Part 1. 1 Basic Theory of Categories. 1.1 Concrete Categories

Transcription

1 Part 1 1 Basic Theory o Cateories 1.1 Concrete Cateories One reason or inventin the concept o cateory was to ormalize the notion o a set with additional structure, as it appears in several branches o mathematics. The structured sets are abstractly considered as objects. The structure preservin maps between objects are called morphisms. Formulatin all items in terms o objects and morphisms without reerence to elements, many concepts, althouh o disparate lavour in dierent mathematical theories, can be expressed in a universal lanuae. The statements about universal concepts provide theorems in each particular theory. 1.1 Deinition (Concrete Cateory). A concrete cateory is a triple C = (O, U, hom), where O is a class; U : O V is a set-valued unction; hom : O O V is a set-valued unction; satisyin the ollowin axioms: 1. hom(a, b) U(b) U(a) ; 2. id U(a) hom(a, a); 3. hom(a, b) hom(b, c) hom(a, c). The members o O are the objects o C. An element hom(a, b) is called a morphism with domain a and codomain b. A morphism in hom(a, a) is an endomorphism. Thus End a = hom(a, a). We call U(a) the underlyin set o the object a O. By Axiom 1, a morphism hom(a, b) is a mappin U(a) U(b). We write : a b or a b to indicate that hom(a, b). Axiom 2 ensures that every object a has the identity unction id U(a) as an endomorphism. Axiom 3 uaranties that morphisms can be composed. 1.2 Example (Concrete Cateories). The ollowin are examples o concrete cateories.

2 1. SET is the cateory o sets. Its class o objects is V. The mappin U : V V is the identity unction ID V. For arbitrary sets x, y we have hom(x, y) = y x. 2. GROUP is the cateory o roups. For any roup, U() is the underlyin set o, that is, with all operations inored. I, h are roups, then the set hom(, h) is the set o all roup homomorphisms h. 3. RIN G is the cateory o rins. 1 I Λ is a rin, then U(Λ) is the underlyin set o Λ. For rins Λ, Σ, the set hom(λ, Σ) is the set o all rin homomorphisms Λ Σ. 4. Λ MOD, where Λ is a rin. It has all let unitary Λ-modules as objects. U(a) is the underlyin set o the module a; hom(a, b) is the set o all Λ-linear homomorphisms The cateory T OP whose object class consists o all topoloical spaces x, U(x) is the set o all points o x; hom(x, y) is the set o continuous unctions x y. Oten we will write C(x, y) or the set o continuous mappins x y. In a similar way one obtains MOD Λ riht Λ-modules and Λ-homomorphisms; ΛMOD Σ ΛΣ-bimodules and ΛΣ-homomorphisms; SGRP semiroups and their homomorphisms; CSGRP commutative semiroups and semiroup homomorphisms; MON monoids and identity-preservin semiroup homomorphisms; CMON abelian monoids and monoid homomorphisms; AB abelian roups and roup homomorphisms; CRIN G commutative rins and rin homomorphisms; ID interal domains and rin homomorphisms; FIELD ields and homomorphisms; ALG R associative R-alebras and alebra homomorphisms (where R CRIN G); CALG R commutative R-alebras and alebra homomorphisms; LIEALG k Lie alebras and Lie homomorphisms (k FIELD); Rel sets with a binary relation and relation preservin maps; POS partially ordered sets and monotone unctions; LAT join semi-lattices and supremum preservin unctions; LAT meet semi-lattices and inimum preservin unctions; LAT lattices and lattice homomorphisms; 3 BA Boolean alebras and Boolean homomorphisms; BR Boolean rins and rin homomorphisms; 4 BS Boolean spaces and continuous unctions; 5 1 Rin always means rin with unit, and a homomorphism o rins preserves units. 2 I Λ is a ield, this ives the cateory o vector spaces over Λ and Λ-linear maps. 3 A lattice homomorphism is a map obeyin the rule (x y) = x y and (x y) = x y. 4 A Boolean rin is a rin R with 1 and idempotent multiplication. Evidently R is commutative and obeys the rule x + x = 0. 5 A Boolean space is a totally disconnected, compact Hausdor space, where totally disconnected means that every point is its own connected component. Equivalently, a space is Boolean, i it is 0-dimensional and compact and Hausdor. Such a space is also called a Stone space. 2

3 T OP T 0 T0-spaces and continuous unctions; T OP T 1 T1-spaces and continuous unctions; T OP T 2 Hausdor spaces and continuous unctions; T OP Comp.T 2 compact Haussdor spaces and continuous unctions; T OP Loc.Comp locally compact Hausdor spaces and continuous unctions; T OP C.Re completely reular spaces and continuous unctions; 6 MET metric spaces and uniormly continuous unctions; CM ET complete metric spaces and uniormly continuous unctions; T OPGROUP topoloical roups and continuous homomorphisms; AB Loc.Comp locally compact abelian Hausdor roups and continuous homomorphisms; N R K normed linear spaces and continuous (=bounded) linear maps, (K = R K = C); BAN K Banach spaces and ccontinuous linear maps, (K = R K = C); C ALG C -alebras and conntinuous involution preservin linear maps; SET sets with base point (pointed sets) and base point preservin unctions; 7 T OP topoloical spaces with base point and continuous base point preservin unctions; T OP bipointed spaces with continuous unctions preservin both base points; T OP 2 topoloical pairs and continuous mappins o pairs; 8 T OP n topoloical n-tupels. 9 It is plain that this list could be continued indeinitely. Its notation will be used throuhout several examples. 1.2 Abstract Cateories In many instances the attempt to build new cateories rom old ones by usual alebraic constructions leaves the scope o concrete cateories. For this reason there is a more eneral concept o cateory not suerin rom this restriction. 1.3 Deinition (Abstract Cateory). An abstract cateory is a quintuple C = (O, M, dom, cod, ), where dom and cod are unctions M O; is a unction D := {(, ) M M dom() = cod()} M, such that the ollowin axioms are satisied: 1. [Matchin condition] (, ) D dom( ) = dom() cod( ) = cod(); 2. [Associativity] (, ) D (h, ) D (h ) = h ( ); 6 A space X is completely reular i all its points are closed and it satisies the T 3.5- separation axiom. T 3.5 claims that, or any closed set A X and any point x X \ A, there is a continuous unction : X [0, 1] such that (x) = 1 and (A) {0}. A completely reular space is also called Tychono space. 7 A pointed set is a pair (X, x 0 ) with x 0 X. A base point preservin unction : (X, x 0 ) (Y, y 0 ) is a map Y X such that (x 0 ) = y 0. 8 Objects are pairs o topoloical spaces (X, A), where A X is a subspace. A morphism : (X, A) (Y, B) is a continuous map : X Y with (A) B. 9 Objects are n-tupels X = (X 1,..., X n), where X 1 is a space, and X k X 1 k = 1,..., n. A morphism : X Y is a continuous unction : X 1 Y 1 such that (X k ) Y k k. 3

4 3. [Identity existence] a O e M ( dom(e) = cod(e) = a [(, e) D e = (e, ) D e = ] ) ; 4. [Morphism set condition] hom(a, b) := { M dom() = a cod() = b} V. O is the class o objects, M is the class o morphisms. The partial unction M M D M is called the composition law. As within concrete cateories we write : a b or a b to indicate that hom(a, b). We call a morphism with domain a and codomain b, an arrow rom a to b, or a map. Composition is a binary operation with domain D turnin M into a partial semiroup. Mostly we write or. Whenever a composition occurs, it is assumed that (, ) D. When dealin with several cateories C, we use a subscript O C, M C etc. When clear rom context, x C means x O C or x M C dependin on the nature o x. Note that by Axiom 1, the hypothesis o Axiom 2 is suicient to assure that its conclusion does make sense. Since we have (h, ) D. Furthermore and thus (h, ) D. Now dom(h) = dom() = cod() dom(h) = cod() = cod() dom(h ) = dom() = dom() = dom(h ) cod(h ) = cod(h) = cod(h) = cod(h ). and So, by the hypothesis o Axiom 2, it is possible that h = h and the axiom postulates that this be the case. Axiom 3 mimics the Concrete Cateories Axiom 2 by orcin the existence o an identity morphism or each object. Axiom 4 is necessary to assure that constructions are possible within the underlyin set-theoretic system. The unctions dom and cod enerate a partition o M into disjoint sets M = hom(a, b). (a,b) O O 4

5 b cod hom(a, b) dom a In practice an abstract cateory oten appears in a vestment ( O, (hom(a, b))(a,b) O O, ). The orm correspondin to Deinition 1.3 is then achieved by declarin M to be the disjoint union o the sets hom(a, b). For each M there is then a unique pair o objects (a, b) with hom(a, b). dom() = a and cod() = b establish the domain and codomain maps. The irst o the ollowin examples deals with this situation. 1.4 Example (Abstract Cateories). 1. The Cateory naturally associated with a Concrete Cateory (O, U, hom) is ( ) O, {a} hom(a, b) {b}, dom, cod,. (a,b) O O Here dom(a,, b) = a, cod(a,, b) = b and (b,, c) (a,, b) = (a,, c). In view o this construction a concrete cateory is a special abstract cateory. Thereore, rom now on, when we shall say cateory, we will mean abstract cateory, a concrete cateory bein tacitly understood in the sense o this construction. 2. The cateory REL o sets and relations Object class is V. For a, b V, hom(a, b) = P(a b). The composition is the usual product o relations. 3. The Cateory o Topoloical Bundles (T OP T OP) The object class consists o all triples (x, π, y), where x, y are objects in T OP and π : x y is a continuous map. hom((x, π, y), (x, π, y )) is the set o pairs o continuous maps (, ) which make the diaram π x y x y π 5

6 commutative. Composition is componentwise, that is, by pastin toether correspondin squares. 4. Quasi-ordered Classes Let (X, ) be a quasi-ordered class (a class equipped with a relexive and transitive relation). (X, ) ives rise to a cateory whose object class O is X. For x, y X, we set { {(x, y)} i x y, hom(x, y) = i x y. I x y and y z then the compsition is (y, z) (x, y) = (x, z). The sets hom(x, y) contain at most one arrow. Any cateory C with the property x, y O : Card(hom(x, y)) 1 can be considered that way. 5. Ordered Classes As an instance o the previous example, partial orders and total orders can be considered as cateories. (X, ) is a partially ordered class x, y O : Card(hom(x, y) hom(y, x)) 1; (X, ) is a totally ordered class x, y O : Card(hom(x, y) hom(y, x)) = Ordinal Numbers Bein well-ordered sets they build a special instance o the last item. Considerin their inite members we obtain the positive inteers as cateories. 0 = 1 = 2 = 3 = 4 = When depictin cateories by such diarams, we omit the requisite identity morphisms. 7. The class On o all ordinals is a cateory. 6

7 8. Monoids A monoid M deines a cateory with exactly one object. The set o morphisms is M, composition is iven by the product rule o M. Any cateory with exactly one object is a monoid. 9. Groups As a special instance o the previous point, every roup is a cateory. C. Deinition The cateory o matrices MAT Λ Let Λ be a rin. We deine the cateory MAT Λ o Λ-matrices: The class o objects is N. For inteers m, n we set hom(m, n) = Λ n m that is, the morphisms are the n m-matrices with entries in Λ. Composition is the usual matrix product. 11. The Cateory Λ GRMOD o Z-raded Λ-Modules Λ RIN G. A Z-raded Λ-module is a sequence (A n ) n Z o Λ-modules. For x A n we set de x = n. A map o deree p (p Z) rom a raded module A to a raded module B is a sequence o Λ-homomorphisms = ( n ) n Z with n : A n B n+p. For maps A B, B C o deree p, q respectively, the composition is the map o deree p + q iven by Then we obtain ( ) n = n+p n. de (x) = de + de x and de( ) = de + de. Graded Λ-modules toether with maps o arbitrary deree build the cateory ΛGRMOD. 12. The Cateory Λ MOD Z Objects are raded Λ-modules as beore, but morphisms are restricted to maps o deree The cateory Λ CC o chain complexes Λ RIN G. A chain complex over Λ is a pair (A, ) where A is a Z- raded Λ-module and : A A is a map with de( ) = 1 such that 2 = 0. The elements o deree n are reerred to as n-dimensional chains, is called the boundary operator o the complex A, or its dierential. A chain map between chain complexes (A, ), (A, ) is a map : A A o deree 0 such that =. This turns the class o chain complexes toether with chain maps into a cateory which is denoted by Λ CC. The ormula 2 = 0 means n n+1 = 0 : A n+1 A n 1. It is equivalent to the condition ker( n ) im( n+1 ). 7

8 14. The Cateory GA o roup actions Its objects are triples (T, G, α) where T, G are roups and α: T Aut G is a homomorphism o roups. The roup action T G G is written as t = α(t)(). The set hom((t, G, α), (T, G, α )) consists o pairs (ϕ, ) where T ϕ T and G G are homomorphisms o roups, such that (t ) = ϕ(t) (). Composition is componentwise, i.e., i then (ψ, ) (ϕ, ) = (ψϕ, ). (T, G, α) (ϕ,) (T, G, α ) (ψ,) (T, G, α ) 15. The cateory MOD o let modules over arbitrary rins Objects are paires (Λ, A), where Λ RIN G and A Λ MOD. A morphism (Λ, A) (Σ, B) is a pair (ϕ, ), where ϕ: Λ Σ is a morphism o rins, and : A B is a morphism o abelian roups, such that (λ a) = ϕ(λ) (a). 16. The cateory RS o rined spaces Objects are paires (X, O), where X is a topoloical space and O is a shea 10 o rins on X. A morphism (X, O X ) (Y, O Y ) is a pair (, φ) consistin o a continuous map : X Y and a morphism o sheaves φ: O Y O X. 17. The cateory o schemes It consists o schemes and morphisms between them. A scheme is a locally rined space which locally looks like a rin spectrum. 1.5 Proposition. Let C be a cateory. 1. For each a O there is exactly one morphism a e a satisyin Axiom 3. It is written 1 a, or just 1, and called the identity o a. 2. For a O, End a carries the structure o a monoid. Its units constitute the automorphism roup Aut a. 1.6 Proposition (Characterization o Identities). Let C be a cateory. For any e M C, the ollowin are equivalent: 1. a O C with e = 1 a ; 2. : (, e) D C e = ; 3. : (e, ) D C e =. The identities o C are exactly the identities 11 o the partial semiroup (M, ). Since, by Proposition 1.5, objects are encoded as identity morphisms, it is possible to ive an object ree axiomatization o cateory theory. 10 See the section on unctors and natural transormations. 11 An identity in a partial semiroup (M, ) where M M D M is an element e M with the property M ([(, e) D e = ] [(e, ) D e = ]). 8

9 1.7 Deinition (Object ree Axiomatization). A cateory is a pair (M, ) where M is a class and M M D M is a binary partial operation on M subject to the conditions: 1. (h, ) D (, ) D = (h, ) D (h, ) D; 2. (h, ) D (h, ) D and in this case (h ) = h ( ); 3. M identities e, e M such that (, e) D (e, ) D; 4. identities e, e M, the class { M (, e) D (e, ) D} is a set. 1.8 Theorem. The two axiomatic descriptions o cateory theory iven by deinitions 1.3 and 1.7 are equivalent. I (O, M, dom, cod, ) is a cateory accordin to Deinition 1.3, then (M, ) is a cateory in the sense o Deinition 1.7. Conversely, when (M, ) is a cateory accordin to Deinition 1.7, then (O, M, dom, cod, ), where O is the class o identities o (M, ), dom() is the unique identity e such that (, e) D, cod() the one with (e, ) D, is a cateory iven in terms o Deinition Duality The symmetry inherent to any o the axiomatic descriptions o cateory theory allows or a simple dualization concept. To each statement A about data o a cateory C, one may build the dual statement A δ. The process, called dualization, proves its useulness in two ways: First, to any cateorical notion there is a dual notion which, passin to a particular cateory, may be quite dierent in character rom the oriinal one. Secondly, to each theorem o cateory theory there is a dual theorem. We discuss here the elementary concept o dualization as a syntactic process applicable to elementary cateory theory. To this end we reormulate cateory theory in irst order lanuae. 1.9 Deinition. Elementary cateory theory is a theory T ormulated in irst order predicate loic with identity. It is iven by the ollowin data. 1. T has two unary predicate symbols O,M and the binary equality symbol =. For terms a and we write a O and M to denote the ormulas O(a) and M() respectively. 2. T has the symbols dom, cod as unary unction symbols and as binary unction symbol. As usual we write or (, ). 3. The axioms o T are the axioms o a irst order predicate loical system with equality, enriched by the ollowin proper axioms: (a) M dom() O cod() O (b), M M (c), M dom() = cod() dom( ) = dom() cod( ) = cod() (d) h,, M dom() = cod() dom(h) = cod() (h ) = h ( ) 9

10 (e) a O e M (dom(e) = cod(e) = a M ( (dom() = a e = ) (cod() = a e = ) ) ). The dual P δ o a statement P o T is deined by induction on the complexity o the expression P: 1.10 Deinition. x δ x or 12 variables x (dom()) δ cod( δ ) or terms (cod()) δ dom( δ ) or terms ( ) δ δ δ or terms, ( M) δ δ M or terms (a O) δ a δ O or terms a ( A) δ (A δ ) or ormulas A (A B) δ (A) δ (B) δ or ormulas A,B and loical connectives ( x A) δ ( ) x A δ or ormulas A ( x A) δ ( ) x A δ or ormulas A. Passin rom P to P δ is called dualization Example. Let P be the statement 13 M h,k (h M k M cod(h) = dom() cod(k) = dom() h = k h = k). Then its dual P δ is M h,k (h M k M dom(h) = cod() dom(k) = cod() h = k h = k) Theorem (Principle o Duality). Let P be a statement o T. Then P is a theorem o T i and only i P δ is a theorem o T. Proo. I P is a theorem o T then there is a proo π o P. This is a inite sequence o ormulas o T each o which either is an axiom or associated by a loical inerence rule to ormulas prior in the list. Since the axioms are sel-dual and the metaunction δ cuts throuh loical inerence rules, applyin δ to each individual ormula, we obtain a proo π δ o P δ. Now P δδ P. So dualizin a proo o P δ ives back a proo o P. Durin its development a ormal theory requently extends its lanuae and enlares its supply o axioms. For instance, in Proposition 1.5, or each object the unique existence o identity morphisms is stated. Thereore a new unary unction symbol 1 is introduced, intended to denote this one morphism dependent on objects. Formally this amounts to introducin a new proper axiom, expressin as a rule the behavior o identity morphisms. It is obvious that the process o dualization and the principle o duality apply to all such conservative extensions o the oriinal theory T. Consider the ollowin example: 12 We use the symbol to indicate identity in the meta lanuae. 13 P expresses the notion o monomorphism. Its dual P δ describes epimorphisms. 10

11 1.13 Example. Let Q be the statement 14 M ( M dom() = cod() cod() = dom() = 1 dom() ). Then statement Q δ is M ( M dom() = cod() cod() = dom() = 1 cod() ). In the sequel we will ormulate several elementary statements o T in usual relaxed notation. Each o these has a dual statement which can easily be produced rom the oriinal. We thereore will in eneral write down only one o the duals. 1.4 Special Morphisms and Objects 1.14 Deinition. Let A be a cateory. 1. A section in A is a morphism havin a let invers, i.e., x y is a section r : y x such that r = 1 x. 2. Dually, is a retraction i it has a riht inverse. 15 An object x is a sect provided that a section x y, i.e., x is the domain o a section. y is a retract i it is the codomain o a retraction. 3. is an isomorphism i it is simultaneously a section and a retraction. In this case the let- and the riht inverse o aree to a unique two-sided inverse Deinition. Let a, b be objects in a cateory A. We say that a and b are isomorphic, denoted a = b, i there exists an isomorphism : a b. The property o bein isomorphic deines an equivalence relation on the class O A Deinition. 1. x y is called a monomorphism (also monic or mono), i it can be cancelled rom the let: i h k x y such that h = k, then h = k. 2. Dually, x y is an epimorphism (also epic or epi) i it can be cancelled rom the riht: x y h k with h = k = h = k. 3. A bimorphism is an arrow that is both, monic and epic. 14 Q describes the property o bein a section. Q δ means is a retraction. C. the next pararaph. 15 Oten a retraction is reerred to as split epimorphism. 11

12 Oten we write in order to emphasize that is epi. The pictoram indicates that is mono. Plainly every section is monic, and every retraction is epic. Consequently, each isomorphism is a bimorphism. In a concrete cateory we have in addition: section = injective = monomorphism; retraction = surjective = epiomorphism. In the cateory T OP consider x x s(x), ξ s(ξ), let s(x) s r y with rs = 1. Write s or the map j y denote inclusion and let p = s r. s s x y r j s(x) p Then s rjs = s rs = 1s. The map s beein surjective is epi whence s rj = 1. Also rjs = 1, that is, s is an isomorphism. In addition pj = 1, i.e., p s(x) = 1. When we call a subspace b y a topoloical retract o y i p: y b with p b = 1, then the section x s y is an embeddin (i.e. an isomorphism) onto a retract o y, while the retraction x r y projects onto that retract. Conversely, i s is an embeddin onto a retract and r = (s ) 1 p, then we obtain s rs = s (s ) 1 ps = pjs = s and thereore rs = 1. s = r s x y j p s(x) This observation is in orce or all cateories whose morphisms x y allow or a actorization x y j z where is an epimorphism (c. Deinition 1.39). This is true e.. or the cateories SET, T OP, Λ MOD, AB, SGRP, MON, GROUP, RIN G, ID, ALG K, CALG k. Observe that or Λ MOD this says that a section is just an embeddin onto a direct summand, since x s y with rs = 1 implies that y = ker r im s. Amon others this is embodied in the ollowin list o examples Example. r 12

13 1. I X then : X Y is a section in SET i is injective. 2. : X Y is a section in T OP i is a topoloical embeddin, i.e., induces a homeomorphism X (X) and (X) is a topoloical retract 16 o Y. 3. : A B is a section in Λ MOD i is injective and im is a direct summand 17 in B. 4. Let C be one o SET, T OP, GROUP, Λ MOD. Let X Y be a morphism in C. Then the raph embeddin X X Y, x (x, x) is a section in C. 5. A morphism in SET is a retraction i and only i it is surjective. 6. X Y is a retraction in T OP i and only i there is a continuous retraction r mappin X onto a subspace S o X (and leavin points in S ixed) and a homeomorphism S h Y such that hr =. 7. A B is a retraction in Λ MOD i and only i there is a projection p o A onto a submodule S A (with p 2 = p as a map A A) and an isomorphism S h B such that hp =. 8. Isomorphisms in SET, T OP, GROUP, Λ MOD are bijections, homeomorphisms, isomorphisms o roups, isomorphisms o modules respectively. 9. In SET, T OP, GROUP, AB, Λ MOD, POS, T OP Comp.T 2 the monomorphisms (epimorphisms) are precisely those morphisms which are injective (surjective) on underlyin sets In T OP T 2 and T OP C.re the monomorphisms are the injective continuous unctions, while the epimorphisms are the continuous unctions with dense imae. 11. In FIELD every morphism is mono. 12. e it : (R, 0) (S 1, 1) is a monomorphism in the cateory o pointed connected spaces and continuous base-point preservin unctions. 13. The map Q π Q/Z is a bimorphism in the cateory o divisible abelian roups. I D h Q π Q/Z with πh = πk, then h k : D Z whence h = k. k 14. The embeddin Z Q is a bimorphism in RIN G and in MON. 16 Remember that A Y is a topoloical retract, i there is a continuous map r : Y A that leaves each point o A ixed. 17 In analoy to the situation in T OP, a retract o a module M is a submodule N such that r : M N with r N = id N. Now we have: : A B is a section in Λ MOD i is injective and im is a retract o B. 18 C. Corollary

14 The proos o these statements are o diverse severity. The assertions concernin monomorphisms are in eneral obvious. I e.. a b is a monomorphism in RIN G then the ree rin provides an appropriate tool: From (x) = (y) Z[t] x y a b we et x = y. This works in all concrete cateories which admit a ree object on at least one enerator. Also it is easy to see that in the cateory Λ MOD the epimorphisms are precisely the surjections: I a b is epi, and π : b b/im is the projection, then π = 0 = 0 a b π 0 b/im whence π = 0, which means that is surjective. We just used the cokernel b/im as a test object. Provin the correspondin statement or the cateories SET or T OP works alon equal lines. The proo or the cateory GROUP is less obvious. Reardin an epimorphism a b, let k be the normal subroup o b enerated by (a). Then π = 0 = 0 a b π 0 b/k but this only ives b = k, which does not prove surjectivity o. Usin the cokernel similar to the case o modules ails. One has to choose another test object Lemma. Let H < G in GROUP. 19 Then K GROUP r, s: G K with r s r H = s H. Proo. Let 2 = {±1} be the roup Z 2 written multiplicatively. A := 2 G. We turn A into a G-module with action a = a ρ where ρ: G op S(G), ρ (x) = x is the riht reular representation. 20 Then ( a)(x) = a(ρ (x)) = a(x). Now let K = A G, and consider the split exact sequence 1 A A G π G 1 s s: G A G is the splittin homomorphism s() = (e, ), where e is neutral in A. Next consider the characteristic unction o H: { x H, χ(x) = 1... x G \ H. 19 The expression H < G means that H is a proper subroup o G. 20 G op is the opposite roup to G, i.e. G with reversed multiplication. S(G) denotes the symmetric roup o the set G. 14

15 Then χ A and the principal derivation obeys the cocycle identity ϕ: G A, [, χ] = ( χ)χ 1. ϕ(xy) = ϕ(x) ( x ϕ(y) ) whence the association (ϕ(), ) deines a homomorphism o roups For arbitrary x G s(x) = t(x) t: G A G. y G : χ(yx) = χ(y). Thereore s(x) = t(x) x H. Consequently s H = t H and s t since H is a proper subroup Corollary. In the cateory GROUP the epimporphisms are exactly the surjective roup homomorphisms. Proo. Let : X G be an epimorphism in GROUP. I (X) < G, there would be a roup K and distinct morphisms s, t: G K with s = t, which is impossible. The remarks precedin the last corollary apply similarly to the cateory o Hausdor spaces. I they are compact then everythin comes riht: For an epimorphism X Y in T OP Comp.T 2 the set (X) is closed in Y. I were not surjective, then takin a point y Y \ (X), the Urysohn Lemma would provide a continuous unction h: Y [0, 1] with h(y) = 1 and h((x)) = {0}. Then h = 0 and we would have derived the contradiction h = 0. Also or completely reular spaces these aruments work. In T OP T 2 they do not. Also Y/(X) - the natural choice or a test object - is not uaranteed to be a Hausdor space Lemma. Let Y T OP T 2 and A Y a closed subspace. Let Z denote the space obtained by luein toether two copies o Y alon the points o A. Then Z T OP T 2. Proo. Consider the disjoint union Y Y = Y {0} Y {1}. Then Z = Y Y/R where R is the equivalence relation enerated by {((a, 0), (a, 1)) a A}. Let ι 0 Y Y Y be the natural embeddins and set h = πι 0, k = πι 1. Thus ι 1 h(y) = [y, 0] and k(y) = [y, 1]. 21 Y could e.. be a non - T3 space. Y h ι 0 Y Y π Z ι 1 k Y 15

16 Consider distinct points z 1, z 2 Z. We shall construct disjoint neihborhoods o z 1, z 2 : Set G = h(y \A), H = k(y \A), K = h(a) = k(a). Then Z = G K H is a partition o Z into the closed subset K and two open sets G, H. I z 1 G and z 2 H then G, H provide disjoint neiborhoods. I z 1, z 2 h(y ), z 1 = h(y 1 ), z 2 = h(y 2 ), then choose disjoint open sets U 1, U 2 Y with y 1 U 1 and y 2 U 2. Then h(u 1 ) k(u 1 ) and h(u 2 ) k(u 2 ) are open disjoint neihborhoods o z 1, z 2 respectively Corollary. In T OP T 2 the epimorphisms are precisely the continuous unctions with dense imae. Proo. Let X Y in T OP T 2. I (X) = Y and Y h k W are such that h = k, then h (X) = k (X). Since W is Hausdor it ollows that h (X) = k (X), i.e. h = k, and thereore is epi. Conversely, assume that is epi. In the notation o the previous lemma set A = (X). Then X Y h = k and thereore h = k. This says that [y, 0] = [y, 1] y Y whence Y = (X). h k Z The ollowin statements are immediate consequences o the deinitions Proposition. Let A be an arbitrary cateory, a b c in A. Then, section section;, retraction retraction., monol mono;, epi epi. section section; retracton retraction. mono mono; epi epi. section mono; retraction epi Corollary. Let a b c in A. Then, bi bi., iso iso and () 1 = 1 1. iso bi mono and epi. iso bi Proposition. In any cateory the ollowin statemens are equivalent: 1. is an isomorphism. 2. is monomorphism and retraction. 3. is section and epimorphism. Proo. Assume that a b is a monic retraction. Then s with s = 1 b. Thereore 1 a = 1 b = s. Consequently s = 1 a a 1 a s a b that is, is an isomorphism. 16

17 1.25 Deinition. A monomorphism a m b is called extremal monomorphism, provided that, or any actorization m = ϕη, i η is an epimorphism then η is an isomorphism. a m b η ϕ An epimorphism a e b is called extremal epimorphism, provided that, or any actorization e = µϕ, i µ is mono then µ is an isomorphism Example. a e b ϕ µ 1. In the cateories SET, GROUP, Λ MOD the extremal monomorphisms (extremal epimorphisms) are exactly the usual monomorphisms (epimorphisms), i.e., the injective (surjective) homomorphisms. 2. In T OP the extremal monomorphisms are embeddins. Extremal epimorphisms are the quotient maps. 3. In T OP T 2 extremal monomorphisms are closed embeddins, extremal epimorphisms are quotient maps. Proo. We very the assertions or T OP. Consider X Y in T OP. Assume that is an extremal monomorphism. Examine the actorization X Y i (X) where i denotes inclusion. is epi, thus it must be a homoeomorphism. Conversely, assume that is an embeddin, and consider a actorization (X) i = X Y ϕ η ϕ Z 17

18 where η epi and thence surjective. Thereore ϕ(z) = ϕη(x) = (x) (X) which means that ϕ actors throuh (X). We et iϕ η = ϕη = = i whence ϕ η =. Since is an isomorphism we obtain ( ) 1 ϕ η = 1 which says that η is a section. Proposition 1.24 exposes η as an isomorphism. This demonstrates that is extremally mono. Let X Y be an extremal epimorphism. Consider the kernel relation x x (x) = (x ). actors throuh X/ π X Y h X/ and since h is mono it must be a homeomorphism. identiyin, i.e., a quotient map. Consequently is As to the converse let be identiyin. Examine a actorization X Y Then µ is a continuous bijection. Let U Z be open. Then 1 (µ(u)) = 1 (U). So µ(u) has an open -preimae whence it is open in Y. This shows that µ is a homeomorphism. Z µ 1.27 Proposition. For any morphism the ollowin are equivalent: 1. is an isomorphism. 2. is a monomorphism and an extremal epimorphism. 3. is an extremal monomorphism and an epimorphism. Proo. I is an isomorphism then it is mono and epi. A actorization = mϕ entails 1 = mϕ 1, so m is a retraction, hence, i m is mono, then it is iso. This demonstrates that an isomorphism is mono and extremally epi. Likewise it is epi and extremally mono. Conversely the identity = 1 shows that is an isomorphism provided that it is extremally mono and epi Deinition. 1. An object a is called initial i or all x O we have Card(hom(a, x)) = Dually: a is terminal i x O : Card(hom(x, a)) = 1. 18

19 3. a is a zero object i it is both, initial and terminal Example. 1. SET, SGRP, T OP:, the empty set (semiroup, space), is initial. The terminal objects are the sinleton sets (semiroups, spaces) {x}. 2. RIN G: Z is initial, 0 is terminal. 3. {0, 1} (where 0 1) is initial in BA. The 1-element Boolean alebra 0 is terminal in BA. 4. FIELD does not own initial or terminal objects, since hom(e, F ) char E = char F. 5. Let (X, ) be a quasi-ordered class. Then Initial object = least element o X; terminal object = reatest element o X is a zero in GROUP, MON, AB, Λ MOD, T OPGROUP. 7. Sinletons are the zeroes in T OP. 8. In a concrete cateory the initial objects are precisely the ree objects on the empty set. Plainly any two initial objects are isomorphic. The same holds or terminal objects and 0-objects. In case it exists, the unique 22 zero object is denoted by the symbol Deinition. 1. A morphism a b is called constant provided that or each object x and or all r, s hom(x, a) we have r = s. 2. Dually, is coconstant, i y u, v hom(b, y) we have u = v. 3. is a zero morphism i it is simultaneously constant and coconstant. We write 0 or a 0-morphism, and when necessary, 0 ab or a 0-morphism a b. A 0-morphism a 0 b behaves similar to the number 0 with respect to multiplication. Given the situation r s a 0 b u v then 0r = 0s and u0 = v Example. 22 In such a context unique always means unique up to isomorphism. 19

20 1. SET and T OP: x y is constant x = η y : (x) = {η}. 2. In a concrete cateory, any morphism which is a constant unction in the usual sense on underlyin sets is a constant morphism. 3. I t O A is terminal then x O A the map x t is constant. Dually, i i O A is initial then x O A the arrow i x is coconstant Proposition. Consider x 0-morphism), then so is c. c y. I c is constant (coconstant, 1.33 Proposition. Let A be a cateory with terminal object t. I x y actors throuh t, then is constant. Conversely, i is constant and hom(t, x), then actors throuh t. Proo. I actors throuh t, then r s x ξ y τ t r = τξr = τξs = s, that is, is constant. Conversely, assume that is constant. Let u: t x. Then = 1 x = uξ, hence actors throuh t Proposition. Let : a b be a morphism in a cateory with zero object 0. The ollowin statements are equivalent: 1. is constant; 2. is coconstant; 3. is a zero morphism; 4. actors throuh 0, i.e.,, h with a b = a 1.35 Proposition. Let, : a b. Then 0 h b. constant coconstant hom(b, a) = Deinition. For an object a let S a denote the class o all monomorphisms with codomain a. Let Q a be the class o all epimorphims with domain a. 1. For m, m S a we write m m h with m h = m. 2. For e, e Q a, e e k with ke = e. m e a k m e h Both relations are relexive and transitive. We set m m m m and m m. Dually, e e e e and e e. The ollowin notions reer to these equivalence relations on S a and Q a respectively. 20

21 An equivalence class o monomorphisms in S a is called a subobject o a. We will write S(a) or the class o all subobjects o a. An equivalence class o epimorphisms in Q a is called a quotient object o a. Q(a) denotes the class o all quotient objects o a. The class S(a) o all subobjects o an object a is ordered by and similarly or quotient objects Deinition. [m] [n] m n 1. The intersection o a amily o subobjects µ i o a is the inimum i µ i in the ordered class S(a). 2. Dual notion: The cointersection o a amily o quotient objects. This is the inimum in Q(a). Obviously a monomorphism equivalent to an extremal monomorphism is extremal, and similarly or quotient objects. This allows us to talk about extremal subobjects and extremal quotient objects Deinition. A subobject deined by an extremal monomorpism is called an extremal subobject. A quotient object deined by an extremal epimorphism is an extremal quotient object Deinition. Let x y be an arrow. 1. The imae o is the smallest subobject s o y such that actors throuh s. We write im or the imae o. 2. The coimae o is the smallest quotient object o x throuh which actors. We write coim or the coimae o. 1.5 Properties o Cateories im x y coim 1.40 Deinition. A cateory C = (O, M, dom, cod, ) is said to be small provided that C is a set; discrete i M = {1 a a O}; 23 O course one has to deine S(a) and Q(a) as a system o representatives wrto. the respective equivalence relation. As it seems closer to ones intuition we will retain the harmless inaccuracy o talkin about the class o all subobjects (quotient objects). 21

22 stronly connected i a, b O : hom(a, b) ; connected provided that or all objects a, b there is a chain a = x 0 x 1 x r 1 x r = b, where x i x i+1 indicates that there is a morphism x i x i+1 or one x i x i+1. A discrete cateory is simply a class. Thereore a set is a small discrete cateory Proposition. The ollowin statements are equivalent. C is small; O V; M V; dom V; cod V; V. In raph theory a directed multiraph is a triple (V, E, ϕ) where V and E are disjoint sets, and ϕ: E V V. V is the set o vertices, E the set o edes. Thus, a small cateory A is a special directed multiraph (O A, M A, ϕ), with ϕ: M A O A O A, (dom(), cod()) Proposition. Any cateory possessin an initial or a terminal object is connected Deinition. A cateory is said to be balanced provided that each o its bimorphisms is an isomorphism Example. 1. SET, GROUP, Λ MOD, T OP Comp.T 2 are balanced cateories. 2. SGRP, RING, T OP, P OS are not balanced. 3. A partially ordered class is balanced i and only i it is discrete Proposition. For any cateory A the ollowin are equivalent: 1. A is balanced. 2. Each monomorphism is extremal. 3. Each epimorphism is extremal. Proo. Assume A bein balanced and consider a actorization e ϕ x y where is mono and e is epi. By Proposition 1.22 e is mono whence it must be an isomorphism. Conversely, i every monomorphism is extremal, the trivial actorization x y shows that every bimorphism is an isomorphism. y 1 22

23 1.46 Deinition. A is said to have inite intersections provided that any two subobjects x, y o an arbitrary object possess an intersection x y. A has intersections i arbitrary set-indexed amilies o subobjects have intersections. In many cateories appearin in nature the subobjects constitute a set Deinition. A cateory A is called well-powered provided that each object a has a representative class o monomorphisms which is a set. This means, the class o all subobjects o a is in 1-1 correspondence with a set. Dually, A is called co-well-powered i each object a has a set o epimorphisms representin all its quotient objects Example. The cateories SET, GROUP, Λ MOD, T OP, T OP T 2 are wellpowered and co-well-powered. The class o ordinal numbers On is well-powered (by the well-order o On). Since On is a proper class it is not co-well-powered. In a well-powered cateory we may consider the subobjects o an object a as an ordered set with a reatest element (1 a ). Dually, in a co-well-powered cateory the quotient objects orm a set with a reatest element Theorem. The ollowin statements are equivalent: 1. a, b O C : hom(a, b) contains a 0-morphism. 2. a, b O C : hom(a, b) contains exactly one 0-morphism. 3. a, b O C : hom(a, b) contains exactly one constant. 4. a, b O C : hom(a, b) contains exactly one coconstant. 5. a, b O C : hom(a, b) contains a constant and a coconstant. 6. There exists a choice unction selectin one element out o each hom(a, b) such that the composition o a selected morphism with any morphism (rom the let and rom the riht) is aain a selected morphism Deinition. The cateory A is pointed provided that each set hom(a, b) contains a 0-morphism 0 ab, that is, A satisies the conditions o Theorem Example. 1. GROUP, MON, Λ MOD, SET, T OP are pointed cateories. 2. SET, SGRP, T OP, POS, LAT are not pointed. Any two composable morphisms a commutative diaram 0 b a b 0 b c c in a pointed cateory provide a 0 23

24 In a pointed cateory the choice unction is uniquely determined and it selects exactly the zero morphisms 24 (0 ab ) (a,b) A A. Every cateory with a zero object 0 is pointed, as a 0 b = 0 ab Deinition. A cateory A is said to be skeletal provided that 1.53 Example. x, y A (x = y = x = y). 1. A class, considered as a discrete cateory, is skeletal. 2. Partially (and totally) ordered classes are skeletal cateories. 3. I k is a ield, then the cateory o k-matrices MAT k is skeletal. In a skeletal cateory all isomorphisms are automorphisms, and distinct objects are never isomorphic. In this respect the ollowin concept is diametrically opposed Deinition. A cateory is a roupoid i all its morphisms are isomorphisms Example. A roup is a roupoid with exactly one object. In a roupoid any monoid End x coincides with Aut x. I there is a morphism x y, then the respective automorphism roups are isomorphic 1.6 Constructions on Cateories Aut x = Aut y, u u 1. (1) 1.56 Deinition (Opposition). Let C be the cateory (O C, M C, dom C, cod C, ). The opposite (or dual) cateory to C is C op = ( O C, M C, cod C, dom C, ) where we deine =, whenever is deined. Obviously (C op ) op = C Example. 1. I C is a roup, the opposite cateory C op is the opposite roup as it appears in the proo o Lemma The opposite cateory o a partially ordered class (X, ) is the inversely ordered class (X, ). 24 When there is no daner o conusion we will omit the reerence to domain and codomain o 0-morphisms, writin them all simply as 0. 24

25 The opposite cateory C op is obtained rom C by revertin all arrows. allows a semantic view upon dualization: C op Given a statement P about some data o a cateory C, we may express this statement or the data o C op and then rephrase it in terms o C. The outcome o this process is the dual statement P δ Deinition. Let A, B be cateories. B is a subcateory o A i O B O A, M B M A, dom B, cod B and composition in B are restrictions o dom A, cod A and composition in A respectivly, every B-identity is an A-identity. We write B A to indicate that B is a subcateory o A. Evidently then x, y O B : hom B (x, y) hom A (x, y) Example. Consider a cateory A that consists o one object and two morphisms End A = {1, x} with composition rule xx = x. Further let B be the cateory with O B = { } and End B = {x}. A 1 x 1 1 x x x x B x x x Due to the last item in Deinition 1.58, B is not a subcateory o A. As A and B are just monoids, it is natural that this should happen Deinition. Let B A. 1. B is a ull subcateory o A x, y O B : hom B (x, y) = hom A (x, y). 2. B is a dense subcateory o A a O A b O B with a = b. 3. B is isomorphism closed in A b B a A : b = a a B Example. 1. The cateory o inite sets is a ull isomorphism closed subcateory o SET. 2. The cateory o sets and injective unctions is a subcateory o SET which is not ull. 3. AB GROUP and GROUP MON, both are ull. MON SGRP is not ull. GROUP SGRP is ull. 4. BA LAT POS; neither is ull. 5. POS Rel is ull. 6. BR CRING is a ull isomorphism closed subcateory. 25

26 7. By Cayley s theorem, the cateory o permutation roups (i.e. subroups o the automorphism roups o the cateory SET ) is a ull dense subcateory o GROUP. 8. The cateory o cardinal numbers is a ull dense subcateory o SET. 9. Let A be an arbitrary cateory. Any subclass X O A deines the ull subcateory ( ) X, hom(x, y), dom, cod, A. x,y X 10. Every ull subcateory o a pointed cateory is pointed Deinition. A skeleton o a cateory A is a maximal ull skeletal subcateory S o A Example. 1. The powers k n (n N) build a skeleton or the cateory o inite-dimensional k-vector spaces. 2. The class o cardinal numbers is a skeleton or SET. 3. Ordinal numbers constitute a skeleton or well-ordered sets and monotone unctions Proposition. Every cateory has a skeleton Deinition (Product). Let A, B be cateories. The product o A and B is the cateory A B = (O A O B, M A M B, dom A dom B, cod A cod B, ) where ( 1, 2 ) ( 1, 2 ) = ( 1 1, 2 2 ). The product o a set-indexed amily o cateories is deined accordinly Deinition (Sum). The sum o two cateories A, B is A B = (O A O B, M A M B, dom A dom B, cod A cod B, A B ) where the dotted symbol denotes disjoint union. The sum o a set-indexed amily o cateories is deined in the same way Example. The small skeletal roupoids are precisely (cateorical) sums o roups Deinition (Quotient). An equivalence relation ρ on the morphism class o a cateory C is a conruence provided that ρ dom() = dom() cod() = cod(); ρ ρ ()ρ( ) whenever these compositions are deined. Given a conruence on C, the quotient cateory is deined as the cateory C/ρ, where O C/ρ = O C, M C/ρ = M C /ρ, and composition is iven by [] [] = []. 25 That is, what the axiom o choice is ood or. 26

27 The irst point o this deinition says that ρ is subordinated to the hom-partition. The second requires compatibility o ρ with composition Example. 1. Let X, Y T OP. A homotopy is a continuous map X [0, 1] Y. For continuous maps X Y we set : h: X [0, 1] Y with h(, 0) = h(, 1) =. We write h : and call h a homotopy rom to. h describes a continuous deormation rom (X) to (X); each point (x) movin alon the continuous path h(x, ) towards (x). The maps, are then called homotopic. Homotopy o continuous maps is a conruence in T OP. The quotient T OP := T OP/ is the homotopy cateory o spaces. We denote the set o homotopy classes hom T OP (X, Y ) by [X, Y ]. 2. Let, : (X, x 0 ) (Y, y 0 ) be morphisms in T OP. and are homotopic in T OP i they are homotopic in T OP by a homotopy h which or all instances t [0, 1] deines a morphism h(, t): (X, x 0 ) (Y, y 0 ) in T OP. So the homotopy h does not move the base point. This deines a conruence in T OP, the respective quotient cateory T OP is called homotopy cateory o pointed spaces. The set o morphisms rom one pointed space to another is written [(X, x 0 ), (Y, y 0 )] = hom T OP ((X, x 0 ), (Y, y 0 )). 3. Two morphisms (X, A) (Y, B) in T OP 2 are called homotopic i they are homotopic in T OP via a homotopy h producin morphisms h(, t) in T OP 2 or all t [0, 1]. Thus the values o points in A must not leave B durin deormation. This concept yields a conruence in the cateory T OP 2. The correspondin quotient is the cateory T OP 2. We write [(X, A), (Y, B)] = hom T OP2 ((X, A), (Y, B)). 4. The homotopy cateory o bipointed spaces is T OP = T OP /. Homotopy between unctions (X, x 0, x 1 ) (Y, y 0, y 1 ) is restricted to such continuous maps h: X [0, 1] Y that produce morphisms h(, t) in T OP t [0, 1], i.e., h(x 0, t) = y 0 and h(x 1, t) = y 1. Aain the set o morphisms in the quotient cateory is denoted by brackets. 5. Let (a, ), (b, ) Λ CC be chain complexes, a b chain maps. Set : h: a b in Λ GRMOD with de h = 1 and h+h =. 27

28 We call and homotopic provided that. The map h: is called a chain homotopy. Homotopy o chain maps is a conruence in the cateory Λ CC. The resultin quotient Λ CC := Λ CC/ is the homotopy cateory o chain complexes. The set o morphisms [a, b] := hom ΛCC (a, b) is the abelian roup hom ΛCC(a, b)/{ 0}. 6. Let, : G H be morphisms in GROUP. Call and conjuate, i they dier by an inner automorphism o H, that is y H x G : (x) = y(x)y 1. This deines a conruence in GROUP; the resultin quotient cateory GROUP is called the cateory o roups and conjuacy classes o homomorphisms. The transition rom a cateory to one o its quotiens may delete certain properties o morphisms Example. Let S 1 denote the unit circle {z C : z = 1} and D = {z C : z 1} the closed unit disk. The embeddin S 1 D 2 is a monomorphism in T OP while its homotopy class [] is not mono in T OP. Similarly, the map e it : R S 1 is epi, while [e it ] is not. Usin homotopy cateories we construct a nontrivial roupoid: 1.71 Example. Let X T OP. The undamental roupoid o the space X is the ollowin small cateory Π 1 (X). Objects o Π 1 (X) are the points o X. For points x, y X we set hom Π1(X)(x, y) := [([0, 1], 0, 1), (X, x, y)]. Thus, a morphism rom x to y is a homotopy class o paths; the homotopy relation bein considered in the cateory o bipointed spaces. I α: ([0, 1], 0, 1) (X, x, y) and β : ([0, 1], 0, 1) (X, y, z) are paths in X (α(1) = β(0)), then we deine the product o α and β as the path { α(2t) 0 t 1 α β : ([0, 1], 0, 1) (X, x, z), (α β)(t) = 2, 1 β(2t 1) 2 t 1. Assume that h: α α and h: β β in T OP. The mappins h and k obey the equations h(s, 0) = α(s); h(s, 1) = α (s); k(s, 0) = β(s); k(s, 1) = β (s); h(0, t) = x; k(0, t) = y; h(1, t) = y; k(1, t) = z. 28

29 For (s, t) [0, 1] [0, 1] set H(s, t) = { h(2s, t) 0 s 1 2, k(2s 1, t) 1 2 s 1. The restriction o H to the closed hal-squares [0, 1 2 ] [0, 1] and [ 1 2, 1] [0, 1] are continuous and coincide on the overlap, whence H is a continuous map [0, 1] [0, 1] X. Furthermore { h(2s, 0) (0 s 1 H(s, 0) = 2 ) α(2s) } k(2s 1, 0) ( 1 = (α β)(s); 2 s 1) β(2s 1) H(s, 1) = { h(2s, 1) (0 s 1 2 ) α (2s) k(2s 1, 1) ( 1 2 s 1) β (2s 1) } = (α β )(s). Thereore H : α β α β. This proves that composition o x y [β] z may be correctly deined as [β] [α] = [α β]. [α] y and Given x [α] y [β] z [γ] w, one easily constructs a homotopy (α β) γ α (β γ) whence ([γ] [β]) [α] = [γ] ([β] [α]). x A point x determines the constant path [0, 1] {x} X. Obviously x α α and α y y. Consequently [x] = 1 x and Π 1 (X) is a small cateory. Every morphism x [α] y has an inverse, iven by Thus Π 1 (X) is indeed a roupoid. t α(1 t). For each point x X, the automorphisms o x orm a roup called the undamental roup o X at point x, also reerred to as irst homotopy roup π 1 (X, x) := Aut Π1(X) x. Points belonin to equal path components have isomorphic undamental roups (c. Example 1.55). A path-connected space has thereore a unique undamental roup π 1 (X). 2 Functors 2.1 Deinition and Examples 2.1 Deinition. Let A, B be cateories. A unctor rom A to B is a map F : M A M B that preserves composition and identities. In detail, F has to satisy the ollowin rules: 29

30 I is deined then so is F ()F (), and F ()F () = F (). I e is an identity in A then F (e) is an identity in B. We write F : A B or A A unctor F : A B induces a map F B when F is a unctor rom A to B. O A O B, a dom(f (1 a )) which we denote by the same symbol. Thus we obtain a commutative diaram With this deinition 26 we have: M A F M B 1 A O A F O B dom F (hom A (x, y)) hom B (F x, F y) and F (1 x ) = 1 F x. The restriction o the map F to a particular hom-set F hom A (x, y): hom A (x, y) hom B (F x, F y) is called a hom-set restriction o F. Since unctors are maps on classes they can be composed in the usual way and the result o composin two unctors ives aain a unctor. F : A B G : B C G F : A C. We will reely use the notation GF or the composite unctor G F. A unctor A A is addressed as an endounctor. 2.2 Example (Covariant Functors). 1. For any cateory A and subcateory B there is an inclusion unctor B A. For A = B this is the identity unctor 1 A. 2. I is a conruence on A then [] deines the canonical projection A A/. 3. I A, B are cateories and b is an object in B, there is the constant unctor C b : A B, 1 b. 4. Let A 1,..., A n be cateories and a = (a 1,..., a n ) A 1 A n a tuple o objects. For 1 i n the association x y (a 1,..., x,..., a n ) (1,...,,...,1) (a 1,..., y,..., a n ) deines a unctor ι a,i : A i A 1 A i A n. 26 We write F (z) or F z or the value o the unctor F at an arument z, no matter whether z denotes a morphism or an object. 30