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On Characterization of Reflection and Coreflection in Categories

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On Characterization of Reflection and Coreflection in Categories Proyecc-iones Vol. 18, T\" 2, pp. 175-182, December 1999 Universidad Católica del Norte Antofagasta - Chile ON CHARACTERIZATION OF REFLECTION AND COREFLECTION IN CATEGORIES HAMZA A. S. ABUJABAL King Abdul Aziz University, Saudi Arabia and S. M. A. ZAIDI Aligrah Muslim University, India Abstract The main purpose of the present paper is to present a criteria of re.flection and its dual through generalized pushout ( G PO) and gener­ alized pullback ( G P B) structures. Severa/ results related to re.flective and core.flective subcategories are obtained. KEY WORDS AND PHRASES : Re.flection, Re.flective sub­ category, dense subcategory, G P B category, G PO category, G P B .functor·, G PO functor, generator. 1991 AMS Subject classification : 18D15. 176 Hamza A. S. Abujabal and S. M. A. Zaidi l. Introduction The concepts of reflection and reflective subcategory and their duals in­ troduced by P.J. Freyd (see [1]). Here we have obtained the existence of reflection and coreflection by generalized pushout (G PO) and generalized pullback ( G P B) respective! y. We have also discussed several other proper­ ties of reflective and coreflective subcategories. 2. Preliminares In this paper, we will clarify only few preliminary notions frequently used in the dcvelopment of our results. For other concepts we refer [2], [3] and [4]. Definition 1 : A non-empty family of morphisms { ai : A ---> Bi} 1 in a category e is called a left cone over J with vertex A and base objects Bi, i E J. We will denote it by [A,ai,Bi]J. Dually, we can define a right cone. Definition 2 : In a category e, a left cone [P, ai, Bi]I is called the generalized pullback (GPB) of the right cone [Bi,/3i,A]¡ if and only if GPB1: /3jo:j = f3ko:k for allj,k E J, GPB2 : For any other left cone [X,/i,Bi]J in e with ,6jfj = !3kfk for all j, k E J, there exists a unique morphism r¡ : X ---> P in e such that air¡ = Ji for all i E J. Dually, we can define generalized pushout (GPO). Definition 3 : Let e be a category andA a subcategory of C. For an object e E e, an object e E A together with a morphism Pe :e---> e will be called a reflection of e, in A, if for every morphism a : e ___, A with A E A, there exists a unique morphism a : e ---> A in A such that e6e~A=e~A. Dually, an object e E A together with a morphism ac : e --t e will be called a coreflection of e in A if for every morphism ,6 : A ---> e with A E A, there exists a unique morphism [}_ : A ---> e in A such that ALe~e=A~e. On Characterization of Refiection and ... 177 Remark 1 : Reflection ( coreflcction) is unique up to isomorphism. Definition 4 : If every object of the category C has a reflection ( core­ flec:tion) in a subc:ategory A, then A will be called a reflective (coreflective) subcategory of e. Definition 5 : If A ~ e is a reflective subcategory, then by associating evcry object of e to its reflection in A, we can define a functor R : e ____, A. This is called a reflector. Dually, if A ~ C is a coreflective subcategory. we can define coreflector R: C ___,A. Remark 2 : Every reflector is a left adjoint of the inc:lusion functor and hence a G PO functor. Dually, every coreflector is a right adjoint of the inclusion functor and hence a G PE functor. 3. Existence and properties Theorem 1: Let A be a subcategory of e, and [A, ai, Ai]¡be a left cone in A.If its GPO's exist in C andA both, then the vertex of GPO [A, ai, Ai]¡ in A, is a reflection of the vertex of GPO [A, ai, Ai]I in C. Proof: Suppose, and GPO [A,ai,A;]¡ = [A,/';,Q]¡ in C. Since A ~ C,there exists a unique morphism (by Axiom GP02)r¡ : Q ____, P E C such that T//'i = ;3; : A ____, P for al! i in 1. Now, we show that the object PE A together with a morphism r¡ : Q ____,Pis a reflection of Q in A. For any morphism f : Q ____, X in C with X E A, there exists a right cone [Ai, JI';, X]¡ in C such that the morphisms (f!';)a; = f( riai) : A ____, A; ____,X 178 Hamza A. S. Abujabal and S. M. A. Zaidi are the same for all i E J. Further, by Axiom GP02, there exists a unique morphism f : P -+ X in A such that /{3; = f¡; :A;-+ X for all i E J, and so f(r¡¡;) = (fr¡)¡'; = J¡; for all i E J. Again, by Axiom GP02, we get Jr¡ =f. Hence, P together with r¡: Q-+ Pis a refiection of Q. Dually, we have Theorem 1 * : Let A be a subcategory of C, and [A;, a;, A]¡ be a right cone in A, if its GP Bts exists in C and A both. Then the vertex of GPB[A;,a;,A]¡ in A is a corefiection of the vertex of GPB[A;,a;,A]1 in C. Theorem 2 : Every GPO-dense subcategory of a GPO-category is refiective. Conversely, every refiective subcategory of a GPO-category is also a G PO-category. Proof : Suppose C is a category and A is a GPO-dense subcategory of C which is also a GPO-category. Let e be any object of C. By dense property, there exists a left cone [A, a;, A;] 1 in A such that e is the vertex of GPO[A, a;, A;]¡ in C. Suppose, GPO [A, a;, A;]¡= [A;, {3;, e]I in C. Further, A is a GPO-category, this implies that GPO[A, a;, A;] 1also exists in A.Let GPO [A, a;, A;]¡= [A,,6;,C]¡ in A. By Theorem 1, C is the refiection of e in A. Hence, Ais a refiective subcategory of e. Conversely, we assume that e is a GPO-category and is a refiective subcategory of C. Consider an arbitrary left cone [A, a;, A;]¡in Aand let GPO [A, a;, A;]¡= [A;, {3;, P]¡ in C. On Characterization of Refiection and ... 179 Let P E A together with a morphism p p : P ----+ P be a reflection of P in A.Consider the right cone [A;, PPf3i, P]¡ in A. It is clear that the morphisms (PP{J;)o:; :A----+ A;----+ p are the same for all i E J in A. Suppose there 1s another right cone [A;, f;, X]1 in A such that the morphisms k:xi: A----+ A; ---+X are the same for all i E J in A. By axiom GP02, there exists a unique morphism r¡ : P ----+ X in C such that r¡¡3; = .f; : Ai ----+ X for all i E J. Further, by definition of reflection, there exists a unique morphism r¡: P----+ X such that iiPp = r¡, r¡ (pP/3;) = (r¡pP) ;3; = r¡¡3; = f; for all i E J, and GPO [A, o:;, A;]¡= [A;, PPf3i, P]¡ in A. Hence, A is a GPO-category. Dually, we have Theorem 2* : Every GPO-dense subcategory which is also a G P B- category is coreflective. Converse! y, every coreflective subcategory of a GPB-category is also a GPB-category. Theorem 3 : Every full refiective subcategory of a GPB- category is al so a G P B- category. Proof : Let C be a G P B-category and A be its full reflective subcate­ gory. Consider an arbitrary right cone [A;, o:;, A]¡ in A. Suppose GPB [A;,o:;,A] 1 = [P,¡3;,A;]¡ in C. Let PE A together with morphism pp : P----+ P be a reflection of P in A. Consider the left cone [P, ~;,A;]¡ in A, where ~iPP = /3; for all i E J. Then 180 Hamza A. S. Abujabal and S. M. A. Zaidi are the same for all i E I (by uniqueness). By axiom GPB2, there exists a unique morphism r¡ : p ----> p in e such that f3ir¡ = !Ji : P ----> A for all i E J. For each i E I, we have Further, by axiom GP B2, we get r¡pp = lp N ow, we will show that p pr¡ = I p. Since A is a full subcategory, we can assume that PP = Ip, R(pp) = Ip, where R is the reflector of e in A, which is a left adjoint of the inclusion functor from A into e. Thus ppr¡ = R(r¡)pp = R(r¡)R(pp) = R(r¡pp) = R(Ip) = Ip. This shows that p p is an isomorphism and consequently, we get Hence A is a GPB-category. From Theorem 3 and Remark 3, we have Corollary 1 : Every full reflective subcategory of a G P B- category is a GPB- subcategory. From Theorem 3 and the definition of replete subcategory, we have Corollary 2 : Every full replete reflective subcategory of a GPB­ category is a GP B- closed subcategory. Dually, we have Theorem 3* : Every full coreflective subcategory of a GPO- category is also a GPO- category. Corollary 1 * : Every full replete coreflective subcategory of a GPO­ category is a GPO- subcategory. On Characterization of Reflection and ... 181 Corollary 2* : Every full replete coreflcctive subcategory of a GPO­ category is a GPO- closed subcategory. Lemma 1 : The reflection of an initial object is also an initial object.
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