Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T08:58:25.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Equational presentations of functors and monads

Published online by Cambridge University Press:  25 March 2011

JIŘÍ VELEBIL  and
ALEXANDER KURZ
JIŘÍ VELEBIL
Affiliation:
Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic E-mail: [email protected]
ALEXANDER KURZ
Affiliation:
Department of Computer Science, University of Leicester, United Kingdom E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study equational presentations of functors and monads defined on a category that is equipped by an adjunction F ˧ U : → of descent type. We present a class of functors/monads that admit such an equational presentation that involves finitary signatures in .

We apply these results to an equational description of functors arising in various areas of theoretical computer science.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Special Issue 2: Coalgebraic Logic , April 2011 , pp. 363 - 381
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adámek, J. (1974) Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolin. 15 589–602.Google Scholar
Adámek, J., Gumm, H.-P. and Trnková, V. (2009) Presentation of set functors: A coalgebraic perspective. J. Logic Comput. 20 (5)991–1015.CrossRefGoogle Scholar
Adámek, J. and Rosický, J. (1994) Locally presentable and accessible categories, Cambridge University Press.Google Scholar
Adámek, J. and Trnková, V. (1990) Automata and algebras in categories, Kluwer Academic Publishers.Google Scholar
de Bakker, J. and de Vink, E. (1996) Control flow semantics, MIT Press.Google Scholar
Bird, G. J. (1984) Limits in 2-categories of locally-presented categories, Ph.D. Thesis, The University of Sydney.Google Scholar
Bonsangue, M. M. and Kurz, A. (2006) Presenting functors by operations and equations. In: Aceto, L. and Ingólfsdóttir, A. (eds.) Proceedings of FOSSACS 2006. Springer-Verlag Lecture Notes in Computer Science 3921 172–186.CrossRefGoogle Scholar
Blackburn, P., de Rijke, M. and Venema, Y. (2001) Modal logic, Cambridge University Press.Google Scholar
Clouston, R. and Pitts, A. (2007) Nominal equational logic. In: Cardelli, L., Fiore, M. and Winskel, G. (eds.) Computation, Meaning and Logic – Articles dedicated to Gordon Plotkin. Electronic Notes in Theoretical Computer Science 172 223–257.Google Scholar
Fiore, M. P. and Staton, S. (2006) Comparing operational models of name-passing process calculi. Information and Computation 204 (4)524–560.Google Scholar
Gabbay, M. J. (2009) Nominal algebra and the HSP theorem, J. Logic Comput. 19 (2)341–367.CrossRefGoogle Scholar
Gabbay, M. J. and Pitts, A. (1999) A new approach to abstract syntax involving binders. In: Proceedings 14th Annual IEEE Symposium on Logic in Computer Science 214–224.Google Scholar
Gabriel, P. and Ulmer, F. (1971) Lokal präsentierbare Kategorien. Springer-Verlag Lecture Notes in Mathematics 221.Google Scholar
Gadducci, F., Miculan, M. and Montanari, U. (2006) About permutation algebras, (pre)sheaves and named sets. Higher-order and Symbolic Computation 19 (2-3)283–304.CrossRefGoogle Scholar
Kapulkin, K., Kurz, A. and Velebil, J. (2010) Expressivity of coalgebraic logic over posets. In: Jacobs, B. P. F., Niqui, M., Rutten, J. J. M. M. and Silva, A. M. (eds.) CMCS'10 Short Contributions: 10th International Workshop On Coalgebraic Methods In Computer Science, CWI 16–17.Google Scholar
Kelly, G. M. (1982a) Basic concepts of enriched category theory. ondon Mathematical Society Lecture Notes Series 64, Cambridge University Press. (Also available as TAC reprint via http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.htmlGoogle Scholar
Kelly, G. M. (1982b) Structures defined by finite limits in the enriched context I. Cahiers de Top. et Géom. Diff. XXIII (1)3–42.Google Scholar
Kelly, G. M. and Power, A. J. (1993) Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads. J. Pure Appl. Algebra 89 163–179.Google Scholar
Kurz, A. and Petrişan, D. (2010) On universal algebra over nominal sets. Mathematical Structures in Computer Science 20 (2)285–318.Google Scholar
Kurz, A., Petrişan, D. and Velebil, J. (2010) Algebraic theories over nominal sets. Available as arXiv preprint at http://front.math.ucdavis.edu/1006.3027.Google Scholar
Kurz, A. and Rosický, J. (2006) Strongly complete logics for coalgebras.Google Scholar
Linton, F. E. J. (1969) Coequalizers in categories of algebras. In: Eckmann, B. (ed.) Seminar on triples and categorical homology theory. Springer-Verlag Lecture Notes in Mathematics 80 61–72. (Also available as TAC reprint via http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html.)Google Scholar
Mac Lane, S. (1998) Categories for the working mathematician, second edition, Springer-Verlag.Google Scholar