David Yetter's comments concerning toposes with free categories contains a misstatement. It is not true that finite limit theories have tripleable categories of models over any topos with NNO. For example, the category of small categories is not regular, hence not tripleable over Sets for any underlying functor (p. 120 of Triples, Toposes, and Theories). --Charles Wells =================================
To try to clarify the remarks of David Yetter and Charles Wells on essentially algebraic theories, let me pass on an extract from the paper "Preframe Presentations Present" which I recently wrote with Peter Johstone (for "CT '90", to appear in Springer Lecture Notes in Mathematics). I ought to say that the section from which this extract is taken, which was intended to be a brief yet helpful account of essentially algebraic theories, is a distillation of Peter's knowledge rather than mine. "For a small essentially algebraic theory T, the forgetful functor from T-models to Set (or Set^n if T is many-sorted) has a left adjoint, just as in the algebraic case: the free T-model on a set X is constructed in the usual way as the set of words (i.e. terms) in the elements of X, modulo T-provable equality. The adjunction will not be monadic unless T is algebraic, but it will be possible to factor it as a tower of monadic adjunctions in the style of MacDonald and Stone ("The tower and regular decomposition", pp. 197-213 in Cahiers Top. Geom. Diff 23 (1982))." I presume, again without being an expert in these fields, that for finitary theories this all works over arbitrary elementary toposes with NNO. Steve Vickers
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Steven John Vickers