On the essentially-algebraic theory generated by a sketch

G.M. Kelly
1982 Bulletin of the Australian Mathematical Society  
By a sketch we here mean a small category 5 together with a small set $ of protective cones in S , each cone ( J > € $ being indexed by a small category L, . A model of S in any category 8 is a functor G : S -*• 8 such that each G is a limit-cone. Let F be any small set of small categories containing all the L, . 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
more » ... all the F-limit-cones. It is an important result of modern universal algebra, due originally to Ehresmann, that each sketch S = (S, $) with every L, € 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 8 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 .
doi:10.1017/s0004972700005591 fatcat:uyorzltt35ed3k27oitjoqn7ei