Springer Verlag, 1971 or [Recommended for advanced students.] 6. S. Mac Lane, I. Moerdijk: Sheaves in Geometry and Logic, Springer Verlag, b) Set theory / basics of mathematics: 1. M. German: Introduction to the basics of mathematics, University of Bremen printing, [Overall somewhat difficult to read , Chapters 5.1 to 5.15 still offer a good overview.] 2. PR Halmos: Naive set theory, Göttingen, [Despite the title, a highly recommended introduction to (axiomatic) set theory.] 3. Chapter II and the appendix of 1 . in a) offer a good overview of the various set-theoretical possibilities for laying the foundations for category theory. For readers with some experience in axiomatic set theory, everything essential can be found there. 9

11 Chapter 2 Categories 2.1 Deinition and examples Deinition A category C is a 6-tuple where C = (O, M, dom, cod, id,) (1) O is a class whose elements are called C objects A, B , C, ... etc. (notation: ob (c): = O); (2) M is a class whose elements are called C-morphisms, g, h, ... etc. (notation: mor (c): = M); (3) dom, cod: mor (c) ob (c) are operations (i.e. mappings between classes); one writes briefly: A B or A B for dom () = A, cod () = B and calls A the domain or B the codomain of; (4) id: ob (c) mor (c) is an operation, called identity, with id (a): A A (notation: id (a) =: id A, sometimes also 1 A); (5): D mor (c) is an operation, called composition, with D: = {(g,) mor (c) mor (c) dom (g) = cod ()} and (g,) =: g or g for short (speaking: g deiniert iff (g,) D); so that the following conditions are fulfilled: (M) Matching condition: (g,) D = dom (g) = dom () cod (g) = cod (g) 10

12 i.e. always commutates; (A) Associativity: i.e. always commutes; AB gg C (g,), (h, g) D = h (g) = (hg) A hg BD gh C g (U) Unit law: (id B,), (g, id B) D = id B = g id B = g ie id B is a unit (or identity) and always commutes; A B g id B B g C (S) Smallness condition: for every pair (A, B) of C-objects the class is a set. C (A, B): = hom C (A, B): = {mor (c): A B} Remark (i) In category theory, diagrams are often used to illustrate or provide evidence. This makes proofs clearer and more structurally understandable in contrast to proofs with long equations. A diagram consists of corners (often provided with or an object name) and bearings (ot marked with morphism names) between corners. Various examples of diagrams have already appeared in the introductory chapter. A diagram is said to be commutative or commutated, and the following applies: For every pair of corners A, B of the diagram, the composition 1 ... n of all morphisms along any path of bearings A n ... 1 B always results in the same morphism A. B. (ii) The objects ob (c) of a category C and the units of C have a one-to-one relationship: 11

13 For every C-object B there is exactly one C-morphism id B that satisfies (U). If u: BB is another such C morphism, then from (U) for id B and u: id B = u id B = u. Therefore there is a 1: 1 correspondence B id B and id B is the uniquely determined C-identity of B. Therefore the operation id is clearly established by the other data. In addition, this correspondence enables an object-free deinition of categories, which appears somewhat counterintuitive to the authors, but also offers advantages, since some proofs are considerably simpler (cf. literature: Pumplün 1999). Deinition A category C is called (1) small, all C (i.e. whether (c) and mor (c)) is a set; (2) discrete, alls mor (c) = id [ob (c)] holds; (3) connected, alls always C (A, B) applies. Examples (1) Set: Category of the sets X, Y, ... (objects) and the images: X Y, ... (morphisms) in between. Of course, dom (): = X, cod (): = Y and the composition is the usual composition of images. As the notation suggests, for an object X, i.e. a set, id X is the identical mapping to X: id X: x x. (2) Grp: Category of groups (objects) and group homomorphisms (morphisms) in between. Remaining data as in set. (3) Top: Category of topological spaces (objects) and continuous mappings (morphisms) in between. Remaining data as in set. (4) R-Mod (R a ring): Category of the left-R modules (objects) and module homomorphisms (morphisms) in between. Remaining data as in set. (Analog: Mod-R.) (5) POS (partially ordered sets): Category of partially ordered sets (objects) and monotonous mappings (morphisms) in between. Remaining data as in set. (6) Lat: Category of associations (objects) and association homomorphisms (morphisms) in between. Remaining data as in set. (7) Ab: Category of Abelian groups (objects) and group homomorphisms (morphisms) in between. Remaining data as in set. (8) Rng: Category of rings (objects) and ring homomorphisms (morphisms) in between. Remaining data as in set. (Analog: Field (body) and Ring (rings with one).) (9) Rel: Category of sets (objects) and relations (morphisms), i.e. for sets A, B r: AB is then and only a Rel morphism, if r AB holds. And the composition? Exercise assignment. 12th

14 (10) Every pre-ordered class (X,) (that is, X is a class and XX a relexive and transitive relation) can be understood as a category X: Set whether (x): = X and for x, y X exists exactly one morphism xy if xy is. This also establishes the composition of morphisms, because according to transitivity, x y and y z also result in x z. Relexivity guarantees the existence of identities x x. In particular, each class X can be given the discrete order X = {(x, x) x X} and omitted as a category. These are precisely the discrete categories. (11) A monoid (M ,, e) can be interpreted as a category M as follows: Set ob (m): = 1 = {} and mor (m): = M, ie every element m M gives an M-morphism m and for m, n M their product mn M is the compound of the morphisms m and n. (12) Of course there are also finite categories, such as 0 = empty category 1 = 2 = = = etc. or about ,,,. Note that examples (1) - (9) are real classes and only (11) and (12) are examples of small categories (in the sense of the deinition above). In the example (10) this depends on whether the X there is a real class or a set. The above examples show how many different mathematical areas are covered by category theory. Quantities and figures certainly had a motivating influence on the definition of the concept of the category, which is also clear from the notation. This then also covers those mathematical areas whose objects of consideration are sets with an additional structure and special, structure-preserving images (such as in example (2) - (8)). Pre-ordered classes and monoids can also be described categorically, as examples (10) and (11) show. In addition, the term category is relatively algebraic and thus allows some standard constructions (see next section). With all this, it should always be kept in mind that for a categorical description there are always two basic concepts object and morphism 13

15 belong. This maxim is often referred to as the categorical imperative among category theorists 1, but does not make the same moral claim as the eponym from philosophy. 2.2 Constructions with categories As we have seen, the term category can be deified relatively algebraically, so that the following constructions can obviously be carried out in a reasonable way: Subcategory B of C: ob (b) ob (c), mor (b) mor ( c) and the operations dom, cod, id, in B are the restrictions of the corresponding operations in C. For A, B ob (b) then B (A, B) C (A, B) holds. If B (A, B) C (A, B) also holds for all pairs A, B of B-objects, then B is called the full sub-category of C. Quotient category with respect to a congruence. Product category B C: The morphism class is mor (b) mor (c), the object class ob (b) ob (c) and it is linked point by point. Sum category B + C (disjoint union). We forego a detailed treatment of these constructions at this point, since in the following only the term sub-category is required, which should be adequately explained by the above sketch. Deinition For each category C = (O, M, dom, cod, id,) the dual category C op is deined by where C op = (O, M, cod, dom, id,), g in C op: g in C. Note that the operations dom and cod are interchanged, ie all bearings are reversed: AB g C in CC g BA in C op. The concept of duality is of particular importance in category theory, as will become clear later. First of all, two constructions should be mentioned, the second being interesting with regard to duality: 1 Based on the famous categorical imperative of Immanuel Kant, of course. 14th

16 DF category C 2: Objects are all C morphisms and the morphisms in C 2 are pairs (a, b): (AB) (C g D) of C morphisms a: AC, b: BD, for the A a C g B b D commutates. The composition of morphisms is deined point by point. Comma category: For each C-object A there is a category AC, whose objects are C-morphisms AB, A g C, ... with domain A and whose morphisms h: (AB) (A g C) C-morphisms h: BC are commutated for AB hg. The composition of the morphisms is in C. C A is denoted in the same way, where the objects are C morphisms B A with codomain A. For this, C A = (A C op) op (exercise task!) Applies. Remark (duality) Clearly, C = (C op) op. An example should clarify how to get a dual proposition S op for a categorical proposition S. First, let P be a property of a category C that can only be expressed in terms of objects and morphisms. The dual property P op is then the property corresponding to P in C op, shown in C. In other words: P op emerges from P by turning bearings. For example, let X be a C-object and P (X) Y ob (c)! Y X mor (c). Then the property corresponding to P (X) is in C op: Y ob (c op)! Y and this in turn expressed in C reads: C X mor (c op) P op (X) Y ob (c)! X Y mor (c). 15th

17 In C = Set, for example, this means: P (X) X = 1 P op (X) X = Often times a dual term is given the prefix Co-, e.g. in the introductory chapter: (A 1 A 2, π 1, π 2) product in C (π 1, π 2, A 1 A 2) coproduct in C op A property P is called self-dual, where P = P op applies. If S is a categorical statement (in which categorical properties may occur), one has by deinition S op applies in CS applies in C op and thus finally the following duality principle: If a categorical statement S applies in every category, then the dual one also applies Statement S op in each category. One should be very careful when dualizing statements, because errors often creep in here, especially if the dual statement looks different than expected. 2.3 Special morphisms and objects Deinition (1) A C-morphism A B is called retraction in C (or also C-retraction), if there is a C-morphism B g A with g = id B (i.e. everything has a right inverse). (2) A C-morphism A B is called a cut in C (or C-cut, co-retraction in C), if there is a C-morphism B g A with g = id A (i.e. all has a left inverse). (3) A C-morphism is called isomorphism in C (or C-isomorphism) because it is a C-cut and -retraction. Remark (i) Obviously, retraction and intersection are dual terms to one another. Namely, if S (, C) the statement is a C-cut, then this reads in C or C op: S (, C) mor (c) g mor (c): g is C-identity S (, C op ) mor (c op) g mor (c op): g is C op identity Then the dual statement is: S op (, C) mor (c) g mor (c): g is C identity which means that is a C retraction. (ii) According to (i), isomorphism is a self-dual term. Examples (1) A morphism in Set is a cut (or a retraction) if and only if is injective and dom () holds (or if is surjective). 16

18 (2) Let A B be a morphism in R-Mod. Then there is a cut in R-Mod if and only if is injective and [A] is a direct summand of B. Furthermore, there is a retraction if and only if there is a projection p: A S au a sub-module S A and an isomorphism h: S B with = h p. (3) Let X Y mor (top). Then there is a top cut if and only if there is a topological embedding and [X] is a retract of Y (i.e. there is continuous r: Y [X] with r [X] = id). On the other hand, there is a top retraction if and only if there is a homeomorphism h and a topological retraction r with = h r. Sentence (1) A B, B g C C cuts = A g C C cuts. (2) A B, B g C C retractions = A g C C retraction. Proof: Simply. From the duality of the term (1) = (2) and thus also (2) = (1). If, for example, (1) applies and g are C retractions as in (2), then g are C op slices and thus also g according to (1). So g is a C retraction. According to the above sentence, isomorphisms are apparently also closed under composition. Theorem If, g mor (c), such that g is deined and a C-cut, then there is also a C-cut. Proof: If g is a C-cut, then there is h mor (c) with (h g) = h (g) = id dom (), i.e. h g is left inverse to. The following theorem is the dual version of the previous one: Theorem are, g mor (c), so that g is deined and a C retraction, then g is also a C retraction. Proof: Follows through dualization from In the course of time we will go into the dual terms less and less. Each deinition or each sentence then tacitly contains a dual deinition or hides a dual statement, which of course also applies and will be used. Theorem For mor (c) are equivalent: (1) is isomorphism in C; (2) has exactly one right inverse h and exactly one left inverse g in C and g = h. Proof: (2) (1): Sure. (1) (2): The following applies: g = g id cod () = g (h) = (g) h = 1 dom () h = h. Remark (i) Because of the uniqueness, one speaks of the inverse morphism of an isomorphism and denotes this with 1. Obviously, 1 is also an isomorphism and (1) 1 = applies. (ii) Two objects A, B are also called C-isomorphic, if there is a C-isomorphism: A B; Notation: A = B. to be isomorphic gives an equivalence relation to ob (c). 17th

19 Deinition Let B be a subcategory of C. (1) B is called a dense subcategory of C, all the following applies: C ob (c) B ob (b): B = C in C. (2) B is called isomorphism-closed (iso - closed), in all cases: C ob (c), B ob (b) and C = B = C ob (b). Examples (1) The isomorphisms in Set are exactly the bijective mappings. (2) The isomorphisms in Grp are exactly the group isomorphisms. (3) The isomorphisms in Top are exactly the homeomorphisms. (4) The category of all cardinal numbers and mappings in between is dense in Set. (5) If K is a field, then the full sub-category of finite powers K n is a dense sub-category of finite-dimensional vector spaces over K. (6) A full sub-category B of C is dense and iso-closed in C if and only if B = C applies. Deinition (1) A morphism A B is called a monomorphism in C (or C-mono for short), all of which applies to all h, k mor (c) with h = k and h = k (i.e. all can be left-shortened). (2) The term dual to (1) is epimorphism in C (or C-Epi for short). (Legal abbreviation) Example In the categories Set, Grp, Ab, R-Mod, POS and Top, the monomorphisms (or epimorphisms) correspond to those morphisms that are injective (or surjective) as mappings to the underlying sets. Only one of these two equivalents applies in Ring. (Why? Exercise.) Sentence The following statements contain the dual statement in brackets: (1) AB, B g C C-Mono (or -Epi) = A g C C-Mono (or -Epi) . (2), g mor (c) and g C-Mono (or -Epi) = C-Mono (or g C-Epi). (3) Every C-cut (or retraction) is a C-mono (or epi). Proof: Consider the following situation: h k g to (1): From (g) h = (g) k olt first h = k, since g is a mono and then h = k, since it is also a mono. 18th

20 to (2): From h = k olt surely follows (g) h = (g) k and thus by assumption h = k. to (3): According to the assumption there is a mor (c) with = id. If now h = k, then also h = k, i.e. id h = id k, i.e. h = k. Theorem In each category C are equivalent: (1) is a C isomorphism; (2) is C mono and retraction (or C epi and cut). Proof: The equivalence of (1) and the statement in brackets is obtained by dualizing. (1) (2): Sure with, (3). (2) (1): Let a C-mono and g mor (c) with g = id. Then olgt (g) = (g) = id = id = g = id, i.e. is a C isomorphism. Deinition (1) A C-epi e is called extremal, all e m is mono commutated = m is C-isomorphism. (2) Dual means a C-mono m means extremal, all Epi e m commutated = e is C-isomorphism. (3) An (extremal) sub-object of an object B is a pair (A,), where: A B is an (extremal) monomorphism. (4) Dual to (3) one deines the (extremal) quotient (, B) of an object A. (5) For sub-objects of an object B one deines: (A,) (C, g): A h B g C (6 ) In the same way, one deines for quotients. (Exercise assignment) 19

21 (7) (A,) (C, g): (A,) (C, g) (C, g) (A,). Theorem For sub-objects A B, C g B of an object B ob (c) the following applies: (A,) (C, g):! Isom. h: A C: g h =. Proof: Because of (A,) (C, g) and (C, g) (A,) one obtains the following commutative diagram: and thus h A k B g C (kh) = gh = = id as well as g (hk) = k = g = g id. Since and g are C monos, k h = id and h k = id, i.e. h is an isomorphism. The uniqueness of also results from the left-shortening of g. Remark (& Deinition) (i) For every object B ob (c) there is an equivalence relation from the class of the (extremal) subobjects of B. With the help of the axiom of choice, one obtains a system of representatives for which is then a subclass of the class all (extremal) sub-objects of B. (ii) A category C is called (extremally) well-powered, as long as there is a system of representatives for every object, which is a set. Dual is called C (extremally) co-well-powered. This means that a category is (extremally) well-powered if and only if every object, except for isomorphism, only has a set of (extremal) sub-objects. Examples (1) Set, Grp and Top are well-powered and co-well-powered. (2) The class of all ordinal numbers (as a category) is well-powered but not co-well-powered. (3) In Set, Grp, and R-Mod, all epimorphisms and monomorphisms are extremal. In Top, the extremal epimorphisms (monomorphisms) are exactly the topological quotients (embeddings). Theorem For mor (c) are equivalent: (1) is an isomorphism; (2) is an extremal epi. and a monomorphism; (3) is an extreme mono. and an epimorphism. 20th

22 Proof: (1) (2): If there is an isomorphism, then there is also an epi- and monomorphism after It remains to show that there is an extremal epimorphism. If = m g with suitable morphisms m, g and m is a monomorphism, then m is also a retraction, because it is a retraction according to assumption. So m is an isomorphism according to (2) (1): Let extremal epimorphism and monomorphism. Then because of the factorization = id is also an isomorphism. (1) (3): Sure, since (1) is self-dual and (3) the statement that is dual to (2). After the proof, the following implications obviously apply: Retraction (cut) = extremal epi. (Mono.) = Epi. (Mono.) Where the inversions generally do not apply. Hence, extreme epi. a real weakening of the term retraction. The characterization is therefore more often used than the similarly sounding sentence The question of whether in a certain category all morphisms that are mono- and epimorphism at the same time are also isomorphisms is structurally related to the question of the distinguishability of extreme and normal epimorphisms: sentence In a category C the following are equivalent: (1) epi- & monomorphism = isomorphism; (2) epimorphism = extremal epimorphism; (3) monomorphism = extremal monomorphism. Proof: (1) (2): Let an epimorphism and = m g be a factorization with a monomorphism m. According to m is also an epimorphism, since there is one. So by assumption m is an isomorphism. (2) (1): Clearly according to (1) (3): Clearly, since (1) is self-dual and (3) the statement that is dual to (2). Deinition (1) I ob (c) is called an initial C-object, all for all A ob (c) the set C (I, A) is one-element (i.e.! I A). (2) Dual to this, T ob (c) is a terminal C-object, all for all A ob (c) the set C (A, T) is one-element (i.e.! A T). (3) X ob (c) is called zero object in C, all X is initial and terminal in C. Theorem Two initial (or terminal) C objects X, Y are C-isomorphic. Proof: Since X is initial, there is exactly one: XY and exactly one id X: X X. There is also exactly one g: YX and exactly one id Y: Y Y. This means that g = id X and g = id Y .Examples (1) The object with underlying set 1 = {} is terminal in Set, Grp, Ab, R-Mod, Rng and Top; is initially in set and top. (2) Z is initial in Rng. In Grp, Ab and R-Mod there are also initial objects. But: Field has neither initial nor terminal objects. (Why? Exercise assignment.) 21

23 Chapter 3 Functors and Natural Transformations Following the categorical imperative, we want to introduce suitable morphisms between categories, functors, in this chapter. 3.1 Functors Deinition Let A and B be categories. A functor F: AB from A to B is a pair of mappings F: ob (a) ob (b), F: mor (a) mor (b) such that: (1) AA mor (a) = FAFFA mor (b). (2) F (id A) = id F A, for all A ob (a). (3) If g is deined in A, then F (g) = F g F. Examples (1) For each category A there is an identity functor id A: AA, with id A (A) = A, for all A ob ( a), and id A () =, for all mor (a). (2) For every object A ob (a) of a category A and every category B there is a functor c A: BA with c A (B) = A, for all B ob (b), and c A () = id A , for all mor (b). This is the constant functor with value A. (3) For A ob (c) there is a so-called hom-functor: C (, A): C op Set with C (, A) (D) = C (D, A ) and C (, A) (DD) = C (, A): C (D, A) C (D, A) with C (, A) (g) = g. This is a contravariant functor: Deinition A functor F: C op D is called a contravariant functor F: CD, this then satisfies F (g) = F g F. (4) A hom-functor also exists for A ob (c) : C (A,): C Set with C (A,) (D) = C (A, D) and 22

24 C (A,) (D D) = C (A,): C (A, D) C (A, D) with C (A,) (g) = g. Remark Ot we denote hom-functors as olgt: or C (A,) = hom C (A,) = hom (a,): C Set C (, A) = hom C (, A) = hom (, A ): C op Set, whereby the latter term is only used if there is no danger of confusion. (5) The category Met of the metric spaces (X, d) and contractions in between gives an example of a so-called forgetful functor U: Met Set, deified by U: (X, d) This forgets the metric structure. (Y, d) X Y. (6) For subcategories B A there is an obvious inclusion function B A. Attention! For a group homomorphism G G, as is well known, its image [G] is a subgroup of G. This does not apply to categories: Consider e.g. ACB g FAFBFAFB = FCDF g F g FFD Here F (B) is not a subcategory of A. (7) For every set A there is a functor A: Set Set with BAB id ABAB where id A (a, b) = (a , (a)). Ot is also simply written A for id A. Deinition Let F: A B be a functor. (1) F means full, all for all A, A whether (a) the restriction is surjective, i.e. all it to FAF AA: A (A, A) B (FA, FA), F g FA always mor (a) gives with F () = g. 23

25 (2) F means true, all for all A, A whether (a) F AA is injective, i.e. all of F = F g always = g olgt. (3) F means embedding, all F is faithful and injective to objects (i.e. from F (A) = F (B) it always follows A = B). Examples Let us review some of the above examples of functors: (1) For each category A, id A is full, true, and one embedding. (2) The forgetful fool U: Met Set is loyal. (3) The inclusion B A of a sub-category an embedding. It is full if and only if B is a full subcategory of A. 3.2 Operations with functors Deinition Let F: A B be a functor. (1) The F op dual to F: A op B op is defined by F op (A) = F (A) for objects and F op () = F () for morphisms. (2) If G: BC is a functor, then GF = GF: AC deined by GF (A) = G (F (A)), GF () = G (F ()) is a functor, the composition of F and G. Remark The composition of functors is associative and in this respect id A is an identity au A. However, in general for a pair of categories A, B the class of functors AB is not a set! Therefore categories as objects and functors as morphisms do not together form a category in our sense. But one can show that for small categories A, B the class of functors A B is a set. Therefore the class of the small categories as objects, together with the class of the functors in between as morphisms, forms a category. We refer to this category of the small categories as Cat. Deinition Let A, B categories. A functor A F B is called isomorphism, if a (uniquely determined) functor B G A exists with F G = id B, GF = id A. A and B are called isomorphic, if there is an isomorphism A B. Then we write A = B. Examples (1) The category of metric spaces and continuous mappings Met c is not isomorphic to the category Top m of metrizable topological spaces: As is well known, the Euclidean metric d E and the maximum metric d M produce the same topology au the R 2. Therefore, the spaces (R 2, d E) and (R 2, d M) different in Met c cannot be distinguished in Top m. An inducing metric cannot therefore be clearly assigned to a given, metrizable space. (2) For a commutative ring R, R-Mod = Mod-R applies. Theorem Let A F B, B G C functors. Then: 24

26 (1) F, G full (or true, embedding) = GF full (or true, embedding) (2) GF full = G full from the image of F (3) GF true (or embedding) = F faithful (or embedding). Proof: We show a part of (1) as an example and leave the rest to the reader as an exercise. Let G and F be full and GF A g GF A be a C morphism. Since G is full, there is an h F A F A with G (h) = g. Since F is full, there is an A A with F () = h, i.e. with GF () = G (h) = g. Theorem Every functor preserves retractions, cuts, isomorphisms and commutative triangles (i.e. retraction (etc.) = F () retraction (etc.)). Proof: Let F: A B be a functor and r an A retraction, i.e. there is an s with r s = id. Then F (r) F (s) = F (r s) = F (id) = id, so F (r) is also a retraction. Now F op: A op B op also preserves retractions, i.e. because of the duality of the concept, F preserves cuts. Since F preserves cuts and retractions, F also preserves isomorphisms. The statement about commutative triangles results directly from the deinition of a functor, since these preserve the composition of morphisms. Theorem Every functor that is full and true relects retractions, cuts and isomorphisms (i.e. F () retraction (etc.) = retraction (etc.)). Proof: Let F: AB be a functor and FAF () FA a cut, ie there is a B-morphism FA h FA with h F () = id F A. Since F is full, there is an A-morphism A g A with F (g) = h, ie with F (g) = F (g) F () = h F () = id FA = F (id A). Since F is true, g = id A, i.e. is a cut. The statement about retractions and isomorphisms follows again with duality. Theorem Every faithful functor relects epimorphisms, monomorphisms, and commutative diagrams. Proof: Let F: A B be a functor and F () a monomorphism. Let g, h A morphisms with g = h. Then F () F (g) = F (g) = F (h) = F () F (h), i.e. F (g) = F (h), since F is a monomorphism. Because F is also faithful, g = h. The statement about epimorphisms results from dualizing. 25th

27 Every commutative diagram is composed of just such triangles. For a commutative triangle F A F F B F h, F (g) = F (g) F () = F h, i.e. g = h, because F is true. Therefore F takes commutative diagrams into account. Deinition A functor F: AB is called iso-dense, as long as there is an A ob (a) for every B ob (b) with F (A) = B. Proposition Every full, faithful and iso-dense functor preserves and relieves: Mono -, epi and isomorphisms, as well as sections, retractions and commutative diagrams. Proof: Let F: A B be a functor. It suffices to show that F preserves monomorphisms. The remaining statements result from dualizing or from the above sentences. Let AA be a monomorphism and h, g mor (b) with F g = F h: FCF g B s = FA gh gs FAFFA hs Since F is iso-dense, there is an A ob (a) and an isomorphism s: FA B. Because F is full, there are morphisms u, v for gs and hs with From F g = F it follows F (u) = gs, F (v) = h s. F (u) = FF u = F gs = F hs = FF v = F (v), so u = v, since F is true. Now is a monomorphism, so u = v and thus g s = F (u) = F (v) = h s, i.e. g = h, because s is an isomorphism. 3.3 Natural Transformations In this section we will deify and examine reasonable morphisms α between parallel functors: A F α B G 26

28 Deinition Let F, G: A B functors. A natural transformation α: F G is a family (F A α A GA) A ob (a) of B morphisms, so that for all A morphisms: A A the diagram F A α A GA F F A α A G GA commutes. For this one sometimes writes AF α B. G Examples (1) For every functor F: AB there exists the identical natural transformation id F: FF, given by (id F) A = id F A. (2) Let H: Grp Ab the functor, which assigns the Abelian group AG = G / [G, G] to each group G, ie actuates the commutator 1. Let E: Ab Grp denote the embedding, and for each G ob (grp) p G: GAG the canonical Projection. Then p: id Grp EH is a natural transformation. (3) Let B be a set and B: Set Set, B: Set Set functors with and Then BB (AB (AA) = BA id BBA, A) = AB id BAB, τ B, deined by (b, a) (b, (a)), (a, b) ((a), b). τ A: B A A B, (b, a) (a, b), a natural transformation. (3) If B, B are sets and: B B are a function, then a natural transformation: B B is given by A: B A B A, (b, a) ((b), a). This is sometimes denoted by, where () A = A = id A. (4) Let D: Vek op Vek be the dual space functor, i.e. V V D: h W W (ϕ ϕ h) G 1 See also (4). A morphism: G G assign H the canonical continuation of G p G A G. 27

29 where V: = {V ϕ R ϕ linear} and W accordingly. Let D: Vek = (Vek op) op D op Vek op D Vek be the double dual space functor. Then α: id Vek D, deed by α V: V D (V) = V, x (ϕ ϕ (x)), is a natural transformation. Here even every α V is an isomorphism. Deinition (1) Let G, G: A B functors, G α G a natural transformation. Then α op: (G) op G op, the natural transformation dual to α. α op A: = α A, (2) Let α: G G as in (1) and F: C A be a functor. Then the natural transformation αf: GF G F is deed by (αf) A: = α F A. (3) Let α: G G as in (1) and H: B D be a functor. Then the natural transformation Hα: HG HG is deined by (Hα) A: = Hα A. The second and third parts of the above deinition can be illustrated as follows: Every situation CFAG α BHDG induces natural transformations C GF GF αf B and A HG HG Hα D. β H natural transor set Let F, G, H: AB functors, f mations. Then β α, deified by (β α) A: = β A α A, α G and G is a natural transformation F H, i.e. natural transformations can be composed as follows: A F α G β H B A F βα B H 28

30 Proof: The assertion follows from the following commutative diagram: AFA α A (βα) a GA β A HA FAFA α AG GA β AH HA (βα) A Remark The composition of natural transformations is associative, and for every functor F is id F neutral regarding this composition. Proposition Let α: F G be a natural transformation. Equivalent are: (1) For every object A, α A is an isomorphism. (2) There is a natural transformation β: G F with β α = id F and α β = id G. Proof: Let F, G: A B functors and α: F G be natural. (1) (2): Set β A: = α 1 A for every A ob (a). Then in id GA A GA β AFA α A GA GA GA β AFFA α AG GA id GA the outer frame and the right square commute, that is, α AF β A = α A β A G. Since α A is an isomorphism by assumption is, then F β A = β AG, which was to be shown. (2) (1): Let A be ob (a). From β α = id it follows β A α A = (β α) A = id F A. Analogously, α A β A = id GA. Deinition Natural transformations that meet the conditions of the above theorem are called natural isomorphisms. 29

31 Example In the above example (4), α: id Vek D is a natural isomorphism from the category of finite-dimensional R-vector spaces. Deinition F, G: A B are of course isomorphic, as long as there is a natural isomorphism α: F G. Remark Being naturally isomorphic is an equivalence relation of the class of functors A B. Theorem Let F, G: A B functors and α: F G be a natural isomorphism. Then F is full (or faithful) if and only if G is full (or faithful). Proof: ad full: For reasons of symmetry it suffices to show the implication. So let F be full and GA h GA a B morphism. Since F is full, there is an A-morphism AA, so that the following diagram commutes: Since α is natural, FAF α A GA h FA GA α 1 AG α A = α AF = α A α 1 A h α A = h α A, ie G = h. ad treu: Exercise task (works analogously). 3.4 Equivalence of categories The concept of isomorphism of categories already introduced in Section 3.2 turns out to be too strong in practice. It turns out that the somewhat weaker concept of the equivalence of categories will be sufficiently strong, i.e. equivalent categories have the same categorical properties, and are therefore indistinguishable in this sense. Proposition Let F: A B be a functor. Equivalents are: (1) F is full, faithful and iso-dense. η (2) There is a functor G: BA and natural isomorphisms id A GF, FG ε id B. Proof: (2) (1): Since id A is true and η is a natural isomorphism, GF is also true by theorem So if F is true to Likewise, id B is full and ε is a natural isomorphism, i.e. FG full to, and thus also F to Be B ob (b), then F (GB) ε BB 30

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