From: Frank.Piessens@cs.kuleuven.ac.be (Frank Piessens) Can somebody give me a reference to a proof of the following result: Let C be a small category and let F:C -> Set be a functor. Then, the slice category Fun(C,Set)/F is equivalent with Fun(G(C,F),Set) where G is the Grothendieck-construction . . .
This result has been known to category theorists for a long time, and probably appears somewhere in SGA 4. For a published reference with two-line proof (given a little knowledge of fibrations) see my Proposition (7.3) page 293 of "Cosmoi of internal categories" Transactions AMS 258 #2 (1980) 271 - 318. This result was used to produce an algorithm for finding all internal full subcategories of a presheaf category. Moreover, I pushed up the equivalence to the case where C supports a sketch (I called sketches "Gabriel theories" in that paper with a footnote giving the other terminology). If F is a model of the sketch, one obtains a sketch on your G(C,F), and an equivalence Mod(C,Set)/F ~~~ Mod(G(C,F),Set). See Proposition 7.21 on page 297. I used this result to produce an algorithm for finding all internal full subcategories of certain locally presentable categories (eg, Cat). I also used it to characterize the sketches whose model categories had cartesian closed slice categories (Theorem 7.25). [I mention the sketch result because sketches seem to be of interest to computer scientists.] An interesting example of the original equivalence is the case where C is a group and F is a G-set, so that G(C,F) is the wreath product; or, if F is a G-module, G(C,F) is the semidirect product. Regards, Ross ++++++++++++++++++++++