Examples

Almost every known example of a mathematical structure with the appropriate structure-preserving map yields a category.

1. The category Set with objects sets and morphisms the usual functions. There are variants here: one can consider partial functions instead, or injective functions or again surjective functions. In each case, the category thus constructed is different
2. The category Top with objects topological spaces and morphisms continuous functions. Again, one could restrict morphisms to open continuous functions and obtain a different category.
3. The category hoTop with objects topological spaces and morphisms equivalence classes of homotopic functions. This category is not only important in mathematical practice, it is at the core of algebraic topology, but it is also a fundamental example of a category in which morphisms are not structure preserving functions.
4. The category Vec with objects vector spaces and morphisms linear maps.
5. The category Diff with objects differential manifolds and morphisms smooth maps.
6. The categories Pord and PoSet with objects preorders and posets, respectively, and morphisms monotone functions.
7. The categories Lat and Bool with objects lattices and Boolean algebras, respectively, and morphisms structure preserving homomorphisms, i.e., (⊤, ⊥, ∧, ∨) homomorphisms.
8. The category Heyt with objects Heyting algebras and (⊤, ⊥, ∧, ∨, →) homomorphisms.
9. The category Mon with objects monoids and morphisms monoid homomorphisms.
10. The category AbGrp with objects abelian groups and morphisms group homomorphisms, i.e. (1, ×, ?) homomorphisms
11. The category Grp with objects groups and morphisms group homomorphisms, i.e. (1, ×, ?) homomorphisms
12. The category Rings with objects rings (with unit) and morphisms ring homomorphisms, i.e. (0, 1, +, ×) homomorphisms.
13. The category Fields with objects fields and morphisms fields homomorphisms, i.e. (0, 1, +, ×) homomorphisms.
14. Any deductive system T with objects formulae and morphisms proofs.

These examples nicely illustrates how category theory treats the notion of structure in a uniform manner. Note that a category is characterized by its morphisms, and not by its objects. Thus the category of topological spaces with open maps differs from the category of topological spaces with continuous maps — or, more to the point, the categorical properties of the latter differ from those of the former.

We should underline again the fact that not all categories are made of structured sets with structure-preserving maps. Thus any preordered set is a category. For given two elements p, q of a preordered set, there is a morphism f : p → q if and only if p ≤ q. Hence a preordered set is a category in which there is at most one morphism between any two objects. Any monoid (and thus any group) can be seen as a category: in this case the category has only one object, and its morphisms are the elements of the monoid. Composition of morphisms corresponds to multiplication of monoid elements. That the monoid axioms correspond to the category axioms is easily verified.

Hence the notion of category generalizes those of preorder and monoid. We should also point out that a groupoid has a very simple definition in a categorical context: it is a category in which every morphism is an isomorphism, that is for any morphism f : X → Y, there is a morphism g : Y → X such that f ○ g = id_X and g ○ f = id_Y.

• –––, 2004, “An Answer to Hellman's Question: Does Category Theory Provide a Framework for Mathematical Structuralism”, Philosophia Mathematica, 12: 54–64.
• –––, 2006, Category Theory, Oxford: Clarendon Press.
• –––, 2007, “Relating First-Order Set Theories and Elementary Toposes”, The Bulletin of Symbolic, 13 (3): 340–358.
• –––, 2008, “A Brief Introduction to Algebraic Set Theory”, The Bulletin of Symbolic, 14 (3): 281–298.
• Awodey, S., et al., 2013, Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program.
• Awodey, S. & Butz, C., 2000, “Topological Completeness for Higher Order Logic”, Journal of Symbolic Logic, 65 (3): 1168–1182.
• Awodey, S. & Reck, E. R., 2002, “Completeness and Categoricity I. Nineteen-Century Axiomatics to Twentieth-Century Metalogic”, History and Philosophy of Logic, 23 (1): 1–30.
• –––, 2002, “Completeness and Categoricity II. Twentieth-Century Metalogic to Twenty-first-Century Semantics”, History and Philosophy of Logic, 23 (2): 77–94.
• Awodey, S. & Warren, M., 2009, “Homotopy theoretic Models of Identity Types”, Mathematical Proceedings of the Cambridge Philosophical Society, 146 (1): 45–55.
• Baez, J., 1997, “An Introduction to n-Categories”, Category Theory and Computer Science, Lecture Notes in Computer Science (Volume 1290), Berlin: Springer-Verlag, 1–33.
• Baez, J. & Dolan, J., 1998a, “Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes”, Advances in Mathematics, 135: 145–206.
• –––, 1998b, “Categorification”, Higher Category Theory (Contemporary Mathematics, Volume 230), Ezra Getzler and Mikhail Kapranov (eds.), Providence: AMS, 1–36.
• –––, 2001, “From Finite Sets to Feynman Diagrams”, Mathematics Unlimited – 2001 and Beyond, Berlin: Springer, 29–50.
• Baez, J. & Lauda, A.D., 2011, “A Pre-history of n-Categorical Physics”, Deep Beauty: Understanding the Quantum World Through Mathematical Innovation, H. Halvorson, ed., Cambridge: Cambridge University Press, 13–128.
• Baez, J. & May, P. J., 2010, Towards Higher Category Theory, Berlin: Springer.
• Baez, J. & Stay, M., 2010, “Physics, Topology, Logic and Computation: a Rosetta Stone”, New Structures for Physics (Lecture Notes in Physics 813), B. Coecke (ed.), New York, Springer: 95–172.
• Baianu, I. C., 1987, “Computer Models and Automata Theory in Biology and Medecine”, in Witten, Matthew, Eds. Mathematical Modelling, Vol. 7, 1986, chapter 11, Pergamon Press, Ltd., 1513–1577.
• Bain, J., 2013, “Category-theoretic Structure and Radical Ontic Structural Realism”, Synthese, 190: 1621–1635.
• Barr, M. & Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
• –––, 1999, Category Theory for Computing Science, Montreal: CRM.
• Batanin, M., 1998, “Monoidal Globular Categories as a Natural Environment for the Theory of Weak n-Categories”, Advances in Mathematics, 136: 39–103.
• Bell, J. L., 1981, “Category Theory and the Foundations of Mathematics”, British Journal for the Philosophy of Science, 32: 349–358.
• –––, 1982, “Categories, Toposes and Sets”, Synthese, 51 (3): 293–337.
• –––, 1986, “From Absolute to Local Mathematics”, Synthese, 69 (3): 409–426.
• –––, 1988, “Infinitesimals”, Synthese, 75 (3): 285–315.
• –––, 1988, Toposes and Local Set Theories: An Introduction, Oxford: Oxford University Press.
• –––, 1995, “Infinitesimals and the Continuum”, Mathematical Intelligencer, 17 (2): 55–57.
• –––, 1998, A Primer of Infinitesimal Analysis, Cambridge: Cambridge University Press.
• –––, 2001, “The Continuum in Smooth Infinitesimal Analysis”, Reuniting the Antipodes — Constructive and Nonstandard Views on the Continuum (Synthese Library, Volume 306), Dordrecht: Kluwer, 19–24.
• –––, 2005, “The Development of Categorical Logic”, in Handbook of Philosophical Logic(Volume 12), 2nd ed., D.M. Gabbay, F. Guenthner (eds.), Dordrecht: Springer, pp. 279–362.

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مواضيع ذات صلة

CATEGORIES-Definition of a category

CATEGORIES-Functions of sets

CATEGORIES- Introduction

Philosophical Significance

Brief Historical Sketch

Fundamental Concepts of the Theory

Definitions

Category Theory