Dear Dimitri,
For example, if I is the empty category, the question becomes "when can you make c an initial object in a faithful way?". If I is the final category, then the cocone alpha amounts to a morphism f : F(*) -> c and the question becomes "when can you make f an isomorphism in a faithful way?".
There are two obvious necessary conditions. 1) Let f,g : c -> d be two morphisms of C. If f alpha_i = g alpha_i for every i in I, then we should have f = g.L 2) Let f : a -> F(i) and g : a -> F(j) be two morphisms of C such that alpha_i f = alpha_j g. Then for every cocone beta : F => d, we should have beta_i f = beta_j g.
In the case of the empty category, the first condition means that for every object d there is at most one arrow c -> d and the second condition is void. In the case of the final category, the first condition means that f is an epi and the second that f is a mono. It is not hard to prove that in both cases, theses conditions are sufficient.
While I'm willing to believe the case of initial objects, the statement is wrong already for the case of isomorphisms. Let's call f a potential isomorphism if there exists a faithful functor (equivalently an embedding) that makes f an isomorphism. Then one has the following property of potential isomorphisms (from my 1999 thesis; in German, I'm afraid): Lemma: Let s be a potential isomorphism, and let f,g,h,j,l,p be morphisms such that fs = sg hg = ks fl = sp. Then kl = hp. Proof: In an extended category where s has a two-sided inverse s^{-1}, we have gs^{-1} = s^{-1}f from fs = sg, and hence kl = kss^{-1}l = hgs^{-1}l = hs^{-1}fl = hs^{-1}sp = hp [] The property of the lemma is not implied by s being both epi and mono (i.e. a bimorphism). It is comparatatively easy to prove this using contrived examples, such as the following. Let the category A consist of objects A, B, C, D, and families of morphisms g_i: A -> A p_i: B -> A s_k: A -> C h_i: A -> D l_i: B -> C f_i: C -> C k_i: B -> D q_i,r: B -> D indexed over i>=0, k>=1, where we identify f_0 and g_0 with the respective identities. Composition is by addition of indices, with the single exception h_0p_0 = r. (This satisfies the associative law, since the exceptional case r occurs only in trivial cases of the law -- it cannot be pre- or postcomposed with a nontrivial morphism, and its two factors do not have proper factorisations.) Then s_1 is a bimorphism but violates the property of the above lemma, and hence is not a potential isomorphism: f_1s_1 = s_1g_1, h_0g_1 = k_0s_1, and f_1l_0 = s_1p_0, but k_0l_0 = q_0 \neq r = h_0p_0. Best regards, Lutz -- -------------------------------------- PD Dr. Lutz Schro"der Senior Researcher DFKI Bremen Safe and Secure Cognitive Systems Cartesium, Enrique-Schmidt-Str. 5 D-28359 Bremen phone: (+49) 421-218-64216 Fax: (+49) 421-218-9864216 mail: Lutz.Schroeder@dfki.de www.dfki.de/sks/staff/lschrode -------------------------------------- ------------------------------------------------------------- Deutsches Forschungszentrum fu"r Ku"nstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Gescha"ftsfu"hrung: Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) Dr. Walter Olthoff Vorsitzender des Aufsichtsrats: Prof. Dr. h.c. Hans A. Aukes Amtsgericht Kaiserslautern, HRB 2313 ------------------------------------------------------------- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]