DOCSLIB.ORG

On the Ring Isomorphism & Automorphism Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Ring Isomorphism & Automorphism Problems On the Ring Isomorphism & Automorphism Problems Neeraj Kayal, Nitin Saxena ∗ National University of Singapore, Singapore and IIT Kanpur, India {kayaln, nitinsa}@cse.iitk.ac.in Abstract We will restrict our attention to finite rings with unity. We assume that the rings are given in terms of the basis of We study the complexity of the isomorphism and auto- their additive group and the multiplication table of basis el- morphism problems for finite rings with unity. ements. Given two rings in this form, the ring isomorphism We show that both integer factorization and graph iso- problem is to test if the rings are isomorphic. We show that morphism reduce to the problem of counting automor- this problem is in NP ∩ coAM and is at least as hard as the phisms of rings. The problem is shown to be in the complex- graph isomorphism problem. Thus, ring isomorphism is a ity class AM ∩ coAM and hence is not NP-complete unless natural algebraic problem whose complexity status is simi- the polynomial hierarchy collapses. Integer factorization lar to that of graph isomorphism. The search version of the also reduces to the problem of finding nontrivial automor- isomorphism problem is to find an isomorphism between phism of a ring and to the problem of finding isomorphism two given rings. We show that integer factoring reduces to between two rings. the search version of the problem. We also show that deciding whether a given ring has a non-trivial automorphism can be done in deterministic Another variant of the problem is to count the number polynomial time. of isomorphisms between two rings. We show that both integer factorization and graph isomorphism reduce to this problem. We also show that this problem is equivalent to that of counting the number of automorphisms in a ring and 1 Introduction lies in the class FPAM∩coAM. This implies that the problem is not NP-hard unless the polynomial hierarchy collapses to ΣP [Sch88]. A ring consists of a set of elements together with addi- 2 tion and multiplication operations. These structures are fun- damental objects of study in mathematics and particularly The ring automorphism problem is to test if a ring has so in algebra and number theory. It has long been recog- a non-trivial automorphism. We prove that this problem is nized that the group of automorphisms of a ring provides in P. This is in contrast to the corresponding problem for valuable information about the structure of the ring. Ga- graphs whose status is still open. On the other hand we lois initiated the study of the group of automorphisms of a show that the problem of finding a nontrivial automorphism field and it was later applied by Abel to prove the celebrated of a given ring is equivalent to integer factoring. This im- theorem that there does not exist any formula for finding plies that the search version of the problem is likely to be the roots of a quintic (degree 5) polynomial. However, to strictly harder than the decision version. the best of our knowledge, the computational complexity of the ring isomorphism and automorphism related problems The most general problem here is to compute the auto- has not been investigated thus far. In this paper, we initiate morphism group of a given ring, in terms of a small set of such a study and show interesting connections to some well generators. It is easy to see that all the above problems re- known problems. duce to it. Also, the proof of upper bound on counting au- ∗ tomorphisms can be adapted to exhibit an AM protocol for This work was done while the authors were visiting Princeton Univer- sity, Princeton, NJ, USA in 2003-04. Also partially supported by research it implying that this problem too is not NP-hard unless PH P funding from Infosys Technologies Limited, Bangalore. = Σ2 . 2 Representation of finite rings Remark. This theorem can be used to check in polyno- mial time whether for two rings, given in basis form, the The complexity of the problems involving finite rings de- additive groups are isomorphic or not. Suppose the two Z ⊕ ··· ⊕ Z pends on the representation used to specify the ring. We will additive groups are G := d1 dn and G := Z ⊕···⊕Z use the following natural representation of a ring: d1 dn . We can take gcd of disanddjs and keep factoring them till we stop getting nontrivial factors. At this Definition 2.1. Basis representation of rings: AringR point G will be isomorphic to G if and only if the multi-sets is given by first describing its additive group in terms of n {di}i and {di}i are equal. generators and then specifying multiplication by giving for each pair of generators, their product as an element of the Proposition 2.5. GivenaringR in terms of generators, additive group. all having prime-power additive orders, we can compute k the number of automorphisms of the additive group of R, (R, +,.):=(d ,d ,d , ··· ,dn), ((a )) ≤i,j,k≤n 1 2 3 ij 1 #Aut(R, +), in polynomial time. k where, for all 1 ≤ i, j, k ≤ n, 0 ≤ a <dk. ij Proof. Refer to appendix. This specifies a ring R generated by n elements e1,e2, ···en with each ei having additive order di and Proposition 2.6. [Structure theorem for rings] If R is a fi- (R, +) = e1⊕e2···⊕en. Thus (R, +) has the struc- nite ring with unity then it can be uniquely (up to permuta- ∼ Z ⊕ Z ⊕···⊕Z ture (R, +) = d1 d1 dn . Moreover multi- tions) decomposed into indecomposable rings R1,...,Rs plication in R is specified by specifying the product of each such that pair of generators as an integer linear combination of the R = R1 ⊗ ...⊗ Rs. ≤ ≤ · n k generators: for 1 i, j n, ei ej = k=1 aij ek. Remark. It follows immediately from the proof Definition 2.2. Representation of maps on rings: Sup- ([McD74]) of above proposition that for a commutative ring pose R is a ring given in terms of its additive generators 1 R its decomposition can be found in polynomial time given e ,...,en and ring R given in terms of f ,...,fn.Inthis 1 2 1 oracles to integer and polynomial factorizations. Observe paper maps on rings would invariably be homomorphisms that any commutative ring R with characteristic n can be on the additive group. Then to specify any map φ : R → 1 expressed as: R2, it is enough to give the images φ(e1),...,φ(en).So we represent φ by an n × n matrix of integers A, such that ∼ R = Zn[x1,...,xm]/(f1(x),...,fl(x)) for all 1 ≤ i ≤ n: n and so if we can factor n into its prime factors and mul- φ(ei)= Aij fj tivariate polynomials fis into irreducible factors (modulo j=1 prime powers) then we can effectively factor ring R into its indecomposable components. and for all 1 ≤ i, j ≤ n, 0 ≤ Aij < additive order of fj. Algebraic structures mostly break into simpler objects. Let us also define the multiplication operation on ideals In the case of rings we get the following simpler rings. which will be useful in the last section. Definition 2.3. Indecomposable or Local ring: AringR Definition 2.7. Let I, J be two ideals of a ring R.We is said to be indecomposable or local if there do not exist define their product as ∼ rings R ,R such that R = R ⊗ R ,where⊗ denotes 1 2 1 2 I·J { | ∈I ∈J} the natural composition of two rings with component wise := ring generated by the elements ij i ,j addition and multiplication. It, for positive integer t, is defined similarly. We collect some of the known results about rings. Their It is easy to see that I·J is again an ideal of R. proofs can be found in algebra texts, e.g. [McD74]. Finally, we define the ring isomorphism and related Proposition 2.4. [Structure theorem for abelian groups] If problems that we are going to explore. R is a finite ring then its additive group (R, +) can be uniquely (up to permutations) expressed as: • The ring isomorphism problem is to check whether two given rings are isomorphic. The corresponding lan- ∼ Z α (R, +) = pi i guage we define as i RI := {(R1,R2) | rings R1,R2 are given in the where pi’s are primes (not necessarily distinct) and αi ∈ ≥ ∼ } Z 1. basis representation and R1 = R2 . • #RI is defined as the functional problem of computing R → R iff ∀iφisomorphically maps Ri to Ri. Thus, wlog the number of isomorphisms between two rings given we can assume that the additive basis of the rings R and R in basis form. is given in the form: • #RA is defined as the functional problem of comput- Zdα1 ⊕ ...⊕ Zdαn where 1 ≤ α1 ≤ ...≤ αn ing the number of automorphisms of a given ring. Its decision version can be viewed as the language Now in this case φ is an isomorphism from R → R iff it satisfies the following conditions: cRA := {(R, k) | R is a ring in basis form • ∈ Z∗ × s.t. #Aut(R) ≥ k} . det(A) d,whereA is the n n integer matrix describing the map φ : R → R. • RA is defined as the problem of determining whether • for all 1 ≤ i ≤ n, additive order of ei is the same as a given ring has a nontrivial ring automorphism.The that of φ(ei). corresponding language is: • ≤ ≤ · n k for all 1 i, j n, φ(ei) φ(ej )= k=1 aij φ(ek), { | k RA := R R is a ring in basis form s.t.
Load more

Recommended publications
  • Chapter 4. Homomorphisms and Isomorphisms of Groups
    Chapter 4. Homomorphisms and Isomorphisms of Groups 4.1 Note: We recall the following terminology. Let X and Y be sets. When we say that f is a function or a map from X to Y , written f : X ! Y , we mean that for every x 2 X there exists a unique corresponding element y = f(x) 2 Y . The set X is called the domain of f and the range or image of f is the set Image(f) = f(X) = f(x) x 2 X . For a set A ⊆ X, the image of A under f is the set f(A) = f(a) a 2 A and for a set −1 B ⊆ Y , the inverse image of B under f is the set f (B) = x 2 X f(x) 2 B . For a function f : X ! Y , we say f is one-to-one (written 1 : 1) or injective when for every y 2 Y there exists at most one x 2 X such that y = f(x), we say f is onto or surjective when for every y 2 Y there exists at least one x 2 X such that y = f(x), and we say f is invertible or bijective when f is 1:1 and onto, that is for every y 2 Y there exists a unique x 2 X such that y = f(x). When f is invertible, the inverse of f is the function f −1 : Y ! X defined by f −1(y) = x () y = f(x). For f : X ! Y and g : Y ! Z, the composite g ◦ f : X ! Z is given by (g ◦ f)(x) = g(f(x)).
    [Show full text]
  • The General Linear Group
    18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices.
    [Show full text]
  • Categories, Functors, and Natural Transformations I∗
    Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0.
    [Show full text]
  • Irreducible Representations of Finite Monoids
    U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately.
    [Show full text]
  • Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1
    Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0.
    [Show full text]
  • Homomorphisms and Isomorphisms
    Lecture 4.1: Homomorphisms and isomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 1 / 13 Motivation Throughout the course, we've said things like: \This group has the same structure as that group." \This group is isomorphic to that group." However, we've never really spelled out the details about what this means. We will study a special type of function between groups, called a homomorphism. An isomorphism is a special type of homomorphism. The Greek roots \homo" and \morph" together mean \same shape." There are two situations where homomorphisms arise: when one group is a subgroup of another; when one group is a quotient of another. The corresponding homomorphisms are called embeddings and quotient maps. Also in this chapter, we will completely classify all finite abelian groups, and get a taste of a few more advanced topics, such as the the four \isomorphism theorems," commutators subgroups, and automorphisms. M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 2 / 13 A motivating example Consider the statement: Z3 < D3. Here is a visual: 0 e 0 7! e f 1 7! r 2 2 1 2 7! r r2f rf r2 r The group D3 contains a size-3 cyclic subgroup hri, which is identical to Z3 in structure only. None of the elements of Z3 (namely 0, 1, 2) are actually in D3. When we say Z3 < D3, we really mean is that the structure of Z3 shows up in D3.
    [Show full text]
  • Friday September 20 Lecture Notes
    Friday September 20 Lecture Notes 1 Functors Definition Let C and D be categories. A functor (or covariant) F is a function that assigns each C 2 Obj(C) an object F (C) 2 Obj(D) and to each f : A ! B in C, a morphism F (f): F (A) ! F (B) in D, satisfying: For all A 2 Obj(C), F (1A) = 1FA. Whenever fg is defined, F (fg) = F (f)F (g). e.g. If C is a category, then there exists an identity functor 1C s.t. 1C(C) = C for C 2 Obj(C) and for every morphism f of C, 1C(f) = f. For any category from universal algebra we have \forgetful" functors. e.g. Take F : Grp ! Cat of monoids (·; 1). Then F (G) is a group viewed as a monoid and F (f) is a group homomorphism f viewed as a monoid homomor- phism. e.g. If C is any universal algebra category, then F : C! Sets F (C) is the underlying sets of C F (f) is a morphism e.g. Let C be a category. Take A 2 Obj(C). Then if we define a covariant Hom functor, Hom(A; ): C! Sets, defined by Hom(A; )(B) = Hom(A; B) for all B 2 Obj(C) and f : B ! C, then Hom(A; )(f) : Hom(A; B) ! Hom(A; C) with g 7! fg (we denote Hom(A; ) by f∗). Let us check if f∗ is a functor: Take B 2 Obj(C). Then Hom(A; )(1B) = (1B)∗ : Hom(A; B) ! Hom(A; B) and for g 2 Hom(A; B), (1B)∗(g) = 1Bg = g.
    [Show full text]
  • 7.3 Isomorphisms and Composition
    392 Linear Transformations 7.3 Isomorphisms and Composition Often two vector spaces can consist of quite different types of vectors but, on closer examination, turn out to be the same underlying space displayed in different symbols. For example, consider the spaces 2 R = (a, b) a, b R and P1 = a + bx a, b R { | ∈ } { | ∈ } Compare the addition and scalar multiplication in these spaces: (a, b)+(a1, b1)=(a + a1, b + b1) (a + bx)+(a1 + b1x)=(a + a1)+(b + b1)x r(a, b)=(ra, rb) r(a + bx)=(ra)+(rb)x Clearly these are the same vector space expressed in different notation: if we change each (a, b) in R2 to 2 a + bx, then R becomes P1, complete with addition and scalar multiplication. This can be expressed by 2 noting that the map (a, b) a + bx is a linear transformation R P1 that is both one-to-one and onto. In this form, we can describe7→ the general situation. → Definition 7.4 Isomorphic Vector Spaces A linear transformation T : V W is called an isomorphism if it is both onto and one-to-one. The vector spaces V and W are said→ to be isomorphic if there exists an isomorphism T : V W, and → we write V ∼= W when this is the case. Example 7.3.1 The identity transformation 1V : V V is an isomorphism for any vector space V . → Example 7.3.2 T If T : Mmn Mnm is defined by T (A) = A for all A in Mmn, then T is an isomorphism (verify).
    [Show full text]
  • Monomorphism - Wikipedia, the Free Encyclopedia
    Monomorphism - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Monomorphism Monomorphism From Wikipedia, the free encyclopedia In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : X → Y such that, for all morphisms g1, g2 : Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism. Contents 1 Relation to invertibility 2 Examples 3 Properties 4 Related concepts 5 Terminology 6 See also 7 References Relation to invertibility Left invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G.
    [Show full text]
  • Group Properties and Group Isomorphism
    GROUP PROPERTIES AND GROUP ISOMORPHISM Evelyn. M. Manalo Mathematics Honors Thesis University of California, San Diego May 25, 2001 Faculty Mentor: Professor John Wavrik Department of Mathematics GROUP PROPERTIES AND GROUP ISOMORPHISM I n t r o d u c t i o n T H E I M P O R T A N C E O F G R O U P T H E O R Y is relevant to every branch of Mathematics where symmetry is studied. Every symmetrical object is associated with a group. It is in this association why groups arise in many different areas like in Quantum Mechanics, in Crystallography, in Biology, and even in Computer Science. There is no such easy definition of symmetry among mathematical objects without leading its way to the theory of groups. In this paper we present the first stages of constructing a systematic method for classifying groups of small orders. Classifying groups usually arise when trying to distinguish the number of non-isomorphic groups of order n. This paper arose from an attempt to find a formula or an algorithm for classifying groups given invariants that can be readily determined without any other known assumptions about the group. This formula is very useful if we want to know if two groups are isomorphic. Mathematical objects are considered to be essentially the same, from the point of view of their algebraic properties, when they are isomorphic. When two groups Γ and Γ’ have exactly the same group-theoretic structure then we say that Γ is isomorphic to Γ’ or vice versa.
    [Show full text]
  • Isomorphisms, Automorphisms, Homomorphisms, Kernels
    Isomorphisms, Automorphisms, Homomorphisms, Kernels September 10, 2009 Definition 1 Let G; G0 be groups and let f : G ! G0 be a mapping (of the underlying sets). We say that f is a (group) homomorphism if f(x · y) = f(x) · f(y) for all x; y 2 G. The composition f G−!G0−!h G" of two homomorphisms (as displayed) is again a homomorphism. For any two groups G, G0 by the trivial homomorphism from G to G0 we mean the mapping G ! G0 sending all elements to the identity element in G0. Recall: Definition 2 A group homomorphism f : G ! G0 is called a group isomorphism|or, for short, an isomorphism|if f is bijective. An isomorphism from a group G to itself is called an automorphism. Exercise 1 Let f : G ! G0 be a homomorphism of groups. The image under the homomorphism 0 f of any subgroup of G is a subgroup of G ; if H ⊂ G is generated by elements fx1; x2; : : : ; xng then its image f(H) is generated by the images ff(x1); f(x2); : : : ; f(xn)g; the image of any cyclic subgroup is cyclic; If f is an isomorphism from G to G0 and x 2 G then the order of x (which|by definition—is the order of the cyclic subgroup generated by x) is equal to the order of f(x). For G a group, let Aut(G) be the set of all automorphisms of G, with `composition of automorphisms" as its `composition law' (denoted by a center-dot (·). As noted in class, Proposition 1 If G is a group then Aut(G) is also a group.
    [Show full text]
  • A Short Introduction to Category Theory
    A SHORT INTRODUCTION TO CATEGORY THEORY September 4, 2019 Contents 1. Category theory 1 1.1. Definition of a category 1 1.2. Natural transformations 3 1.3. Epimorphisms and monomorphisms 5 1.4. Yoneda Lemma 6 1.5. Limits and colimits 7 1.6. Free objects 9 References 9 1. Category theory 1.1. Definition of a category. Definition 1.1.1. A category C is the following data (1) a class ob(C), whose elements are called objects; (2) for any pair of objects A; B of C a set HomC(A; B), called set of morphisms; (3) for any three objects A; B; C of C a function HomC(A; B) × HomC(B; C) −! HomC(A; C) (f; g) −! g ◦ f; called composition of morphisms; which satisfy the following axioms (i) the sets of morphisms are all disjoints, so any morphism f determines his domain and his target; (ii) the composition is associative; (iii) for any object A of C there exists a morphism idA 2 Hom(A; A), called identity morphism, such that for any object B of C and any morphisms f 2 Hom(A; B) and g 2 Hom(C; A) we have f ◦ idA = f and idA ◦ g = g. Remark 1.1.2. The above definition is the definition of what is called a locally small category. For a general category we should admit that the set of morphisms is not a set. If the class of object is in fact a set then the category is called small. For a general category we should admit that morphisms form a class.
    [Show full text]