I note that Andree Ehresman has replied to the questions (copied below) of Hendryk Pfeiffer, drawing his attention to the pioneering works of her late husband Charles, and in particular to a joint paper of theirs. Let me add something to that. Pfeiffer writes as if using the "essentially algebraic" theory of categories restricts him to the level of categories, rather than 2-categories. He may therefore find interesting my article on enriched "essentially algebraic theories": [G.M. Kelly, Structures defined by finite limits in the enriched context, Cahiers de Topologie et Ge'om. Diffe'rentielle 23 (1982), 3 - 42] ; which was dedicated to the memory of Charles Ehresmann following his death, and which , for the general theory of enriched categories, refers to my book [Basic Concepts of Enriched categories, London Math. Soc. Lecture Notes Series 64, Cambridge Univ. Press 1982]. Max Kelly. Hendryk Pfeiffer wrote:
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