alexa Weak topological functors 1 | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on
Medical, Pharma, Engineering, Science, Technology and Business

Weak topological functors 1

Patrik LUNDSTRÖM*

Department of Engineering Science, University West, Trollh¨attan, Sweden

*Corresponding Author:
Patrik LUNDSTRÖM
Department of Engineering Science
University West, Trollh¨attan, Sweden
E-mail:
[email protected]

Received date: November 27, 2007; Revised date: March 13, 2008

Visit for more related articles at Journal of Generalized Lie Theory and Applications

Abstract

We introduce weak topological functors and show that they lift and preserve weak limits and weak colimits. We also show that if A ! B is a topological functor and J is a category, then the induced functor AJ ! BJ is topological. These results are applied to a generalization of Wyler’s top categories and in particular to functor categories of fuzzy maps, fuzzy relations, fuzzy topological spaces and fuzzy measurable spaces.

Introduction

Almost forty years ago, Zadeh [13] introduced the category of fuzzy sets. The objects in this category are maps from ordinary sets to the unit interval and the morphisms are ordinary nondecreasing maps with respect to the fuzzy sets. Two years later, Goguen [5] replaced the unit interval by an arbitrary complete ordered lattice. Since then a lot of work has been devoted to proposing different versions of what fuzzy algebras of distinct types may be, e.g. fuzzy semigroups, fuzzy groups, fuzzy rings, fuzzy ideals, fuzzy semirings, fuzzy near-rings (see [8] for an overview). Fuzziness has also been introduced in various topological settings, e.g. fuzzy topological spaces [4], fuzzy measurable spaces [9] and fuzzy topological groups [2]. Categories of fuzzy structures are often topological over their corresponding base categories, that is, the category of modules in the case of fuzzy modules [10], the category of groups in the case with fuzzy topological groups [3] and so on. This is an important fact since it implies that many properties of the top category, such as completeness and cocompleteness, are automatically inherited from the ground category. However, the theory of topological functors (see e.g. [1]) does not cover the lifting of weak limits and colimits (that is, the uniqueness property of the mediating morphism is dropped). To cover these cases, we introduce weak topological functors (see Definition 2.1) and show a lifting result for such functors (see Proposition 2.2). To simultaneously treat all fuzzy algebraical constructions, we show a result (see Proposition 2.3) concerning the topologicality of functors between functor categories. In the end of the article (see Section 3), these results are applied to a generalization (see Proposition 2.5) of Wyler [12] to prove (weak) completeness and cocompleteness results for functor categories of categories of fuzzy maps, fuzzy relations, fuzzy continuous maps and fuzzy measurable maps.

Topological functors

Let A be a category. The family of objects and the family of morphisms in A is denoted ob(A) and mor(A) respectively. The domain and codomain of a morphism α in A is denoted d(α) and c(α) respectively. The composition of two morphisms α and β in A with d(α) = c(β) is denoted αβ. The identity morphism at a 2 ob(A) is denoted ida. We let homC(a, b) denote the collection of morphisms from a to b. Let Cat denote the category with small categories as objects and functors between such categories as morphisms. If B is another category, then let BA denote the category with functors from A to B as objects and natural transformations between such functors as morphisms. Ifequation then we let Ub denote the corresponding fibre category in the sense of Grothendieck [6], that is, the subcategory of A having as objects all equation and as morphisms allequation in mor(A) between such objects with equation A (simple) source in B is a pairequation consisting of an object b in B and a family of morphisms equation in B indexed by some class I (with cardinality one). Ifequation for some objectsequation then S is called U-structured. A U-lift of such a U-structured source S is a source equation in A whereequation satisfyequationequation We say that such a U-lift of S is weakly initial if for any U-liftequation of a U-structured source equation equipped with aequation in B with the property thatequation there is aequation in A withequation and equation A weak initial U-lift is called initial if the morphism equation is unique. Concepts dual to ”initial” are called ”final”.

Proposition 2.1. Let U : A B be a functor. (a) If each U-structured source (sink) in B has a weak initial (final) U-lift, then U is faithful. (b) Each U-structured source in B has a weak initial U-lift if and only if each U-structured sink in B has a weak final U-lift. (c) Each U-structured source in B has a unique initial U-lift if and only if each U-structured sink in B has a unique final U-lift.

Proof. Adapt the proofs of Theorems 21.3 and 21.9 in [1] to the weak situation.

Definition 2.1. We call a functor satisfying either of the equivalent criteria in Proposition 2.1(c) (or (b)) (weakly) topological. Note that our definition of topological functor coincides with the one given by Herrlich [7].

Proposition 2.2. Let U : A → B be a weakly topological functor. (a) If F : J → A is a functor with weak (co)limit L, then U(L) is a weak (co)limit of UF; (b) If L is a weak (co)limit of UF, then there is a weak (co)limit equation of F such thatequation is weakly (co)complete if and only if B is weakly (co)complete; (d) If U is topological, then (a) and (b) hold with weakness removed. Furthermore, in that case, the (co)limit equation is unique subject to the condition equation

Proof. We only show the ”limit” part of the result. The proof of the ”colimit” part is dual and is therefore left to the reader. (a) Suppose that L is a limit of F : J → A with morphisms equation such thatequation in J. Suppose that there areequation with the property thatequation By weak topologicality of U, there is a U-liftequation withequation By Proposition 2.1(a) U is faithful. Therefore the equalityequationequation implies thatequation Then there isequation withequation This implies thatequation is the desired map. Hence U(L) is a weak limit of UF. Now we show (b). Suppose that L is a weak limit of UF with morphisms equation such thatequation for all α : i → j in J. Let equation be a weak initial U-lift of the U-structured source equation. By U-faithfullness,equation for all morphisms α : i → j in J. Now suppose that there is an object X in A and morphisms equation such thatequation for all morphisms α : i → j. Since L is a weak limit of UF, there is a morphism equation By weak initiality there is a morphismequation in A such thatequation which implies thatequation is a weak limit of F. (c) follows directly from (a) and (b). (d) is Proposition 21.5 in [1].

Proposition 2.3. If J is a category and U : A → B is a topological functor, then the induced functor equation is topological.

Proof. Letequation structured source in BJ . Associated to this source, we define a functor F : J → A in the following way. For each equationequation be the unique initial lift of the U-structured sourceequationequation Take a morphismequation in J. Since,equation there is, by initiality of the lift, a unique equationwith the property that equation Take another morphismequation in J. By uniqueness and initiality ofequation and the fact that the following chain of equalities holds equationequation we get thatequation Also note that since U is topological, the functor F is unique subject to the condition that equation Define a collection of natural transformations equation by the morphismsequation By the construction of theequation we get that they are indeed natural transformations. Suppose that equation andequation are natural transformations satisfyingequation Suppose that there are natural transformationsequation withequation Now we define a natural transformationequation withequation in the following way. Since U is topological there is for eachequation a unique morphismequation in A subject to the conditions thatequation andequation Define p by the morphisms equation Take a morphismequation in J. By the definition of F, we get that equation Therefore,equation Since eachequation is a natural transformation, we get thatequation and hence thatequation Sinceequation is the unique initial lift of equation is a lift ofequation and the morphismsequation satisfyequation we get that s1 = s2 and hence that p is natural transformation.

Let U : A → B be a functor. If U is weakly topological then each fibre of U has weak (co)products. In fact, every simple U-structured source in B has a weak U-initial lift. Take equation and suppose that we have a setequation of objects in Ub . Then the identity morphisms equation considered as a U-structured source, has a weak initial U-liftequation The weak initiality of this lift is equivalent to the condition thatequation is a weak product ofequation In the same way one can show that if U is topological, then the weak (co)product above makes each fibre of U a complete ordered lattice (see Proposition 21.11 in [1]). If we make two additional assumptions, then this condition is sufficient for U to be weakly topological:

Proposition 2.4. (a) A functor U : A B is weakly topological if the following three properties hold: (i) Each simple U-structured source in B has a weak initial lift; (ii) Each fibre of U has weak products; (iii) The weak products in the fibres of U are weak products in A. (b) A functor U : A B is topological if the following three properties hold: (i) Each simple U-structured source in B has a unique initial lift; (ii) Each fibre of U is a complete ordered lattice; (iii) The infima in the fibres of U are products in A.

Proof. Suppose that (i), (ii) and (iii) in (a) hold. Suppose that equation Ustructured source with equation For eachequation let aj be a weak U-initial lift of bj with morphisms equation Let a be a weak product inequation with morphisms equation Putequation Suppose thatequation satisfyequation Suppose thatequation is a U-lift ofequation By simple U-initiality, there are morphisms equation By (iii) there is a morphismequation in A such thatequation Thereforeequation Therefore (a) holds. (b) is a modification of (a).

Now we recall a folkloristic generalization of Wyler’s [12] construction of top categories. Let equation denote a contravariant functor. The categoryequation has as objects all pairsequation and as morphisms all pairsequation whereequation in A andequation The composition ofequation andequation is defined byequation The identity morphisms are defined by equation In the sequel, the forgetful functorequation will be denoted U. By the discussion preceding Proposition 2.4, we get that if U is weakly topological, then each equation has weak (co)products and that if U is topological, then each equation is a complete ordered lattice. If we make an additional assumption, then this condition is sufficient for U to be weakly topological:

Proposition 2.5. (a) If each equation has weak products and eachequation respects weak products, then U is weakly topological.

(b) If each equation is a complete ordered lattice and each equation respects infima, then U is topological.

Proof. (a) Conditions (ii) and (iii) of Proposition 2.4(a) are immediate by the assumptions. Now we show condition (i) of Proposition 2.4(a). Suppose that equation is a morphism in A and equation Clearlyequation is a U-lift of α. Suppose now that equation are morphisms in A satisfyingequation Suppose that equation are lifts ofequation respectively, satisfyingequation Since the left hand side of this equation simplifies toequation. we have thatequation This calculation shows thatequation is a U-initial lift of α. (b) is a modification of the proof of (a) using Proposition 2.4(b).

Applications

The category CfzSet. Let X be a set and C a category. As a generalization of Goguen’s [5] definition of fuzzy sets, we say that a C-fuzzy set on X is a function equation Letequation be another C-fuzzy set. We say that a fuzzy function from μ to v is a pair (f, α) where f : X → Y is a function and equation This is indicated by writing equation be a third fuzzy set andequation a second fuzzy function. Let the composition ofequation be defined byequation where equation It is easy to check that the collection of C-fuzzy sets and C-fuzzy functions form a category which we denote equation denote the forgetful functor. Now we wish to present equation as a top category. For each set X, letequation Note that the objects inequation are the C-fuzzy sets on X and the morphisms in equation are all ordered triplesequation where μ and v are C-fuzzy sets on X and equationequation is a function, letequation be defined by equation andequation whereequation The correspondence equation is a contravariant functor from Set to Cat and there is an isomorphism equation of categories. Hence, by Proposition 2.5 and a straightforward argument, the forgetful functor equation is (weakly) topological if C is (has weak products) a complete ordered lattice. Therefore, by Proposition 2.2, equation is (weakly) complete and cocomplete if C is a complete ordered lattice (has products or coproducts respectively). Furthermore, if we assume that C is a complete ordered lattice and J is a category, then, by Proposition 2.3, the category equation is (co)complete whenever SetJ is (co)complete.

The category LfzRel. For the rest of the article, let L be a complete ordered lattice. Suppose thatequation are L-fuzzy sets. We say that a relationequation is thatequation are L-fuzzy sets. We say that a relationequation is L-fuzzy if equation The collection of L-fuzzy sets and L-fuzzy relations form a category LfzRel. Define a functor equation on objects in the same way as for LfzSet and on relations equation where the infimum is taken over all equation withequation The correspondence equation is a contravariant functor and there is an isomorphism equation of categories. Therefore, by Proposition 2.5(b), the forgetful functor U : equation is topological. Hence, by Proposition 2.2(c), LfzRel is weakly complete and cocomplete.

The categories LT, LTop and LMeas. We define the category LT as the category with objects equation where X is a set and μ is a subset of equation the set of maps from X to L. A morphism equation in LT is a function f : X → Y satisfying equation Now we present LT as a top category. For each set X, letequation be the collection of subsets of equation ordered under reversed inclusion. Ifequation is a function and v is a subset of equation It is easy to see that there is an isomorphismequation of categories. Therefore, by Proposition 2.5(b), the forgetful functor U : LT → Set is topological. Therefore, by Propositions 2.2 and 2.3, LTJ is (co)complete whenever SetJ is (co)complete, for any category J. The category LTop of L-topological spaces (Chang [4]) is the full subcategory of LT with objects (X, μ) where μ is an L-fuzzy topology on X, that is, subsets of LX closed under finite pointwise infima and arbitrary pointwise suprema. If L is a complemented lattice, then the full subcategory LMeas of L-measurable spaces (Klement [9]) is defined as the full subcategory of LT with objects (X, μ) where μ is an L-sigma algebra, that is, a subset of LX closed under countable pointwise suprema and pointwise complements. By following the argument above, it is a straightforward task to show that the forgetful functors U : LTop → Set and U : LMeas → Set are topological. Therefore, by Propositions 2.2 and 2.3, LTopJ and LMeasJ are (co)complete whenever SetJ is (co)complete, for any category J.

References

Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Relevant Topics

Recommended Conferences

  • 7th International Conference on Biostatistics and Bioinformatics
    September 26-27, 2018 Chicago, USA
  • Conference on Biostatistics and Informatics
    December 05-06-2018 Dubai, UAE
  • Mathematics Congress - From Applied to Derivatives
    December 5-6, 2018 Dubai, UAE

Article Usage

  • Total views: 11520
  • [From(publication date):
    September-2008 - Jun 23, 2018]
  • Breakdown by view type
  • HTML page views : 7758
  • PDF downloads : 3762
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh

Peer Reviewed Journals
 
Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
International Conferences 2018-19
 
Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings

Contact Us

Agri & Aquaculture Journals

Dr. Krish

[email protected]

+1-702-714-7001Extn: 9040

Biochemistry Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Business & Management Journals

Ronald

[email protected]

1-702-714-7001Extn: 9042

Chemistry Journals

Gabriel Shaw

[email protected]

1-702-714-7001Extn: 9040

Clinical Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Engineering Journals

James Franklin

[email protected]

1-702-714-7001Extn: 9042

Food & Nutrition Journals

Katie Wilson

[email protected]

1-702-714-7001Extn: 9042

General Science

Andrea Jason

[email protected]

1-702-714-7001Extn: 9043

Genetics & Molecular Biology Journals

Anna Melissa

[email protected]

1-702-714-7001Extn: 9006

Immunology & Microbiology Journals

David Gorantl

[email protected]

1-702-714-7001Extn: 9014

Materials Science Journals

Rachle Green

[email protected]

1-702-714-7001Extn: 9039

Nursing & Health Care Journals

Stephanie Skinner

[email protected]

1-702-714-7001Extn: 9039

Medical Journals

Nimmi Anna

[email protected]

1-702-714-7001Extn: 9038

Neuroscience & Psychology Journals

Nathan T

[email protected]

1-702-714-7001Extn: 9041

Pharmaceutical Sciences Journals

Ann Jose

[email protected]

1-702-714-7001Extn: 9007

Social & Political Science Journals

Steve Harry

[email protected]

1-702-714-7001Extn: 9042

 
© 2008- 2018 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version
Leave Your Message 24x7