Michael Abbott asked whether Set is lfp, intuitionistically.
Or, I hope this is the same question, is an elementary topos with a natural numbers object internally locally finitely presentable? Are there any references for this?
Peter Johnstone said that
the answer is yes, but (like a great many such things) I don't think it is written down anywhere. Finite cardinals are internally finitely presentable (the proof of this is similar to the proof that they are internally projective, see 6.25 in "Topos Theory"), and the fact that every object is internally a filtered colimit of finite cardinals is implicit in the construction of the object classifier (cf. 6.32 in the same reference).
Regarding the notion of LFP category as one way (amongst many) of formulating a generalised (but finitary) algebraic theory, Set is the category of models of the theory with one sort, no operations and (of course) no equations. However, one must be careful. Is the category Set^N lfp? The idea is that an object X is finitely presentable if homming from it preserves filtered colimits. (It is also interesting to investigate directed unions, and filtered colimits of surjections.) But which homming functor do we mean? - the external one C(X,-) : C -> Set - or the internal one (-)^X : C -> C (if C is a CCC or a topos). An object X=(X_n) of Set^N is - externally FP iff sum_n X_n is finite, so Some m.All n>n. X_n=0, - internally FP iff each X_n is finite. So there are far more internally FP objects than externally FP ones. This comes from Section 6.6 of "Practical Foundations of Mathematics", http://www.dcs.qmul.ac.uk/~pt/Practical_Foundations/html/s66.html Paul