Abstract
Category is a mathematical structure consisting of objects and morphisms between objects with some specific properties. Categories examine in abstract way the properties of particular mathematical concepts by formalizing them as collections of objects and morphisms. Categorical structures are widely used in computer science for exact mathematical modeling. This paper highlights the most typical use of categories for constructing the model of part of differential calculus by using special category named arrow category; and codomain and domain functors.
[1] S. Awodey, Category Theory (Carnegie Mellon University, 2005) 10.1093/acprof:oso/9780198568612.001.0001Search in Google Scholar
[2] M. Barr, C. Wells, Category Theory for Computing Science (Prentice Hall International, 1990) Search in Google Scholar
[3] Cs. Szabó, V. Slodičák, Software Engineering Tasks Instrumentation by Category Theory, SAMI 2011, Proceedings of the 9th IEEE International Symposium on Applied Machine Intelligence and Informatics, Smolenice, Slovakia, 27.–29.1.2011, Košice, elfa, s.r.o., 2011, 195–199 http://dx.doi.org/10.1109/SAMI.2011.573887410.1109/SAMI.2011.5738874Search in Google Scholar
[4] V. Novitzká, V. Slodičák, Categorical structures and their application in informatics, Equilibria (Košice, 2010) (in Slovak) Search in Google Scholar
[5] V. Slodičák, P. Macko, New approaches in functional programming using algebras and coalgebras, In European Joint Conferrences on Theory and Practise of Software-ETAPS 2011, Universität des Saarlandes, Saarbrücken, Germany, 2011, 13–23 Search in Google Scholar
[6] J. Adámek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories (John Wiley & Sons, 1990) Search in Google Scholar
[7] N. Bourbaki, Integration I (Springer-Verlag, 2004) 10.1007/978-3-662-07931-7Search in Google Scholar
[8] H.J. Keisler, Elementary Calculus: An Infinitesimal Approach (On-line Edition) University of Wisconsin, Copyright 2000/revised 2012, http://www.math.wisc.edu/keisler/calc.html Search in Google Scholar
[9] T. Harasthy, J. Turán, Ľ. Ovseník, Road line detection based on optical correlator, In: Mipro 2013: 36th international convention: conference proceedings: May 20–24, 2013, Opatija, Croatia. — Rijeka: MIPRO, 2013, 320–322 Search in Google Scholar
[10] M. Fernandez, Models of Computation, An Introduction to Computability Theory (Springer, 2009) 10.1007/978-1-84882-434-8Search in Google Scholar
[11] S. Aleksić, S. Ristić, I. Luković, M. Čeliković, A Design Specification and a Server Implementation of the Inverse Referential Integrity Constraints, Comput. Sci. Inform. Sys. 10(1), 2013, 283–320 http://dx.doi.org/10.2298/CSIS111102003A10.2298/CSIS111102003ASearch in Google Scholar
[12] P. Selinger, Lecture Notes on the Lambda Calculus (Dalhousie University, Halifax, Canada, 2007) Search in Google Scholar
[13] A. Tarlecki, Categories, Institutions, Abstract Model Theory, and Software Specification, Workshop on Applied and Computational Category Theory — ACCAT/ETAPS 2013, Rome, Italy, 2013 Search in Google Scholar
[14] B. Jacobs, Categorical Logic and Type Theory, No. 141 in Studies in Logic and the Foundations of Mathematics (North Holland, Amsterdam, 1999) Search in Google Scholar
[15] W. Steingartner, D. Galinec, The Rôle of Categorical Structures in Infinitesimal Calculus, J. Appl. Math. Comput. Mechanics 12(1), 107–119, 2013 10.17512/jamcm.2013.1.11Search in Google Scholar
[16] D.M. Burton, The History of Mathematics: An Introduction, 6th ed. (McGraw-Hill, 2005) Search in Google Scholar
[17] O. Forster, Analysis 1, Differential- und Integralrechnung einer Vernderlichen 7, (Au. Vieweg-Verlag, 2004) Search in Google Scholar
[18] J. Taufer, O. Vraštilová, Mathematics 1 — Differential calculus for engineers (Montanex, a.s., 1998) (in Czech) Search in Google Scholar
[19] E.E. Rosinger, Nel’s category theory based differential and integral Calculus, or did Newton know category theory? source: http://arxiv.org/abs/math/0504565 (2005) Search in Google Scholar
[20] J. Kock, Polynomial functors (On-line Edition), University of Barcelona, Copyright 2013/revised 2014, http://mat.uab.es/kock/cat/polynomial.html Search in Google Scholar
[21] I. Luković, S. Ristić, A. Popović, P. Mogin, An approach to the platform independent specification of a business application, In: Proceedings of the 23rd Central European Conference on Information and Intelligent Systems — CECIIS’2012, 19th–21st Sept 2012, University of Zagreb, Varaždin, Croatia, 2012, 449–456 Search in Google Scholar
[22] C. Thomas, Introduction to Differential Calculus (University of Sydney, 1997) Search in Google Scholar
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