Abstract
In order to facilitate the study of 2-categories with structure, we state and prove an n-categorical pasting theorem. This is based upon a new definition of n-pasting scheme that generalises Johnson's definition of a well-formed loop-free pasting scheme by weakening his no direct loops condition. We define n-pasting, prove the theorem, and show that for n=3, it incorporates all possible composites of n-cells. We show that that is not true for higher n. We define the horizontal n-category of an (n+1)-category to generalise that of a 2-category, we define horizontal and vertical composition for an (n+1)-category and we state and prove an interchange law. We also study further conditions on a pasting diagram and their impact upon how one may evaluate a composite, and we express Street's free n-categories in terms of left adjoints.
This research was supported by the Australian Research Council and by ESPRIT Basic Research Action 3245: Logical Frameworks Design, Implementation and Experiment.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
9 Bibliography
F.A.Al-Agl and R.Steiner, Nerves of multiple categories, preprint (1990)
J.W. Gray, Formal Category Theory: Adjointness for 2-Categories, in Lecture Notes in Math 391 (Springer, Berlin, 1971)
M.Johnson, Pasting diagrams in n-categories with applications to coherence theorems and categories of paths, Doctoral thesis, University of Sydney (1987)
Michael Johnson, The combinatorics of n-categorical pasting, J. Pure Appl Algebra 62 (1989) 211–225
Michael Johnson, Finding a representable family and the free construction on 1, International Category Theory Meeting 1990 (lecture)
M.M.Kapranov and V.A.Voevodsky, Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders, preprint, Steklov Mathematical Institute (1990)
G.M. Kelly and R.H. Street, Review of the elements of 2-categories, in Lecture Notes in Math 420 (Springer, Berlin, 1974) 75–103
A.J. Power, A 2-categorical pasting theorem, J. Algebra 129 (1990) 439–445
A.J. Power, An algebraic formulation for data refinement, in Lecture Notes in Comp. Science 442 (Springer, Berlin, 1990) 390–401
A.J.Power, Coherence for bicategories with finite bilimits II (in preparation)
M.B. Smyth and G.D. Plotkin, The category-theoretic solution to recursive domain equations, SIAM J. Computing 11 (1982) 761–783
Ross Street, Limits indexed by category-valued 2-functors, J. Pure Appl Algebra 8 (1976) 149–181
Ross Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283–336
Author information
Authors and Affiliations
Editor information
Additional information
dedicated to Max Kelly on the occasion of his 60th birthday.
Rights and permissions
Copyright information
© 1991 Springer-Verlag
About this paper
Cite this paper
Power, A.J. (1991). An n-categorical pasting theorem. In: Carboni, A., Pedicchio, M.C., Rosolini, G. (eds) Category Theory. Lecture Notes in Mathematics, vol 1488. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084230
Download citation
DOI: https://doi.org/10.1007/BFb0084230
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54706-8
Online ISBN: 978-3-540-46435-8
eBook Packages: Springer Book Archive