Dear all, I cannot type any form of LaTeX, and do not know the "standard" ways to introduce indices,exponentials and so on, using only the typing which is admitted on this list. Thus I shall use "non standard" notations, very simple, which I shall explain precisely. If A is a category an object a of A can be identified with a functor "name of a" which I denote by "a": 1 --> A . If F: A --> C and G: B --> C are functors I denote by F/G the comma category they define, and by F//G their 2-pull-back sometimes called their pseudo pull-back. I shall call "weak equivalence" a functor F: A --> B full and faithful and essentially surjective (ff-es) and say that A is weakly equivalent (we) to B if there is such an F. This defines a preorder relation which I denote by W(A,B). It is symmetric iff the Axiom of choice (AC) holds. A strong equivalence between A and B is a pair of adjoint functors F: A --> B and F': B --> A such that the adjunction morphisms are isos. I shall say that A and B are strongly equivalent if such a pair exists and denote by SE(A,B) this equivalence relation. If S is a category I denote by Fib/S the 2-category of fibrations over S I denote by S° the dual of S . An S indexed category C is a pseudo functor C: S° --> Cat . For each map f: s --> t of S , I denote by f*: C(t) --> C(s) the value of C on f. If g: t --> u is another map of S, I denote by c(f,g) the isomorphism f*g* --> (gf)*. I shall assume C normalized i.e. for each object s of S, id(s)* = id(C(s)) For an indexed functor F: C --> C' i use the following notations: If s is an object of S, F(s): C(s) --> C'(s) , if f: s --> t is a map of S is the iso of functors F(s)f* --> f*F(t) (I'm aware of the ambiguity of denoting by the same f* the "re-indexing functors" defined by C and C' but it is the best I can do with the limited typographic means I use, without having too cumbersome notations) If S is has pull-backs, I denote by Cat(S) the category of categories internal to S Answer to Toby Bartels: (i) You propose to compose the spans by pullbacks. Here is an example of two spans A <-- X --> B and B <-- Y --> C such that the functors A <--X and Y --> C are isos, the functors X --> B and B <-- Y have unique quasi inverses, and if Z is the pullback none of the functors A <-- Z and Z --> B is a weak equivalence: Take A=X=Y=C=1, take for B the coarse category with two objects a and b , for X --> B and B <-- Y the functors "a": 1 --> B and B <-- 1: "b" . The pullback Z is 0 How stronger a counter example do you need? And using zig-zags will make the situation even worse. (ii) you define equivalences by spans, not "up to anything". With this definition an equivalence between 1 and 1 is any non empty coarse category. Every non empty set determines up to isomorphism such a category. thus there are at least as many equivalences from 1 to 1 in your sense as there are non empty sets. A bit much don't you think? Answer to David Roberts. I quote you: "Also, for the purposes of further discussion, the terminology "evil" has been demoted to a mere footnote at the nLab, as it was probably always meant to be by its coiners, and replaced by the morally neutral and more informative name 'principle of equivalence'. Interestingly, Voevodsky's Univalence Axiom is a way of ensuring the principle is always respected (and without awkward acrobatics)." I am a very thorough person, so I have looked at the instances of "evil" in nLab and found hundreds of them, not mere footnotes. I shall concentrate on the article: "Grothendieck fibrations in nLab". It has been revised by Mike Shulman on october 11, 2012, i.e. not very long ago, and it has many interesting features. I shall comment only those of them which have some bearings to the present discussion. Although it is supposed to deal with Grothendieck's fibrations there is a long paragraph,almost a whole page, with the title: "Non evil version". The very first line is: "There is something evil about the notion of fibration.." and the word "evil" is even underlined. Another interesting feature is the definition of internal fibration in a strict 2-category K (why "strict"? a 2-category is a 2-category. Probably because the notion of 2-category is unbearably evil). If you use this definition for K = Cat, it is easy to see that such internal fibrations are Grothendieck fibrations equipped with a cleavage. The situation is even worse if K = Cat(S) Last but not least, I recommend reading the deep discussion between Shridar Ramesh and Mike Shulman about "indexing" the fibration Z --> Z/2Z. (but hurry, because after this mail there might very well be a quick revision of the article). I happen to like very much Grothendieck's fibrations, this would probably qualify me as an evil person, let me say that i'm proud of this kind of evilness. Preliminary answer to Marta Bunge: Many thanks for your mail. Before I give a complete answer, I have some questions: (1) - What do you mean precisely by a full, faithful essentially surjective indexed functor. Does this notion depend on the indexings?(this is why I took the pain to give precise notations for such indexed functors) (2) - I suppose that if C is internal to S, your [C] is the canonical (split) indexing defined by [C](s) = S(s,C) thus your answer to (1) will tell me what you mean by [C] and [D] are weakly equivalent. Before getting your answers, I can already make a few comments about your mail. (a) Assuming you have a notion of equivalence E(A,B) reflexive transitive and symmetric, your answer to (ii) is a bit surprising. Given any A, suppose someone asks how many B's are equivalent to A? If B and B' are such B's, they are equivalent, hence if you work up to equivalence, the answer is 1. What do stack completions have to do with this answer? (b) You say that you could have worked with fibered categories instead of indexed ones, and I believe you. But why didn't you do it? What is at stake in this this discussion is the equivalence of categories with or without AC. You know that the relation between indexed categories and fibered ones depends essentially on AC. I have seen very strange things about this relation. The strangest one can be found in the Elephant: p.4, one can read: "we should make the smallest possible demands on the metatheory within which we interpret the theory of categories (and in particular we shall not assume that it satisfies any form of the axiom of choice)" True to this very strong statement, the author gives the proof, due to Grothendiek, that indexed categories are "the same thing" as fibered categories equipped with a cleavage. And that is O.K. for me. But then, without any transition or warning he starts saying: "Let P: C --> S be a fibration and CC be the associated indexed category .." It is not an accidental slip. It can be found in Example 1.3.13, Lemma 1.4.1, Lemma 1.4.5, Lemma 1.4.9, Lemma 1.4.10, Lemma 1.4.11, Theorem 1.4.12, etc.. He does not say "an" associated, but " the" associated. This means not only that every fibration has a cleavage (which is equivalent to AC when restricted to small fibrations) but also comes equipped with a cleavage, which is equivalent to huge form of AC for classes. Not bad considering the strong statement I have quoted. But even assuming this huge AC, "the" associated indexed category is astounding. It well known, and was remarked by Grothendieck more than 50 years ago, that a group homomorphism P: E --> B is a fibration iff it is surjective, and that a cleavage is a section S of (the underlying map of) P. You won't find this fact in the Elephant. "The" associated indexed category means that such a P comes equipped with a section. Imagine the reaction of the students if, in order to explain periodic functions via the surjection P: R --> R/Z , you had to assume that a section S of P is already given, and that S has any influence on the notion of periodic function! I believe that indexed categories have been a disaster for the mathematicians, some of whom very serious, who have adopted them. Let me give two more arguments. (a) They forbid the study of functors P: E --> B more general than fibrations, but which keep enough of the features of the fibrations to be able to prove about them many of the results of fibered category theory. The simplest case is pre-fibrations. (b) The theory of fibrations is a first order theory, therefore they can be internalized, e.g. in a topos (this is much too strong) Can anybody give a definition of internal indexed categories and functors?To Excuse me for such a long mail, there are many more things to say about equivalences of categories, but they will have to be postponed to another mail. (a) You work with S-indexed or [For admin and other information see: http://www.mta.ca/~cat-dist/ ]