Abstract
The significance of the 2-dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street. Independently, Burroni has introduced a general notion of n-dimensional presentation and he has shown that the equational logic of terms is a special case of 2-dimensional calculus.
Here, we propose a combinatorial definition of 2-dimensional diagrams and a simple method for proving that certain monoidal categories are finitely 2-presentable. In particular, we consider Burroni's presentation of finite maps and we extend it to the case of finite relations.
This paper should serve as a reference for our future work on symbolic computation, including a theory of 2-dimensional rewriting and the design of software for interactive diagrammatic reasoning.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Burroni, Higher Dimensional Word Problem, Theoretical Computer Science 115, 1993, pp. 43–62.
S. Eilenberg & J. B. Wright, Automata in general algebras, Information and Control 115, 1967, pp. 452–470.
P.J. Freyd & D.N. Yetter, Braided Compact Closed Categories with Application to Low Dimensional Topology, Advances in Mathematics 77, 1989, pp. 156–182.
A. Joyal & R. Street, The Geometry of Tensor Calculus, Advances in Mathematics 88, 1991, pp. 55–112.
Y. Lafont, Penrose diagrams and 2-dimensional rewriting, Applications of Categories in Computer Science (ed. M.P. Fourman, P.T. Johnstone & A.M. Pitts), LMSLNS 177, Cambridge University Press, 1992, pp. 191–201.
F.L. Lawvere, Functorial Semantics of Algebraic Theories, Proc. Nat. Acad. Sci. USA, 1963.
S. Mac Lane, Categories for the Working Mathematician, GTM 5, Springer-Verlag, 1971.
R. Penrose & W. Rindler, Spinors and space-time, Vol. 1: Two-spinor calculus and relativistic fields, Cambridge University Press, 1986.
A. Poigné, Algebra Categorically, Category Theory and Computer Programming (ed. D. Pitt, S. Abramsky, A. Poigné & D. Rydeheard), LNCS 240, Springer-Verlag, 1985, pp. 76–102.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lafont, Y. (1995). Equational reasoning with 2-dimensional diagrams. In: Comon, H., Jounnaud, JP. (eds) Term Rewriting. TCS School 1993. Lecture Notes in Computer Science, vol 909. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59340-3_13
Download citation
DOI: https://doi.org/10.1007/3-540-59340-3_13
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59340-9
Online ISBN: 978-3-540-49237-5
eBook Packages: Springer Book Archive