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On the essentially-algebraic theory generated by a sketch

  • G.M. Kelly (a1)
Abstract

By a sketch we here mean a small category S together with a small set φ of projective cones in S, each cone φ ∈ φ being indexed by a small category Lφ. A model of S in any category B is a functor G: SB such that each Gφ is a limit-cone. Let F be any small set of small categories containing all the Lφ. A small category T admitting all F-limits (that is, an F-complete small T ) is called an F-theory; it is considered as a sketch in which the distinguished cones are all the F-limit-cones. It is an important result of modern universal algebra, due originally to Ehresmann, that each sketch S = (S, φ) with every LφF determines an F-theory T, with a generic model M: ST of S, such that composition with M induces an equivalence M* between the category of T-models in B and that of S-models in B, whenever B is F-complete. We give a simple proof of this result – one which generalizes directly to the case of enriched categories and indexed limits; and we make the new observation that the inverse to M* is given by (pointwise) right Kan extension along M.

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Copyright
References
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[1]Bastiani, Andrée and Ehresmann, Charles, “Categories of sketched structures”, Cahiers Topologie Géom. Différentielle 13 (1972), 103214.
[2]Coste, M., “Localisation dans les catégories de modèles“ (Thesis, Université Paris Nord, Paris, 1977).
[3]Diers, Yves, “Type de densité d'une sous-catégorie pleine”, Ann. Soc. Sci. Bruxelles Sér. I 90 (1976), 2547.
[4]Gabriel, Peter, Ulmer, Friedrich, Lokal präsentierbare Kategorien (Lecture Notes in Mathematics, 221. Springer-Verlag, Berlin, Heidelberg, New York, 1971).
[5]Kelly, G.M., “A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on”, Bull. Austral. Math. Soc. 22 (1980), 183.
[6]Kelly, G.M., Basic concepts of enriched category theory (London Mathematical Society Lecture Notes, 64. Cambridge University Press, London, New York, Sydney, 1982).
[7]Kock, Anders, Synthetic differential geometry (London Mathematical Society Lecture Notes, 51. Cambridge University Press, London, 1981).
[8]Kock, A., Reyes, G.E., “Doctrines in categorical logic”, Handbook of mathematical logic, 283313 (Studies in Logic and the Foundations of Mathematics, 90. North-Holland, Amsterdam, New York, Oxford, 1977).
[9]Mac Lane, S., Categories for the working mathematician (Graduate Texts in Mathematics, 5. Springer-Verlag, New York, Heidelberg, Berlin, 1971).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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