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Finitary sketches

Published online by Cambridge University Press:  12 March 2014

J. Adámek ,
P. T. Johnstone ,
J. A. Makowsky  and
J. Rosický
J. Adámek
Affiliation:
Institut für Theoretische Informatik, Technical University Braunschweig, Postfach 3329, 38023 Braunschweig, Germany, E-mail: adamek@iti.cs.tu-bs.de
P. T. Johnstone
Affiliation:
Department of Pure Mathematics, Cambridge University, 16 Mill Lane, Cambridge CB2 1SB, England, E-mail: ptj@pmms.cam.ac.uk
J. A. Makowsky
Affiliation:
Faculty of Computer Science, Technion-Israel Institute of Technology, Haifa 32000, Israel, E-mail: janos@csa.cs.technion.ac.il
J. Rosický
Affiliation:
Department of Algebra and Geometry, Masaryk University, Janáčkovo Nám. 2A, 662 95 Brno, Czech Republic, E-mail: rosicky@math.muni.cz
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Abstract

Finitary sketches, i.e., sketches with finite-limit and finite-colimit specifications, are proved to be as strong as geometric sketches, i.e., sketches with finite-limit and arbitrary colimit specifications. Categories sketchable by such sketches are fully characterized in the infinitary first-order logic: they are axiomatizable by σ-coherent theories, i.e., basic theories using finite conjunctions, countable disjunctions, and finite quantifications. The latter result is absolute; the equivalence of geometric and finitary sketches requires (in fact, is equivalent to) the non-existence of measurable cardinals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 699 - 707
Copyright
Copyright © Association for Symbolic Logic 1997

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References

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