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On the essentially-algebraic theory generated by a sketch

Published online by Cambridge University Press:  17 April 2009

G.M. Kelly
G.M. Kelly
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales 2006, Australia.
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Abstract

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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: S → B 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 Lφ ∈ F determines an F-theory T, with a generic model M: S → T 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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 26 , Issue 1 , August 1982 , pp. 45 - 56
Copyright
Copyright © Australian Mathematical Society 1982

References

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