Abstract
We give an exposition of a unified study of categories with extra structure that arise in computer science and mathematics. We consider several examples of structures that arise, showing that with a precise formulation of the notion of category with algebraic structure, they are categories with algebraic structure. We then outline the central results and issues in the study of categories with algebraic structure, with an account of why those issues are of computational interest. We illustrate general theorems that yield substantial results in examples given by familiar structures. In particular, we explain known mathematics that supports the idea of a programming language being freely generated by basic data and specified algebraic structure. We then show in detail how the concept of algebraic structure may be used in defining new category theoretic structures by showing how it affected the precise formulation of premonoidal category, as has recently been proposed to account for contexts.
This work is supported by EPSRC grant GR/J84205: Frameworks for programming language semantics and logic. The writing was done on a visit to the Electrotechnical Laboratory in Japan, with partial funding from MITI's Cooperative Architecture Project.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
M. Barr and C. Wells, Toposes, Triples, and Theories, Grundlehren der math. Wissenschaften 278, Springer (1985).
M. Barr and C. Wells, Category Theory for Computing Science, Prentice-Hall (1990).
R. Blackwell, G.M. Kelly, and A.J. Power, Two-dimensional monad theory, J. Pure Appl. Algebra 59 (1989) 1–41.
J. Harland and D.J. Pym, The uniform proof-theoretic foundation of linear logic programming, Proc. International Symp. on Logic Programming, MIT Press (1991) 304–318.
G.M. Kelly, On Mac Lane's conditions for coherence of natural associativities, commutativities, etc. J. Algebra 1 (1964) 397–402.
G.M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Notes Series 64, Cambridge University Press (1982).
G.M. Kelly, Structures defined by finite limits in the enriched context I, Cahiers de Topologie and Geometrie Differentielle 23 (1982) 3–41.
G.M. Kelly and A.J. Power, Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, J. Pure Appl. Algebra 89 (1993) 163–179.
G.M. Kelly and A.J. Power, Algebraic structure in the enriched context, (1988) (draft).
Y. Kinoshita, P.J. O'Hearn, A.J. Power, M. Takeyama, and R.D. Tennent, Logical relations, algebraic structure, and data refinement, Proc. Theoretical Aspects of Computer Science 97 (to appear).
Y. Kinoshita, A.J. Power, and M. Takeyama, Sketches, Proc. Math. Foundations of Programming Languages '97, Electronic Notes in Theoretical Computer Science 6.
J.Lambek and P.J.Scott, Introduction to higher-order categorical logic, Cambridge Studies in Advanced Mathematics 7, Cambridge University Press (1986).
S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer (1971).
J. Meseguer, Conditional rewriting logic as a unified model of concurrency, Theoretical Computer Science 96 (1992) 73–155.
J. Meseguer and U. Montanari, Petri nets are monoids, Information and Computation 88 (1990) 105–155.
R. Milner, Action calculi, or concrete action structures, Proc. Math. Foundations of Computer Science, Lecture Notes in Computer Science 711 (1993) 105–121.
E. Moggi, Notions of computation and monads, Information and Computation 93 (1991) 55–92.
M. Makkai and R. Pare, Accessible categories: the foundations of categorical model theory, Contemporary Mathematics 104 (1989).
A.J. Power, Why tricategories? Information and Computation 120 (1995) 251–262.
A.J. Power, Premonoidal categories as categories with algebraic structure (submitted).
A.J. Power and E.P. Robinson, A characterization of pie-limits, Math. Proc. Camb. Phil. Soc. 110 (1991) 33–47.
A.J. Power and E.P. Robinson, Premonoidal categories and notions of computation, Proc. LDPL 95, Math Structure in Computer Science (to appear).
A.J. Power and H. Thielecke, Continuation semantics and indexed categories, Proc. Theoretical Aspects of Computer Science 97 (to appear).
Edmund Robinson, Variations on algebra: monadicity and generalisations of algebraic theories, Math. Structures in Computer Science (to appear).
R.D.Tennent, Semantics of Programming Languages, Prentice-Hall (1991)
Hayo Thielecke, Continuations semantics and self-adjointness, Proc Math. Foundations of Programming Semantics '97, Electronic Notes in Theoretical Computer Science 6.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Power, J. (1998). Categories with algebraic structure. In: Nielsen, M., Thomas, W. (eds) Computer Science Logic. CSL 1997. Lecture Notes in Computer Science, vol 1414. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028027
Download citation
DOI: https://doi.org/10.1007/BFb0028027
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64570-2
Online ISBN: 978-3-540-69353-6
eBook Packages: Springer Book Archive