Dear List, Has the following elementary problem been already studied? Let C be a category, I a small category, F : I -> C a functor and alpha : F => c a cocone (c is an object of C). When does there exist a category D and a faithful functor G : C -> D taking alpha to a universal cocone? 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. 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. Question: are they sufficient in the general case? Regards, -- Dimitri [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
Dear Lutz,
While I'm willing to believe the case of initial objects, the statement is wrong already for the case of isomorphisms.
You are right. I didn't checked very carefully the local confluence of my rewriting system. To get confluence, I need to add the fact that the bimorphism f satisfies "fu = vf implies u = 1 and v = 1". So this is enough for f to be a potential isomorphism. While the general statement I gave is wrong, I'm still interested in condition to make the conclusion true (for example, in the case of the sum or of the amalgated sum). Regards, -- Dimitri [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Dimitri,
You are right. I didn't checked very carefully the local confluence of my rewriting system. To get confluence, I need to add the fact that the bimorphism f satisfies "fu = vf implies u = 1 and v = 1". So this is enough for f to be a potential isomorphism.
Yes, that does the trick, and is also the best sufficient condition that I have been able to come up with in my thesis. I would guess something similar might work for the general case, but the condition is actually so strong as to be somewhat unsatisfactory. Note in particular that it fails for actual (nontrivial) isomorphisms. I poked around this problem for quite a bit back then but haven't been able to isolate a satisfactory criterion (other than the obvious one that talks explicitly about equality in a non-confluent axiom system). Good luck, Lutz -- -------------------------------------- PD Dr. Lutz Schröder 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 für Künstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Geschäftsführung: 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/ ]
Dear Dimitri, I think that finding useful necessary and sufficient conditions in general is a very hard problem. But here is something that might be useful in specific cases. There will exist some faithful functor G:C-->D sending alpha to a colimit cocone if and only if the universal functor G:C-->D sending alpha to a colimit cocone is faithful. This universal functor can be constructed as follows (this forms part of the general theory of sketches). Let M be the full subcategory of [C^op,Set] consisting of those functors F which send alpha to a limit cone. Then M is reflective in [C^op,Set], via a left adjoint L, and the composite LY:C-->M does indeed send alpha to a colimit cocone. The universal G is obtained by factorizing LY as a bijective on objects functor G:C-->D followed by a fully faithful J:D-->M. Clearly G will be faithful if and only if LY is, and this can be determined once one has calculated L on representables. Calculating L explicitly is in general still a hard problem, but in specific cases you may be able to do it sufficiently explicitly to solve your problem. The connection with sketches is that you can think of your cocone as giving a limit sketch on C^op, then the category M defined above is the category of models of the sketch. Regards, Steve Lack. On 4/08/09 2:37 AM, "Dimitri Ara" <dimitri.ara@gmail.com> wrote:
Dear List,
Has the following elementary problem been already studied?
Let C be a category, I a small category, F : I -> C a functor and alpha : F => c a cocone (c is an object of C). When does there exist a category D and a faithful functor G : C -> D taking alpha to a universal cocone?
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. 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.
Question: are they sufficient in the general case?
Regards, -- Dimitri
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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Dimitri Ara -
Lutz Schroeder -
Steve Lack