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Simply Connected Limits

Published online by Cambridge University Press:  20 November 2018

Robert Paré*
Affiliation:
Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, N.S., B3H 3J5
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The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Bénabou, J., Introduction to bicategories, Lecture Notes in Math., no. 47 (1967), 177. Springer- Verlag.Google Scholar
2. Gabriel, P. and Ulmer, F., Lokal präsentierbare Kategorien, (Lecture Notes in Math., no. 221, Springer- Verlag, 1971).CrossRefGoogle Scholar
3. Kan, D.M., Adjoint functors, Trans. Amer. Math. Soc, 87 (1958), 294329.Google Scholar
4. Kelly, G.M. and Paré, R., A note on the Albert-Kelly paper “The closure of a class ofcolimits”,, J. Pure Appl. Algebra 57 (1988), 1925.Google Scholar
5. Kennison, J., Semi-geometric Functors and Limits of Topoi, preprint.Google Scholar
6. Kock, A. and Wraith, G., Elementary toposes, (Aarhus Lecture Notes 30 (1971)).Google Scholar
7. Makkai, M. and Paré, R., Accessible Categories: The Foundations of Categorical Model Theory, (Contemporary Mathematics, vol. 104, AMS, Providence, 1989).CrossRefGoogle Scholar
8. Paré, R., Connected Components and Colimits, J. Pure Appl. Algebra 3 (1973), 2142.Google Scholar
9. Rosebrugh, R. and Wood, R., Pullback preserving functors, preprint.Google Scholar
10. Street, R.H., The comprehensive construction of free colimits, Sydney Category Seminar Reports (Macquarie University, 1979).Google Scholar
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