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Simply Connected Limits

  • Robert Paré (a1)
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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].

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References
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1. Bénabou, J., Introduction to bicategories, Lecture Notes in Math., no. 47 (1967), 177. Springer- Verlag.
2. Gabriel, P. and Ulmer, F., Lokal präsentierbare Kategorien, (Lecture Notes in Math., no. 221, Springer- Verlag, 1971).
3. Kan, D.M., Adjoint functors, Trans. Amer. Math. Soc, 87 (1958), 294329.
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.
5. Kennison, J., Semi-geometric Functors and Limits of Topoi, preprint.
6. Kock, A. and Wraith, G., Elementary toposes, (Aarhus Lecture Notes 30 (1971)).
7. Makkai, M. and Paré, R., Accessible Categories: The Foundations of Categorical Model Theory, (Contemporary Mathematics, vol. 104, AMS, Providence, 1989).
8. Paré, R., Connected Components and Colimits, J. Pure Appl. Algebra 3 (1973), 2142.
9. Rosebrugh, R. and Wood, R., Pullback preserving functors, preprint.
10. Street, R.H., The comprehensive construction of free colimits, Sydney Category Seminar Reports (Macquarie University, 1979).
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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