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On the Limitations of Sketches

Published online by Cambridge University Press:  20 November 2018

Michael Barr  and
Charles Wells
Michael Barr
Affiliation:
Department of Mathematics and Statistics McGill University
Charles Wells
Affiliation:
Department of Mathematics and Statistics Case Western Reserve University
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Abstract

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Call a category "sketchable" if it is the category of models in sets of some sketch. This paper explores the subtle boundary between sketchable and non-sketchable categories. We show that the category of small categories that have at least one initial object and functors that take an initial object to an initial object is sketchable. The same is true for weak initial objects, but is false for subinitial objects (that every object has at most one arrow to). Analogous results hold if we substitute finite limits for terminal object. We also show that the category of groups and center-preserving homomorphisms is not sketchable. We describe briefly how "higher-order" sketches can fill these gaps.

Keywords

18C1018A10accessible categoriesregular categoriesembeddings
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 35 , Issue 3 , 01 September 1992 , pp. 287 - 294
Copyright
Copyright © Canadian Mathematical Society 1992

References

Barr, M. and Wells, C., Toposes, Triples and Theories , Grundlehren der math. Wissenschaften 278, Springer- Verlag, Berlin, Heidelberg, New York, 1985.Google Scholar
Barr, M. and Wells, C., Category Theory for Computing Science , Prentice Hall International, Hemel Hempstead, 1990.Google Scholar
Lair, C., Catégories modelables et catégories esquissables , Diagrammes, 6, 1981.Google Scholar
Makkai, M. and Paré, R., Accessible Categories: The Foundations of Categorical Model Theory , Contemporary Mathematics, 104, 1989, Amer. Math. Soc, Providence, RI, USA.Google Scholar
Power, A. J. and Wells, C., A formalism for the specification of essentially-algebraic structures in 2-categories, Preprint, Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106 (1990).Google Scholar
Wells, C., A generalization of the concept of sketch, Theoretical Computer Science 70 (1990), 159–178.Google Scholar
Wells, C. and Barr, M., The formal description of data types using sketches , In Mathematical Foundations of Programming Language Semantics, M. Main, A. Melton, M. Mislove and D. Schmidt, editors, Lecture Notes in Computer Science 298, Springer-Verlag, 1988.Google Scholar