This is a continuation of last's week talk on joint work with Simona Paoli. It deals with the case of finite limit structures other than internal categories.
For locally finitely presentable K, there is a corresponding finite limit theory (the opposite of the full subcategory Kf of K consisting of the finitely presentable objects) and K is the category of models in Set of the theory. For an arbitrary category A with finite limits, write K(A) for the cateogry of models in A of the theory Kopf
Let V be a symmetric monoidal closed category, and suppose that (i) K(V) is reflective in the functor category [Kopf,V], and (ii) K is cartesian closed. Then there is a symmetric monoidal closed structure on K(V) for which the category Op(K(V)) of operads in K(V) is equivalent to K(Op(V)). Furthermore, for any operad T in V, there is an operad k(T) in K(V) for which K(VT)=K(V)k(T).
As in last week's talk, the case where V is abelian is particularly interesting; typical cases of K would now include categories, double categories, n-tuple categories, 2-categories, n-categories, and so on.