Hi, I would be most grateful if somebody could point me to the appropriate literature on internal categories. I am interested in the following. Let C be a category with finite limits. Cat(C) denotes the 2-category whose objects are internal categories in C, whose morphisms are internal functors in C, and whose 2-morphisms are internal natural transformations. If C,D have finite limits and F:C->D is a functor that preserves finite limits, then there is a 2-functor Cat(F):Cat(C)->Cat(D). Similarly, for a natural transformation e:F=3D>G between two such functors F,G:C->D, one obtains a 2-natural transformation Cat(e):Cat(F)=3D>Cat(G). In total, there is a 2-functor Cat(-):LECat -> 2Cat where LECat is the 2-category whose objects are categories with finite limits, whose morphisms are finite-limit preserving functors and whose 2-morphisms are natural transformations. I am interested in the following. If the category with finite limits C has additional properties or additiona= l structure, what does this imply for the 2-category Cat(C)? In particlar if, * C is enriched * C is symmetric|braided monoidal * C is symmetric monoidal and closed Maybe the following point of view (which views Cat(C) as a 1-category) is more familiar. If one studies essentially algebraic theories, one writes, (1) Cat(C)=3DLEFunc(Th(Cat),C). Here LEFunc(D,C) is the category of finite-limit preserving functors D->C and natural transformations. C,D are categories with finite limits. Th(Cat) is a suitable category with finite limits (`theory of categories') such tha= t (1) holds. Cat(C) is then called a model of the theory of categories in the category C. Then my above questions come down to the following: If C is enriched/monoidal/..., what does this imply for LEFunc(D,C)? I am also interested in the 2-categorical structure of Cat(C) which is not visible in this picture. I guess all this is known and has been written somewhere. But where? I woul= d appreciate any sort of comments. Hendryk Pfeiffer --=20 NEU: WLAN-Router f=FCr 0,- EUR* - auch f=FCr DSL-Wechsler! GMX DSL =3D superg=FCnstig & kabellos http://www.gmx.net/de/go/dsl